# Whirlpool, a maelstrom of emotion disambiguation et Y'becca

## Whirlpool, a maelstrom of emotion disambiguation et Y'becca

A maelstrom is a powerful whirlpool.

Definition of maelstrom

1

: a powerful often violent whirlpool sucking in objects within a given radius

tried to shoot the canoe across a stretch of treacherous maelstrom — Harper's

2

: something resembling a maelstrom in turbulence the maelstrom enveloping the country a maelstrom of emotions

A whirlpool is a body of swirling water produced by the meeting of opposing currents. The vast majority of whirlpools are not very powerful and very small whirlpools can easily be seen when a bath or a sink is draining. More powerful ones in seas or oceans may be termed maelstroms. Vortex is the proper term for any whirlpool that has a downdraft.

In oceans, in narrow straits, with fast flowing water, whirlpools are normally caused by tides; there are few stories of large ships ever being sucked into such a maelstrom, although smaller craft are in danger.[1][2] Smaller whirlpools also appear at the base of many waterfalls[3] and can also be observed downstream from manmade structures such as weirs and dams. In the case of powerful waterfalls, like Niagara Falls, these whirlpools can be quite strong.

Notable whirlpools

The maelstrom off Norway, as illustrated by Olaus Magnus on the Carta Marina, 1539

Moskstraumen

Main article: Moskstraumen

Moskstraumen is an unusual system of whirlpools in the open seas in the Lofoten Islands off the Norwegian coast.[4] It is the second strongest whirlpool in the world with flow currents reaching speeds as high as 32 km/h (20 mph). It finds mention in several books and movies.[5]

The Moskstraumen is formed by the combination of powerful semi-diurnal tides and the unusual shape of the seabed, with a shallow ridge between the Moskenesøya and Værøy islands which amplifies and whirls the tidal currents.[6]

The fictional depictions of the Maelstrom by Edgar Allan Poe and Jules Verne describe it as a gigantic circular vortex that reaches the bottom of the ocean, when in fact it is a set of currents and crosscurrents with a rate of 18 km/h (11 mph).[7] Poe described this phenomenon in his short story A Descent into the Maelstrom, which in 1841 was the first to use the word "maelstrom" in the English language;[6] in this story related to the Lofoten Maelstrom, two fishermen are swallowed by the maelstrom while one survives miraculously.[8]

Saltstraumen

Main article: Saltstraumen

Saltstraumen

The maelstrom of Saltstraumen is the Earth's strongest maelstrom, and is located close to the Arctic Circle,[5] 33 km (20 mi) round the bay on the Highway 17, south-east of the city of Bodø, Norway. The strait at its narrowest is 150 m (490 ft) in width and water "funnels" through the channel four times a day.[9] It is estimated that 400 million cubic meters of water passes the narrow strait during this event.[10] The water is creamy in colour and most turbulent during high tide, which is witnessed by thousands of tourists.[9] It reaches speeds of 40 km/h (25 mph),[6] with mean speed of about 13 km/h (8.1 mph). As navigation is dangerous in this strait only a small slot of time is available for large ships to pass through.[5] Its impressive strength is caused by the world's strongest tide occurring in the same location during the new and full moon. A narrow channel of 3 km (2 mi) length connects the outer Saltfjord with its extension, the large Skjerstadfjord, causing a colossal tide which in turn produces the Saltstraumen maelstrom.[11]

Corryvreckan

Main article: Corryvreckan

Corryvreckan whirlpool

The Corryvreckan is a narrow strait between the islands of Jura and Scarba, in Argyll and Bute, on the northern side of the Gulf of Corryvreckan, Scotland. It is the third-largest whirlpool in the world.[5] Flood tides and inflow from the Firth of Lorne to the west can drive the waters of Corryvreckan to waves of over 9 metres (30 ft), and the roar of the resulting maelstrom, which reaches speeds of 18 km/h (11 mph), can be heard 16 kilometres (9.9 mi) away. Though it was initially classified as non-navigable by the British navy it was later categorized as "extremely dangerous".[5]

A documentary team from Scottish independent producers Northlight Productions once threw a mannequin into the Corryvreckan ("the Hag") with a life jacket and depth gauge. The mannequin was swallowed and spat up far down current with a depth gauge reading of 262 metres (860 ft) with evidence of being dragged along the bottom for a great distance.[12]

Other notable maelstroms and whirlpools

Naruto whirlpools

Old Sow whirlpool is located between Deer Island, New Brunswick, Canada, and Moose Island, Eastport, Maine, USA. It is given the epithet "pig-like" as it makes a screeching noise when the vortex is at its full fury. The smaller whirlpools around this Old Sow are known as "Piglets.[5] and reaches speeds of up to 27.6 km/h (17.1 mph).[6]

The Naruto whirlpools are located in the Naruto Strait near Awaji Island in Japan, which have speeds of 26 km/h (16 mph).[6]

Skookumchuck Narrows is a tidal rapids that develops whirlpools, on the Sunshine Coast, Canada with current speeds exceeding 30 km/h (19 mph).[6]

French Pass (Te Aumiti) is a narrow and treacherous stretch of water that separates D'Urville Island from the north end of the South Island of New Zealand. In 2000 a whirlpool there caught student divers, resulting in multiple fatalities.[13]

There was a short-lived whirlpool that sucked in a portion of the Lake Peigneur of 1300 acre area in Louisiana, United States after a drilling mishap in November 1980. This was not a naturally occurring whirlpool, but a man-made disaster caused by breaking through the roof of a salt mine. This mishap resulted in destruction of five houses, loss of nineteen barges and eight tug boats, oil rigs, a mobile home and most of the botanical garden and 10 percent area of the nearby Jefferson Island. A crater of 0.5-mile was created. The lake then drained, until the mine filled and the water levels equalized but the ten-foot deep lake was now 1,300 feet deep. Nine of the barges which had sunk floated back.[14][15][16]

A more recent example of a man-made whirlpool that received significant media coverage was in early June 2015, when an intake vortex formed in Lake Texoma, on the Oklahoma–Texas border, near the floodgates of the dam that forms the lake. At the time of the whirlpool's formation, the lake was being drained after reaching its highest level ever. The Army Corps of Engineers, which operates the dam and lake, expected that the whirlpool would last until the lake reached normal seasonal levels by late July.[17]

Dangers

An illustration from Jules Verne's essay "Edgard Poë et ses oeuvres" (Edgar Poe and his Works,1862) drawn by Frederic Lix or Yan' Dargent

Powerful whirlpools have killed unlucky seafarers, but their power tends to be exaggerated by laymen.[18] There are virtually no stories of large ships ever being sucked into a whirlpool. Tales like those by Paul the Deacon, Edgar Allan Poe, and Jules Verne are entirely fictional.[19]

In literature and popular culture

Apart from Poe and Verne other literary source is of the 1500s, of Olaus Magnus, a Swedish Bishop, who had stated that the maelstrom which was more powerful than The Odyssey destroyed ships which sank to the bottom of the sea, and even whales were sucked in. Pytheas, the Greek historian, also mentioned that maelstroms swallowed ships and threw them up again.[1]

Charybdis in Greek mythology was later rationalized as a whirlpool, which sucked entire ships into its fold in the narrow coast of Sicily, a disaster faced by navigators.[20]

In the 8th century, Paul the Deacon, who had lived among the Belgii, described tidal bores and the maelstrom for a Mediterranean audience unused to such violent tidal surges:[21]

Not very far from this shore … toward the western side, on which the ocean main lies open without end, is that very deep whirlpool of waters which we call by its familiar name "the navel of the sea." This is said to suck in the waves and spew them forth again twice every day. … They say there is another whirlpool of this kind between the island of Britain and the province of Galicia, and with this fact the coasts of the Seine region and of Aquitaine agree, for they are filled twice a day with such sudden inundations that any one who may by chance be found only a little inward from the shore can hardly get away. I have heard a certain high nobleman of the Gauls relating that a number of ships, shattered at first by a tempest, were afterwards devoured by this same Charybdis. And when one only out of all the men who had been in these ships, still breathing, swam over the waves, while the rest were dying, he came, swept by the force of the receding waters, up to the edge of that most frightful abyss. And when now he beheld yawning before him the deep chaos whose end he could not see, and half dead from very fear, expected to be hurled into it, suddenly in a way that he could not have hoped he was cast upon a certain rock and sat him down.

— Paul the Deacon, History of the Lombards, i.6

Three of the most notable literary references to the Lofoten Maelstrom date from the nineteenth century. The first is the Edgar Allan Poe short story "A Descent into the Maelström" (1841). The second is 20,000 Leagues Under the Sea (1870), the famous novel by Jules Verne. At the end of this novel, Captain Nemo seems to commit suicide, sending his Nautilus submarine into the Maelstrom (although in Verne's sequel Nemo and Nautilus survived). The "Norway maelstrom" is also mentioned in Herman Melville's Moby-Dick.[22]

In the 'Life of St Columba', the author, Adomnan of Iona', attributes to the saint miraculous knowledge of a particular bishop who ran into a whirlpool off the coast of Ireland. In Adomnan's narrative, he quotes Columba saying[23]

Cólman mac Beognai has set sail to come here, and is now in great danger in the surging tides of the whirlpool of Corryvreckan. Sitting in the prow, he lifts up his hands to heaven and blesses the turbulent, terrible seas. Yet the Lord terrifies him in this way, not so that the ship in which he sits should be overwhelmed and wrecked by the waves, but rather to rouse him to pray more fervently that he may sail through the peril and reach us here.

Etymology

One of the earliest uses in English of the Scandinavian word (malström or malstrøm) was by Edgar Allan Poe in his short story "A Descent into the Maelström" (1841). In turn, the Nordic word is derived from the Dutch maelstrom, modern spelling maalstroom, from malen (to grind) and stroom (stream), to form the meaning grinding current or literally "mill-stream", in the sense of milling (grinding) grain.[24]

See also

iconEnvironment portal iconWater portal

Coriolis effect

Eddy (fluid dynamics)

Rip current

In English, the word originally referred to the Moskstraumen.

Maelstrom may also refer to:

Contents

1 Amusement rides

2 Characters

3 Film and television

4 Games

5 Literature

6 Music

7 Other

Amusement rides

Maelstrom (ride), a dark ride that ran from 1988 to 2014 at one of 4 Walt Disney World Resort theme parks, Epcot, in Orlando, Florida, US

Maelstrom, a gyro swing ride at Drayton Manor Theme Park, Staffordshire, UK

Characters

Maelstrom (comics), fictional character that appears in comic books published by Marvel Comics

Maelstrom (Ice Age), a Pliosaurus

Mael Strom, aka Yuki Yoshida, from the anime/manga Kore wa Zombie Desu ka?

Film and television

Maelström (film), a 2000 Canadian film

Maelstrom (TV series), a 1985 BBC television drama serial

"Maelstrom" (Battlestar Galactica)

Games

Maelstrom (1992 video game), an Apple Macintosh game

Maelstrom (role playing game), a role-playing game by Alexander Scott

Maelstrom (video game), a 2007 PC game

VOR: The Maelstrom, a miniature wargame

Maelstrom (Live Action Roleplaying), a live action roleplaying game by Profound Decisions

Maelstrom, a lightning-based hammer item in the video game "Dota 2"

Literature

A Descent into the Maelström, an 1841 short story by Edgar Allan Poe

Maelstrom (Timms novel), a novel by E.V. Timms

Maelstrom, a 2001 novel by Peter Watts

Maelstrom, a 2006 novel in the Petaybee Series by Anne McCaffrey and Elizabeth Ann Scarborough

Maelstrom (Destroyermen novel), the third book of the Destroyermen series

Music

Maelstrom, an album by JR Ewing

"Maelstrom," a song by The Steve Miller Band on the album Living in the 20th Century

"Maelstrom," a composition by Julian Cochran

"The Maelstrom," a composition by Robert W. Smith

"Maelstrom 2010," a song by Kataklysm on the album Temple of Knowledge

Malström Graintable Synthesizer, a component of the musical software Reason

Maelstrom, a Boston-based thrash metal band active from the late 1980s to the early 1990s that released an LP on Taang! Records

"Ocean - II. Maelstrom" song by Canadian instrumental progressive metal band "Pomegranate Tiger" released on their 2013 album "Entities"

Other

Maelstrom, a Chromium-based browser by San Francisco-based BitTorrent Inc. with an integrated BitTorrent engine

Whirlpool à Amiens : un dossier industriel sous haute tension

In fluid dynamics, a vortex is a region in a fluid in which the flow rotates around an axis line, which may be straight or curved.[1][2] The plural of vortex is either vortices or vortexes.[3][4] Vortices form in stirred fluids, and may be observed in phenomena such as smoke rings, whirlpools in the wake of boat, or the winds surrounding a tornado or dust devil.

Vortices are a major component of turbulent flow. The distribution of velocity, vorticity (the curl of the flow velocity), as well as the concept of circulation are used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis.

In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries with it some angular and linear momentum, energy, and mass.

Properties

Vorticity

The Crow Instability of a jet aeroplane's contrail visually demonstrates the vortex created in the atmosphere (gas fluid medium) by the passage of the aircraft.

A key concept in the dynamics of vortices is the vorticity, a vector that describes the local rotary motion at a point in the fluid, as would be perceived by an observer that moves along with it. Conceptually, the vorticity could be observed by placing a tiny rough ball at the point in question, free to move with the fluid, and observing how it rotates about its center. The direction of the vorticity vector is defined to be the direction of the axis of rotation of this imaginary ball (according to the right-hand rule) while its length is twice the ball's angular velocity. Mathematically, the vorticity is defined as the curl (or rotational) of the velocity field of the fluid, usually denoted by ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} and expressed by the vector analysis formula ∇ × u → {\displaystyle \nabla \times {\vec {\mathit {u}}}} \nabla \times {\vec {{\mathit {u}}}}, where ∇ {\displaystyle \nabla } \nabla is the nabla operator and u → {\displaystyle {\vec {\mathit {u}}}} {\vec {{\mathit {u}}}} is the local flow velocity.[5]

The local rotation measured by the vorticity ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} must not be confused with the angular velocity vector of that portion of the fluid with respect to the external environment or to any fixed axis. In a vortex, in particular, ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis.

Vortex types

In theory, the speed u of the particles (and, therefore, the vorticity) in a vortex may vary with the distance r from the axis in many ways. There are two important special cases, however:

A rigid-body vortex

If the fluid rotates like a rigid body – that is, if the angular rotational velocity Ω is uniform, so that u increases proportionally to the distance r from the axis – a tiny ball carried by the flow would also rotate about its center as if it were part of that rigid body. In such a flow, the vorticity is the same everywhere: its direction is parallel to the rotation axis, and its magnitude is equal to twice the uniform angular velocity Ω of the fluid around the center of rotation.

Ω → = ( 0 , 0 , Ω ) , r → = ( x , y , 0 ) , {\displaystyle {\vec {\Omega }}=(0,0,\Omega ),\quad {\vec {r}}=(x,y,0),} {\vec {\Omega }}=(0,0,\Omega ),\quad {\vec {r}}=(x,y,0),

u → = Ω → × r → = ( − Ω y , Ω x , 0 ) , {\displaystyle {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\Omega y,\Omega x,0),} {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\Omega y,\Omega x,0),

ω → = ∇ × u → = ( 0 , 0 , 2 Ω ) = 2 Ω → . {\displaystyle {\vec {\omega }}=\nabla \times {\vec {u}}=(0,0,2\Omega )=2{\vec {\Omega }}.} {\vec \omega }=\nabla \times {\vec {u}}=(0,0,2\Omega )=2{\vec {\Omega }}.

An irrotational vortex

If the particle speed u is inversely proportional to the distance r from the axis, then the imaginary test ball would not rotate over itself; it would maintain the same orientation while moving in a circle around the vortex axis. In this case the vorticity ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} is zero at any point not on that axis, and the flow is said to be irrotational.

Ω → = ( 0 , 0 , α r − 2 ) , r → = ( x , y , 0 ) , {\displaystyle {\vec {\Omega }}=(0,0,\alpha r^{-2}),\quad {\vec {r}}=(x,y,0),} {\vec {\Omega }}=(0,0,\alpha r^{{-2}}),\quad {\vec {r}}=(x,y,0),

u → = Ω → × r → = ( − α y r − 2 , α x r − 2 , 0 ) , {\displaystyle {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\alpha yr^{-2},\alpha xr^{-2},0),} {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\alpha yr^{{-2}},\alpha xr^{{-2}},0),

ω → = ∇ × u → = 0. {\displaystyle {\vec {\omega }}=\nabla \times {\vec {u}}=0.} {\vec {\omega }}=\nabla \times {\vec {u}}=0.

Irrotational vortices

Pathlines of fluid particles around the axis (dashed line) of an ideal irrotational vortex. (See animation)

In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern[citation needed], where the flow velocity u is inversely proportional to the distance r. Irrotational vortices are also called free vortices.

For an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis; and has a fixed value, Γ, for any contour that does enclose the axis once.[6] The tangential component of the particle velocity is then u θ = Γ 2 π r {\displaystyle u_{\theta }={\tfrac {\Gamma }{2\pi r}}} {\displaystyle u_{\theta }={\tfrac {\Gamma }{2\pi r}}}. The angular momentum per unit mass relative to the vortex axis is therefore constant, r u θ = Γ 2 π {\displaystyle ru_{\theta }={\tfrac {\Gamma }{2\pi }}} {\displaystyle ru_{\theta }={\tfrac {\Gamma }{2\pi }}}.

However, the ideal irrotational vortex flow is not physically realizable, since it would imply that the particle speed (and hence the force needed to keep particles in their circular paths) would grow without bound as one approaches the vortex axis. Indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as r goes to zero. Within that region, the flow is no longer irrotational: the vorticity ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} becomes non-zero, with direction roughly parallel to the vortex axis. The Rankine vortex is a model that assumes a rigid-body rotational flow where r is less than a fixed distance r0, and irrotational flow outside that core regions. The Lamb-Oseen vortex model is an exact solution of the Navier–Stokes equations governing fluid flows and assumes cylindrical symmetry, for which

u θ = ( 1 − e − r 2 4 ν t ) Γ 2 π r . {\displaystyle u_{\theta }=\left(1-e^{\frac {-r^{2}}{4\nu t}}\right){\frac {\Gamma }{2\pi r}}.} {\displaystyle u_{\theta }=\left(1-e^{\frac {-r^{2}}{4\nu t}}\right){\frac {\Gamma }{2\pi r}}.}

Rotational vortices

The cloud vortex Saturn's hexagon is at the north pole of the planet Saturn.

A rotational vortex – one which has non-zero vorticity away from the core – can be maintained indefinitely in that state only through the application of some extra force, that is not generated by the fluid motion itself.

For example, if a water bucket is spun at constant angular speed w about its vertical axis, the water will eventually rotate in rigid-body fashion. The particles will then move along circles, with velocity u equal to wr.[6] In that case, the free surface of the water will assume a parabolic shape.

In this situation, the rigid rotating enclosure provides an extra force, namely an extra pressure gradient in the water, directed inwards, that prevents evolution of the rigid-body flow to the irrotational state.

Vortex geometry

In a stationary vortex, the typical streamline (a line that is everywhere tangent to the flow velocity vector) is a closed loop surrounding the axis; and each vortex line (a line that is everywhere tangent to the vorticity vector) is roughly parallel to the axis. A surface that is everywhere tangent to both flow velocity and vorticity is called a vortex tube. In general, vortex tubes are nested around the axis of rotation. The axis itself is one of the vortex lines, a limiting case of a vortex tube with zero diameter.

According to Helmholtz's theorems, a vortex line cannot start or end in the fluid – except momentarily, in non-steady flow, while the vortex is forming or dissipating. In general, vortex lines (in particular, the axis line) are either closed loops or end at the boundary of the fluid. A whirlpool is an example of the latter, namely a vortex in a body of water whose axis ends at the free surface. A vortex tube whose vortex lines are all closed will be a closed torus-like surface.

A newly created vortex will promptly extend and bend so as to eliminate any open-ended vortex lines. For example, when an airplane engine is started, a vortex usually forms ahead of each propeller, or the turbofan of each jet engine. One end of the vortex line is attached to the engine, while the other end usually stretches out and bends until it reaches the ground.

When vortices are made visible by smoke or ink trails, they may seem to have spiral pathlines or streamlines. However, this appearance is often an illusion and the fluid particles are moving in closed paths. The spiral streaks that are taken to be streamlines are in fact clouds of the marker fluid that originally spanned several vortex tubes and were stretched into spiral shapes by the non-uniform flow velocity distribution.

Pressure in a vortex

A Plughole vortex

The fluid motion in a vortex creates a dynamic pressure (in addition to any hydrostatic pressure) that is lowest in the core region, closest to the axis, and increases as one moves away from it, in accordance with Bernoulli's Principle. One can say that it is the gradient of this pressure that forces the fluid to follow a curved path around the axis.

In a rigid-body vortex flow of a fluid with constant density, the dynamic pressure is proportional to the square of the distance r from the axis. In a constant gravity field, the free surface of the liquid, if present, is a concave paraboloid.

In an irrotational vortex flow with constant fluid density and cylindrical symmetry, the dynamic pressure varies as P∞ − K/r2, where P∞ is the limiting pressure infinitely far from the axis. This formula provides another constraint for the extent of the core, since the pressure cannot be negative. The free surface (if present) dips sharply near the axis line, with depth inversely proportional to r2. The shape formed by the free surface is called a hyperboloid, or "Gabriel's Horn" (by Evangelista Torricelli).

The core of a vortex in air is sometimes visible because of a plume of water vapor caused by condensation in the low pressure and low temperature of the core; the spout of a tornado is an example. When a vortex line ends at a boundary surface, the reduced pressure may also draw matter from that surface into the core. For example, a dust devil is a column of dust picked up by the core of an air vortex attached to the ground. A vortex that ends at the free surface of a body of water (like the whirlpool that often forms over a bathtub drain) may draw a column of air down the core. The forward vortex extending from a jet engine of a parked airplane can suck water and small stones into the core and then into the engine.

Evolution

Vortices need not be steady-state features; they can move and change shape. In a moving vortex, the particle paths are not closed, but are open, loopy curves like helices and cycloids. A vortex flow might also be combined with a radial or axial flow pattern. In that case the streamlines and pathlines are not closed curves but spirals or helices, respectively. This is the case in tornadoes and in drain whirlpools. A vortex with helical streamlines is said to be solenoidal.

As long as the effects of viscosity and diffusion are negligible, the fluid in a moving vortex is carried along with it. In particular, the fluid in the core (and matter trapped by it) tends to remain in the core as the vortex moves about. This is a consequence of Helmholtz's second theorem. Thus vortices (unlike surface and pressure waves) can transport mass, energy and momentum over considerable distances compared to their size, with surprisingly little dispersion. This effect is demonstrated by smoke rings and exploited in vortex ring toys and guns.

Two or more vortices that are approximately parallel and circulating in the same direction will attract and eventually merge to form a single vortex, whose circulation will equal the sum of the circulations of the constituent vortices. For example, an airplane wing that is developing lift will create a sheet of small vortices at its trailing edge. These small vortices merge to form a single wingtip vortex, less than one wing chord downstream of that edge. This phenomenon also occurs with other active airfoils, such as propeller blades. On the other hand, two parallel vortices with opposite circulations (such as the two wingtip vortices of an airplane) tend to remain separate.

Vortices contain substantial energy in the circular motion of the fluid. In an ideal fluid this energy can never be dissipated and the vortex would persist forever. However, real fluids exhibit viscosity and this dissipates energy very slowly from the core of the vortex. It is only through dissipation of a vortex due to viscosity that a vortex line can end in the fluid, rather than at the boundary of the fluid.

Two-dimensional modeling

When the particle velocities are constrained to be parallel to a fixed plane, one can ignore the space dimension perpendicular to that plane, and model the flow as a two-dimensional flow velocity field on that plane. Then the vorticity vector ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} is always perpendicular to that plane, and can be treated as a scalar. This assumption is sometimes made in meteorology, when studying large-scale phenomena like hurricanes.

The behavior of vortices in such contexts is qualitatively different in many ways; for example, it does not allow the stretching of vortices that is often seen in three dimensions.

Further examples

The visible core of a vortex formed when a C-17 uses high engine power (reverse-thrust) at slow speed on a wet runway.

In the hydrodynamic interpretation of the behaviour of electromagnetic fields, the acceleration of electric fluid in a particular direction creates a positive vortex of magnetic fluid. This in turn creates around itself a corresponding negative vortex of electric fluid. Exact solutions to classical nonlinear magnetic equations include the Landau-Lifshitz equation, the continuum Heisenberg model, the Ishimori equation, and the nonlinear Schrödinger equation.

Bubble rings are underwater vortex rings whose core traps a ring of bubbles, or a single donut-shaped bubble. They are sometimes created by dolphins and whales.

The lifting force of aircraft wings, propeller blades, sails, and other airfoils can be explained by the creation of a vortex superimposed on the flow of air past the wing.

Aerodynamic drag can be explained in large part by the formation of vortices in the surrounding fluid that carry away energy from the moving body.

Large whirlpools can be produced by ocean tides in certain straits or bays. Examples are Charybdis of classical mythology in the Straits of Messina, Italy; the Naruto whirlpools of Nankaido, Japan; and the Maelstrom at Lofoten, Norway.

Vortices in the Earth's atmosphere are important phenomena for meteorology. They include mesocyclones on the scale of a few miles, tornados, waterspouts, and hurricanes. These vortices are often driven by temperature and humidity variations with altitude. The sense of rotation of hurricanes is influenced by the Earth's rotation. Another example is the Polar vortex, a persistent, large-scale cyclone centered near the Earth's poles, in the middle and upper troposphere and the stratosphere.

Vortices are prominent features of the atmospheres of other planets. They include the permanent Great Red Spot on Jupiter, the intermittent Great Dark Spot on Neptune, the polar vortices of Venus, the Martian dust devils and the North Polar Hexagon of Saturn.

Sunspots are dark regions on the Sun's visible surface (photosphere) marked by a lower temperature than its surroundings, and intense magnetic activity.

The accretion disks of black holes and other massive gravitational sources.

Taylor-Couette flow occurs in a fluid between two nested cylinders, one rotating, the other fixed.

See also

iconPhysics portal

Artificial gravity

Batchelor vortex

Biot–Savart law

Coordinate rotation

Cyclonic separation

Eddy

Gyre

Helmholtz's theorems

History of fluid mechanics

Horseshoe vortex

Hurricane

Kelvin–Helmholtz instability

Quantum vortex

Rankine vortex

Shower-curtain effect

Strouhal number

Vile Vortices

Von Kármán vortex street

Vortex engine

Vortex tube

Vortex cooler

VORTEX projects

Vortex shedding

Vortex stretching

Vortex induced vibration

Vorticity

Wormhole

References

Notes

Ting, L. (1991). Viscous Vortical Flows. Lecture notes in physics. Springer-Verlag. ISBN 3-540-53713-9.

Kida, Shigeo (2001). Life, Structure, and Dynamical Role of Vortical Motion in Turbulence (PDF). IUTAMim Symposium on Tubes, Sheets and Singularities in Fluid Dynamics. Zakopane, Poland.

"vortex". Oxford Dictionaries Online (ODO). Oxford University Press. Retrieved 2015-08-29.

"vortex". Merriam-Webster Online. Merriam-Webster, Inc. Retrieved 2015-08-29.

Vallis, Geoffrey (1999). Geostrophic Turbulence: The Macroturbulence of the Atmosphere and Ocean Lecture Notes (PDF). Lecture notes. Princeton University. p. 1. Retrieved 2012-09-26.

Clancy 1975, sub-section 7.5

Other

Loper, David E. (November 1966). An analysis of confined magnetohydrodynamic vortex flows (PDF) (NASA contractor report NASA CR-646). Washington: National Aeronautics and Space Administration. LCCN 67060315.

Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge Univ. Press. Ch. 7 et seq. ISBN 9780521098175.

Falkovich, G. (2011). Fluid Mechanics, a short course for physicists. Cambridge University Press. ISBN 978-1-107-00575-4.

Clancy, L.J. (1975). Aerodynamics. London: Pitman Publishing Limited. ISBN 0-273-01120-0.

De La Fuente Marcos, C.; Barge, P. (2001). "The effect of long-lived vortical circulation on the dynamics of dust particles in the mid-plane of a protoplanetary disc". Monthly Notices of the Royal Astronomical Society. 323 (3): 601–614. Bibcode:2001MNRAS.323..601D. doi:10.1046/j.1365-8711.2001.04228.x.

External links

Wikimedia Commons has media related to Vortex.

Optical Vortices

Video of two water vortex rings colliding (MPEG)

Chapter 3 Rotational Flows: Circulation and Turbulence

Vortical Flow Research Lab (MIT) – Study of flows found in nature and part of the Department of Ocean Engineering.

Whirlpool à Amiens

RAPPORT DE

Y'becca

sur

Le dossier Whirlpool d'Amiens est industriellement complexe et

s'est imposé comme le symbole de la course à la compétitivité

que se livrent les pays de l'Union Européenne...

Definition of maelstrom

1

: a powerful often violent whirlpool sucking in objects within a given radius

tried to shoot the canoe across a stretch of treacherous maelstrom — Harper's

2

: something resembling a maelstrom in turbulence the maelstrom enveloping the country a maelstrom of emotions

A whirlpool is a body of swirling water produced by the meeting of opposing currents. The vast majority of whirlpools are not very powerful and very small whirlpools can easily be seen when a bath or a sink is draining. More powerful ones in seas or oceans may be termed maelstroms. Vortex is the proper term for any whirlpool that has a downdraft.

In oceans, in narrow straits, with fast flowing water, whirlpools are normally caused by tides; there are few stories of large ships ever being sucked into such a maelstrom, although smaller craft are in danger.[1][2] Smaller whirlpools also appear at the base of many waterfalls[3] and can also be observed downstream from manmade structures such as weirs and dams. In the case of powerful waterfalls, like Niagara Falls, these whirlpools can be quite strong.

Notable whirlpools

The maelstrom off Norway, as illustrated by Olaus Magnus on the Carta Marina, 1539

Moskstraumen

Main article: Moskstraumen

Moskstraumen is an unusual system of whirlpools in the open seas in the Lofoten Islands off the Norwegian coast.[4] It is the second strongest whirlpool in the world with flow currents reaching speeds as high as 32 km/h (20 mph). It finds mention in several books and movies.[5]

The Moskstraumen is formed by the combination of powerful semi-diurnal tides and the unusual shape of the seabed, with a shallow ridge between the Moskenesøya and Værøy islands which amplifies and whirls the tidal currents.[6]

The fictional depictions of the Maelstrom by Edgar Allan Poe and Jules Verne describe it as a gigantic circular vortex that reaches the bottom of the ocean, when in fact it is a set of currents and crosscurrents with a rate of 18 km/h (11 mph).[7] Poe described this phenomenon in his short story A Descent into the Maelstrom, which in 1841 was the first to use the word "maelstrom" in the English language;[6] in this story related to the Lofoten Maelstrom, two fishermen are swallowed by the maelstrom while one survives miraculously.[8]

Saltstraumen

Main article: Saltstraumen

Saltstraumen

The maelstrom of Saltstraumen is the Earth's strongest maelstrom, and is located close to the Arctic Circle,[5] 33 km (20 mi) round the bay on the Highway 17, south-east of the city of Bodø, Norway. The strait at its narrowest is 150 m (490 ft) in width and water "funnels" through the channel four times a day.[9] It is estimated that 400 million cubic meters of water passes the narrow strait during this event.[10] The water is creamy in colour and most turbulent during high tide, which is witnessed by thousands of tourists.[9] It reaches speeds of 40 km/h (25 mph),[6] with mean speed of about 13 km/h (8.1 mph). As navigation is dangerous in this strait only a small slot of time is available for large ships to pass through.[5] Its impressive strength is caused by the world's strongest tide occurring in the same location during the new and full moon. A narrow channel of 3 km (2 mi) length connects the outer Saltfjord with its extension, the large Skjerstadfjord, causing a colossal tide which in turn produces the Saltstraumen maelstrom.[11]

Corryvreckan

Main article: Corryvreckan

Corryvreckan whirlpool

The Corryvreckan is a narrow strait between the islands of Jura and Scarba, in Argyll and Bute, on the northern side of the Gulf of Corryvreckan, Scotland. It is the third-largest whirlpool in the world.[5] Flood tides and inflow from the Firth of Lorne to the west can drive the waters of Corryvreckan to waves of over 9 metres (30 ft), and the roar of the resulting maelstrom, which reaches speeds of 18 km/h (11 mph), can be heard 16 kilometres (9.9 mi) away. Though it was initially classified as non-navigable by the British navy it was later categorized as "extremely dangerous".[5]

A documentary team from Scottish independent producers Northlight Productions once threw a mannequin into the Corryvreckan ("the Hag") with a life jacket and depth gauge. The mannequin was swallowed and spat up far down current with a depth gauge reading of 262 metres (860 ft) with evidence of being dragged along the bottom for a great distance.[12]

Other notable maelstroms and whirlpools

Naruto whirlpools

Old Sow whirlpool is located between Deer Island, New Brunswick, Canada, and Moose Island, Eastport, Maine, USA. It is given the epithet "pig-like" as it makes a screeching noise when the vortex is at its full fury. The smaller whirlpools around this Old Sow are known as "Piglets.[5] and reaches speeds of up to 27.6 km/h (17.1 mph).[6]

The Naruto whirlpools are located in the Naruto Strait near Awaji Island in Japan, which have speeds of 26 km/h (16 mph).[6]

Skookumchuck Narrows is a tidal rapids that develops whirlpools, on the Sunshine Coast, Canada with current speeds exceeding 30 km/h (19 mph).[6]

French Pass (Te Aumiti) is a narrow and treacherous stretch of water that separates D'Urville Island from the north end of the South Island of New Zealand. In 2000 a whirlpool there caught student divers, resulting in multiple fatalities.[13]

There was a short-lived whirlpool that sucked in a portion of the Lake Peigneur of 1300 acre area in Louisiana, United States after a drilling mishap in November 1980. This was not a naturally occurring whirlpool, but a man-made disaster caused by breaking through the roof of a salt mine. This mishap resulted in destruction of five houses, loss of nineteen barges and eight tug boats, oil rigs, a mobile home and most of the botanical garden and 10 percent area of the nearby Jefferson Island. A crater of 0.5-mile was created. The lake then drained, until the mine filled and the water levels equalized but the ten-foot deep lake was now 1,300 feet deep. Nine of the barges which had sunk floated back.[14][15][16]

A more recent example of a man-made whirlpool that received significant media coverage was in early June 2015, when an intake vortex formed in Lake Texoma, on the Oklahoma–Texas border, near the floodgates of the dam that forms the lake. At the time of the whirlpool's formation, the lake was being drained after reaching its highest level ever. The Army Corps of Engineers, which operates the dam and lake, expected that the whirlpool would last until the lake reached normal seasonal levels by late July.[17]

Dangers

An illustration from Jules Verne's essay "Edgard Poë et ses oeuvres" (Edgar Poe and his Works,1862) drawn by Frederic Lix or Yan' Dargent

Powerful whirlpools have killed unlucky seafarers, but their power tends to be exaggerated by laymen.[18] There are virtually no stories of large ships ever being sucked into a whirlpool. Tales like those by Paul the Deacon, Edgar Allan Poe, and Jules Verne are entirely fictional.[19]

In literature and popular culture

Apart from Poe and Verne other literary source is of the 1500s, of Olaus Magnus, a Swedish Bishop, who had stated that the maelstrom which was more powerful than The Odyssey destroyed ships which sank to the bottom of the sea, and even whales were sucked in. Pytheas, the Greek historian, also mentioned that maelstroms swallowed ships and threw them up again.[1]

Charybdis in Greek mythology was later rationalized as a whirlpool, which sucked entire ships into its fold in the narrow coast of Sicily, a disaster faced by navigators.[20]

In the 8th century, Paul the Deacon, who had lived among the Belgii, described tidal bores and the maelstrom for a Mediterranean audience unused to such violent tidal surges:[21]

Not very far from this shore … toward the western side, on which the ocean main lies open without end, is that very deep whirlpool of waters which we call by its familiar name "the navel of the sea." This is said to suck in the waves and spew them forth again twice every day. … They say there is another whirlpool of this kind between the island of Britain and the province of Galicia, and with this fact the coasts of the Seine region and of Aquitaine agree, for they are filled twice a day with such sudden inundations that any one who may by chance be found only a little inward from the shore can hardly get away. I have heard a certain high nobleman of the Gauls relating that a number of ships, shattered at first by a tempest, were afterwards devoured by this same Charybdis. And when one only out of all the men who had been in these ships, still breathing, swam over the waves, while the rest were dying, he came, swept by the force of the receding waters, up to the edge of that most frightful abyss. And when now he beheld yawning before him the deep chaos whose end he could not see, and half dead from very fear, expected to be hurled into it, suddenly in a way that he could not have hoped he was cast upon a certain rock and sat him down.

— Paul the Deacon, History of the Lombards, i.6

Three of the most notable literary references to the Lofoten Maelstrom date from the nineteenth century. The first is the Edgar Allan Poe short story "A Descent into the Maelström" (1841). The second is 20,000 Leagues Under the Sea (1870), the famous novel by Jules Verne. At the end of this novel, Captain Nemo seems to commit suicide, sending his Nautilus submarine into the Maelstrom (although in Verne's sequel Nemo and Nautilus survived). The "Norway maelstrom" is also mentioned in Herman Melville's Moby-Dick.[22]

In the 'Life of St Columba', the author, Adomnan of Iona', attributes to the saint miraculous knowledge of a particular bishop who ran into a whirlpool off the coast of Ireland. In Adomnan's narrative, he quotes Columba saying[23]

Cólman mac Beognai has set sail to come here, and is now in great danger in the surging tides of the whirlpool of Corryvreckan. Sitting in the prow, he lifts up his hands to heaven and blesses the turbulent, terrible seas. Yet the Lord terrifies him in this way, not so that the ship in which he sits should be overwhelmed and wrecked by the waves, but rather to rouse him to pray more fervently that he may sail through the peril and reach us here.

Etymology

One of the earliest uses in English of the Scandinavian word (malström or malstrøm) was by Edgar Allan Poe in his short story "A Descent into the Maelström" (1841). In turn, the Nordic word is derived from the Dutch maelstrom, modern spelling maalstroom, from malen (to grind) and stroom (stream), to form the meaning grinding current or literally "mill-stream", in the sense of milling (grinding) grain.[24]

See also

iconEnvironment portal iconWater portal

Coriolis effect

Eddy (fluid dynamics)

Rip current

In English, the word originally referred to the Moskstraumen.

Maelstrom may also refer to:

Contents

1 Amusement rides

2 Characters

3 Film and television

4 Games

5 Literature

6 Music

7 Other

Amusement rides

Maelstrom (ride), a dark ride that ran from 1988 to 2014 at one of 4 Walt Disney World Resort theme parks, Epcot, in Orlando, Florida, US

Maelstrom, a gyro swing ride at Drayton Manor Theme Park, Staffordshire, UK

Characters

Maelstrom (comics), fictional character that appears in comic books published by Marvel Comics

Maelstrom (Ice Age), a Pliosaurus

Mael Strom, aka Yuki Yoshida, from the anime/manga Kore wa Zombie Desu ka?

Film and television

Maelström (film), a 2000 Canadian film

Maelstrom (TV series), a 1985 BBC television drama serial

"Maelstrom" (Battlestar Galactica)

Games

Maelstrom (1992 video game), an Apple Macintosh game

Maelstrom (role playing game), a role-playing game by Alexander Scott

Maelstrom (video game), a 2007 PC game

VOR: The Maelstrom, a miniature wargame

Maelstrom (Live Action Roleplaying), a live action roleplaying game by Profound Decisions

Maelstrom, a lightning-based hammer item in the video game "Dota 2"

Literature

A Descent into the Maelström, an 1841 short story by Edgar Allan Poe

Maelstrom (Timms novel), a novel by E.V. Timms

Maelstrom, a 2001 novel by Peter Watts

Maelstrom, a 2006 novel in the Petaybee Series by Anne McCaffrey and Elizabeth Ann Scarborough

Maelstrom (Destroyermen novel), the third book of the Destroyermen series

Music

Maelstrom, an album by JR Ewing

"Maelstrom," a song by The Steve Miller Band on the album Living in the 20th Century

"Maelstrom," a composition by Julian Cochran

"The Maelstrom," a composition by Robert W. Smith

"Maelstrom 2010," a song by Kataklysm on the album Temple of Knowledge

Malström Graintable Synthesizer, a component of the musical software Reason

Maelstrom, a Boston-based thrash metal band active from the late 1980s to the early 1990s that released an LP on Taang! Records

"Ocean - II. Maelstrom" song by Canadian instrumental progressive metal band "Pomegranate Tiger" released on their 2013 album "Entities"

Other

Maelstrom, a Chromium-based browser by San Francisco-based BitTorrent Inc. with an integrated BitTorrent engine

Whirlpool à Amiens : un dossier industriel sous haute tension

In fluid dynamics, a vortex is a region in a fluid in which the flow rotates around an axis line, which may be straight or curved.[1][2] The plural of vortex is either vortices or vortexes.[3][4] Vortices form in stirred fluids, and may be observed in phenomena such as smoke rings, whirlpools in the wake of boat, or the winds surrounding a tornado or dust devil.

Vortices are a major component of turbulent flow. The distribution of velocity, vorticity (the curl of the flow velocity), as well as the concept of circulation are used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis.

In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries with it some angular and linear momentum, energy, and mass.

Properties

Vorticity

The Crow Instability of a jet aeroplane's contrail visually demonstrates the vortex created in the atmosphere (gas fluid medium) by the passage of the aircraft.

A key concept in the dynamics of vortices is the vorticity, a vector that describes the local rotary motion at a point in the fluid, as would be perceived by an observer that moves along with it. Conceptually, the vorticity could be observed by placing a tiny rough ball at the point in question, free to move with the fluid, and observing how it rotates about its center. The direction of the vorticity vector is defined to be the direction of the axis of rotation of this imaginary ball (according to the right-hand rule) while its length is twice the ball's angular velocity. Mathematically, the vorticity is defined as the curl (or rotational) of the velocity field of the fluid, usually denoted by ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} and expressed by the vector analysis formula ∇ × u → {\displaystyle \nabla \times {\vec {\mathit {u}}}} \nabla \times {\vec {{\mathit {u}}}}, where ∇ {\displaystyle \nabla } \nabla is the nabla operator and u → {\displaystyle {\vec {\mathit {u}}}} {\vec {{\mathit {u}}}} is the local flow velocity.[5]

The local rotation measured by the vorticity ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} must not be confused with the angular velocity vector of that portion of the fluid with respect to the external environment or to any fixed axis. In a vortex, in particular, ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis.

Vortex types

In theory, the speed u of the particles (and, therefore, the vorticity) in a vortex may vary with the distance r from the axis in many ways. There are two important special cases, however:

A rigid-body vortex

If the fluid rotates like a rigid body – that is, if the angular rotational velocity Ω is uniform, so that u increases proportionally to the distance r from the axis – a tiny ball carried by the flow would also rotate about its center as if it were part of that rigid body. In such a flow, the vorticity is the same everywhere: its direction is parallel to the rotation axis, and its magnitude is equal to twice the uniform angular velocity Ω of the fluid around the center of rotation.

Ω → = ( 0 , 0 , Ω ) , r → = ( x , y , 0 ) , {\displaystyle {\vec {\Omega }}=(0,0,\Omega ),\quad {\vec {r}}=(x,y,0),} {\vec {\Omega }}=(0,0,\Omega ),\quad {\vec {r}}=(x,y,0),

u → = Ω → × r → = ( − Ω y , Ω x , 0 ) , {\displaystyle {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\Omega y,\Omega x,0),} {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\Omega y,\Omega x,0),

ω → = ∇ × u → = ( 0 , 0 , 2 Ω ) = 2 Ω → . {\displaystyle {\vec {\omega }}=\nabla \times {\vec {u}}=(0,0,2\Omega )=2{\vec {\Omega }}.} {\vec \omega }=\nabla \times {\vec {u}}=(0,0,2\Omega )=2{\vec {\Omega }}.

An irrotational vortex

If the particle speed u is inversely proportional to the distance r from the axis, then the imaginary test ball would not rotate over itself; it would maintain the same orientation while moving in a circle around the vortex axis. In this case the vorticity ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} is zero at any point not on that axis, and the flow is said to be irrotational.

Ω → = ( 0 , 0 , α r − 2 ) , r → = ( x , y , 0 ) , {\displaystyle {\vec {\Omega }}=(0,0,\alpha r^{-2}),\quad {\vec {r}}=(x,y,0),} {\vec {\Omega }}=(0,0,\alpha r^{{-2}}),\quad {\vec {r}}=(x,y,0),

u → = Ω → × r → = ( − α y r − 2 , α x r − 2 , 0 ) , {\displaystyle {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\alpha yr^{-2},\alpha xr^{-2},0),} {\vec {u}}={\vec {\Omega }}\times {\vec {r}}=(-\alpha yr^{{-2}},\alpha xr^{{-2}},0),

ω → = ∇ × u → = 0. {\displaystyle {\vec {\omega }}=\nabla \times {\vec {u}}=0.} {\vec {\omega }}=\nabla \times {\vec {u}}=0.

Irrotational vortices

Pathlines of fluid particles around the axis (dashed line) of an ideal irrotational vortex. (See animation)

In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern[citation needed], where the flow velocity u is inversely proportional to the distance r. Irrotational vortices are also called free vortices.

For an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis; and has a fixed value, Γ, for any contour that does enclose the axis once.[6] The tangential component of the particle velocity is then u θ = Γ 2 π r {\displaystyle u_{\theta }={\tfrac {\Gamma }{2\pi r}}} {\displaystyle u_{\theta }={\tfrac {\Gamma }{2\pi r}}}. The angular momentum per unit mass relative to the vortex axis is therefore constant, r u θ = Γ 2 π {\displaystyle ru_{\theta }={\tfrac {\Gamma }{2\pi }}} {\displaystyle ru_{\theta }={\tfrac {\Gamma }{2\pi }}}.

However, the ideal irrotational vortex flow is not physically realizable, since it would imply that the particle speed (and hence the force needed to keep particles in their circular paths) would grow without bound as one approaches the vortex axis. Indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as r goes to zero. Within that region, the flow is no longer irrotational: the vorticity ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} becomes non-zero, with direction roughly parallel to the vortex axis. The Rankine vortex is a model that assumes a rigid-body rotational flow where r is less than a fixed distance r0, and irrotational flow outside that core regions. The Lamb-Oseen vortex model is an exact solution of the Navier–Stokes equations governing fluid flows and assumes cylindrical symmetry, for which

u θ = ( 1 − e − r 2 4 ν t ) Γ 2 π r . {\displaystyle u_{\theta }=\left(1-e^{\frac {-r^{2}}{4\nu t}}\right){\frac {\Gamma }{2\pi r}}.} {\displaystyle u_{\theta }=\left(1-e^{\frac {-r^{2}}{4\nu t}}\right){\frac {\Gamma }{2\pi r}}.}

Rotational vortices

The cloud vortex Saturn's hexagon is at the north pole of the planet Saturn.

A rotational vortex – one which has non-zero vorticity away from the core – can be maintained indefinitely in that state only through the application of some extra force, that is not generated by the fluid motion itself.

For example, if a water bucket is spun at constant angular speed w about its vertical axis, the water will eventually rotate in rigid-body fashion. The particles will then move along circles, with velocity u equal to wr.[6] In that case, the free surface of the water will assume a parabolic shape.

In this situation, the rigid rotating enclosure provides an extra force, namely an extra pressure gradient in the water, directed inwards, that prevents evolution of the rigid-body flow to the irrotational state.

Vortex geometry

In a stationary vortex, the typical streamline (a line that is everywhere tangent to the flow velocity vector) is a closed loop surrounding the axis; and each vortex line (a line that is everywhere tangent to the vorticity vector) is roughly parallel to the axis. A surface that is everywhere tangent to both flow velocity and vorticity is called a vortex tube. In general, vortex tubes are nested around the axis of rotation. The axis itself is one of the vortex lines, a limiting case of a vortex tube with zero diameter.

According to Helmholtz's theorems, a vortex line cannot start or end in the fluid – except momentarily, in non-steady flow, while the vortex is forming or dissipating. In general, vortex lines (in particular, the axis line) are either closed loops or end at the boundary of the fluid. A whirlpool is an example of the latter, namely a vortex in a body of water whose axis ends at the free surface. A vortex tube whose vortex lines are all closed will be a closed torus-like surface.

A newly created vortex will promptly extend and bend so as to eliminate any open-ended vortex lines. For example, when an airplane engine is started, a vortex usually forms ahead of each propeller, or the turbofan of each jet engine. One end of the vortex line is attached to the engine, while the other end usually stretches out and bends until it reaches the ground.

When vortices are made visible by smoke or ink trails, they may seem to have spiral pathlines or streamlines. However, this appearance is often an illusion and the fluid particles are moving in closed paths. The spiral streaks that are taken to be streamlines are in fact clouds of the marker fluid that originally spanned several vortex tubes and were stretched into spiral shapes by the non-uniform flow velocity distribution.

Pressure in a vortex

A Plughole vortex

The fluid motion in a vortex creates a dynamic pressure (in addition to any hydrostatic pressure) that is lowest in the core region, closest to the axis, and increases as one moves away from it, in accordance with Bernoulli's Principle. One can say that it is the gradient of this pressure that forces the fluid to follow a curved path around the axis.

In a rigid-body vortex flow of a fluid with constant density, the dynamic pressure is proportional to the square of the distance r from the axis. In a constant gravity field, the free surface of the liquid, if present, is a concave paraboloid.

In an irrotational vortex flow with constant fluid density and cylindrical symmetry, the dynamic pressure varies as P∞ − K/r2, where P∞ is the limiting pressure infinitely far from the axis. This formula provides another constraint for the extent of the core, since the pressure cannot be negative. The free surface (if present) dips sharply near the axis line, with depth inversely proportional to r2. The shape formed by the free surface is called a hyperboloid, or "Gabriel's Horn" (by Evangelista Torricelli).

The core of a vortex in air is sometimes visible because of a plume of water vapor caused by condensation in the low pressure and low temperature of the core; the spout of a tornado is an example. When a vortex line ends at a boundary surface, the reduced pressure may also draw matter from that surface into the core. For example, a dust devil is a column of dust picked up by the core of an air vortex attached to the ground. A vortex that ends at the free surface of a body of water (like the whirlpool that often forms over a bathtub drain) may draw a column of air down the core. The forward vortex extending from a jet engine of a parked airplane can suck water and small stones into the core and then into the engine.

Evolution

Vortices need not be steady-state features; they can move and change shape. In a moving vortex, the particle paths are not closed, but are open, loopy curves like helices and cycloids. A vortex flow might also be combined with a radial or axial flow pattern. In that case the streamlines and pathlines are not closed curves but spirals or helices, respectively. This is the case in tornadoes and in drain whirlpools. A vortex with helical streamlines is said to be solenoidal.

As long as the effects of viscosity and diffusion are negligible, the fluid in a moving vortex is carried along with it. In particular, the fluid in the core (and matter trapped by it) tends to remain in the core as the vortex moves about. This is a consequence of Helmholtz's second theorem. Thus vortices (unlike surface and pressure waves) can transport mass, energy and momentum over considerable distances compared to their size, with surprisingly little dispersion. This effect is demonstrated by smoke rings and exploited in vortex ring toys and guns.

Two or more vortices that are approximately parallel and circulating in the same direction will attract and eventually merge to form a single vortex, whose circulation will equal the sum of the circulations of the constituent vortices. For example, an airplane wing that is developing lift will create a sheet of small vortices at its trailing edge. These small vortices merge to form a single wingtip vortex, less than one wing chord downstream of that edge. This phenomenon also occurs with other active airfoils, such as propeller blades. On the other hand, two parallel vortices with opposite circulations (such as the two wingtip vortices of an airplane) tend to remain separate.

Vortices contain substantial energy in the circular motion of the fluid. In an ideal fluid this energy can never be dissipated and the vortex would persist forever. However, real fluids exhibit viscosity and this dissipates energy very slowly from the core of the vortex. It is only through dissipation of a vortex due to viscosity that a vortex line can end in the fluid, rather than at the boundary of the fluid.

Two-dimensional modeling

When the particle velocities are constrained to be parallel to a fixed plane, one can ignore the space dimension perpendicular to that plane, and model the flow as a two-dimensional flow velocity field on that plane. Then the vorticity vector ω → {\displaystyle {\vec {\omega }}} {\vec {\omega }} is always perpendicular to that plane, and can be treated as a scalar. This assumption is sometimes made in meteorology, when studying large-scale phenomena like hurricanes.

The behavior of vortices in such contexts is qualitatively different in many ways; for example, it does not allow the stretching of vortices that is often seen in three dimensions.

Further examples

The visible core of a vortex formed when a C-17 uses high engine power (reverse-thrust) at slow speed on a wet runway.

In the hydrodynamic interpretation of the behaviour of electromagnetic fields, the acceleration of electric fluid in a particular direction creates a positive vortex of magnetic fluid. This in turn creates around itself a corresponding negative vortex of electric fluid. Exact solutions to classical nonlinear magnetic equations include the Landau-Lifshitz equation, the continuum Heisenberg model, the Ishimori equation, and the nonlinear Schrödinger equation.

Bubble rings are underwater vortex rings whose core traps a ring of bubbles, or a single donut-shaped bubble. They are sometimes created by dolphins and whales.

The lifting force of aircraft wings, propeller blades, sails, and other airfoils can be explained by the creation of a vortex superimposed on the flow of air past the wing.

Aerodynamic drag can be explained in large part by the formation of vortices in the surrounding fluid that carry away energy from the moving body.

Large whirlpools can be produced by ocean tides in certain straits or bays. Examples are Charybdis of classical mythology in the Straits of Messina, Italy; the Naruto whirlpools of Nankaido, Japan; and the Maelstrom at Lofoten, Norway.

Vortices in the Earth's atmosphere are important phenomena for meteorology. They include mesocyclones on the scale of a few miles, tornados, waterspouts, and hurricanes. These vortices are often driven by temperature and humidity variations with altitude. The sense of rotation of hurricanes is influenced by the Earth's rotation. Another example is the Polar vortex, a persistent, large-scale cyclone centered near the Earth's poles, in the middle and upper troposphere and the stratosphere.

Vortices are prominent features of the atmospheres of other planets. They include the permanent Great Red Spot on Jupiter, the intermittent Great Dark Spot on Neptune, the polar vortices of Venus, the Martian dust devils and the North Polar Hexagon of Saturn.

Sunspots are dark regions on the Sun's visible surface (photosphere) marked by a lower temperature than its surroundings, and intense magnetic activity.

The accretion disks of black holes and other massive gravitational sources.

Taylor-Couette flow occurs in a fluid between two nested cylinders, one rotating, the other fixed.

See also

iconPhysics portal

Artificial gravity

Batchelor vortex

Biot–Savart law

Coordinate rotation

Cyclonic separation

Eddy

Gyre

Helmholtz's theorems

History of fluid mechanics

Horseshoe vortex

Hurricane

Kelvin–Helmholtz instability

Quantum vortex

Rankine vortex

Shower-curtain effect

Strouhal number

Vile Vortices

Von Kármán vortex street

Vortex engine

Vortex tube

Vortex cooler

VORTEX projects

Vortex shedding

Vortex stretching

Vortex induced vibration

Vorticity

Wormhole

References

Notes

Ting, L. (1991). Viscous Vortical Flows. Lecture notes in physics. Springer-Verlag. ISBN 3-540-53713-9.

Kida, Shigeo (2001). Life, Structure, and Dynamical Role of Vortical Motion in Turbulence (PDF). IUTAMim Symposium on Tubes, Sheets and Singularities in Fluid Dynamics. Zakopane, Poland.

"vortex". Oxford Dictionaries Online (ODO). Oxford University Press. Retrieved 2015-08-29.

"vortex". Merriam-Webster Online. Merriam-Webster, Inc. Retrieved 2015-08-29.

Vallis, Geoffrey (1999). Geostrophic Turbulence: The Macroturbulence of the Atmosphere and Ocean Lecture Notes (PDF). Lecture notes. Princeton University. p. 1. Retrieved 2012-09-26.

Clancy 1975, sub-section 7.5

Other

Loper, David E. (November 1966). An analysis of confined magnetohydrodynamic vortex flows (PDF) (NASA contractor report NASA CR-646). Washington: National Aeronautics and Space Administration. LCCN 67060315.

Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge Univ. Press. Ch. 7 et seq. ISBN 9780521098175.

Falkovich, G. (2011). Fluid Mechanics, a short course for physicists. Cambridge University Press. ISBN 978-1-107-00575-4.

Clancy, L.J. (1975). Aerodynamics. London: Pitman Publishing Limited. ISBN 0-273-01120-0.

De La Fuente Marcos, C.; Barge, P. (2001). "The effect of long-lived vortical circulation on the dynamics of dust particles in the mid-plane of a protoplanetary disc". Monthly Notices of the Royal Astronomical Society. 323 (3): 601–614. Bibcode:2001MNRAS.323..601D. doi:10.1046/j.1365-8711.2001.04228.x.

External links

Wikimedia Commons has media related to Vortex.

Optical Vortices

Video of two water vortex rings colliding (MPEG)

Chapter 3 Rotational Flows: Circulation and Turbulence

Vortical Flow Research Lab (MIT) – Study of flows found in nature and part of the Department of Ocean Engineering.

Whirlpool à Amiens

RAPPORT DE

Y'becca

sur

Le dossier Whirlpool d'Amiens est industriellement complexe et

s'est imposé comme le symbole de la course à la compétitivité

que se livrent les pays de l'Union Européenne...

**yanis la chouette**- Messages : 575

Date d'inscription : 24/02/2017

## Re: Whirlpool, a maelstrom of emotion disambiguation et Y'becca

Kármán vortex street

In fluid dynamics, a Kármán vortex street (or a von Kármán vortex sheet) is a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies. It is named after the engineer and fluid dynamicist Theodore von Kármán,[1] and is responsible for such phenomena as the "singing" of suspended telephone or power lines, and the vibration of a car antenna at certain speeds

DONC

Le groupe électroménager a décidé de délocaliser la production en Pologne et de fermer l'usine de sèche-linges en juin 2018. 40 millions d’euros d’investissements ont été réalisés sur le site

Duel surprise Macron-Le Pen ce mercredi sur le site de Whirlpool . Comme il s'y était engagé, Emmanuel Macron s'est rendu à Amiens pour rencontrer les salariés de l'usine d'électroménager, dont la production de sèche-linge sera délocalisée en Pologne en juin 2018. Mais pendant que le candidat d'En Marche s'entretenait en ville avec une délégation d'une dizaine de représentants des salariés à la Chambre de commerce et d'industrie, Marine Le Pen, son adversaire du FN, a effectué une visite surprise sur le site même de l'usine, là où les ouvriers tiennent un piquet de grève depuis deux jours.

Du coup, le candidat d'En Marche, qui avait justifié son choix de ne pas aller à l'usine, en expliquant ne pas vouloir exploiter à son profit la détresse des salariés, est allé lui aussi aller à la rencontre des salariés grévistes . «Il se moque de nous. Nous lui avons écrit il y a trois mois. Aujourd'hui, il vient dans l'entre-deux-tours. De toute façon, la plupart des ouvriers ont voté Marine », avait commenté Frédéric Chantrelle, délégué CFDT de l'usine.

Délocalisation en Pologne

Le dossier Whirlpool d'Amiens est industriellement complexe et s'est imposé comme le symbole de la course à la compétitivité que se livrent les pays de l'Union Européenne. C'est en Pologne, à Lodz, que le géant américain de l'électroménager veut, d'ici juin 2018, délocaliser sa production de sèche-linge réalisée dans son usine picarde. Cette fermeture entraînera la suppression de 290 emplois directs, auxquels s'ajoutent des dizaines d'intérimaires, ainsi qu'une soixantaine de postes du sous-traitant in situ Prima. En janvier, le groupe a expliqué que cette fermeture était nécessaire au maintien de sa compétitivité à long terme. Il a indiqué qu'il mettrait tout en oeuvre pour trouver un repreneur.

Une usine moderne

L'usine est en très bon état. Ces cinq dernières années, Whirlpool a investi pas moins de 40 millions d'euros dans sa modernisation. Il a imaginé en faire, un temps, un pôle européen d'excellence de la production de sèche-linge. En contre-partie de cette promesse, les salariés avaient accepté de renégocier leur temps de travail. Le projet s'était finalement soldé par un échec, le modèle innovant lancé par Whirlpool ne répondant pas aux normes de qualité énergétique voulues. En 2012, le site comptait plus de 1300 personnes.

A ce flop industriel, est venu s'ajouter, en 2015, le rachat par l'Américain de l'Italien Indesit et de sa quinzaine d'usine, réparties sur le Vieux Continent : en Italie, mais aussi en Turquie, en Russie et en Pologne. A Lodz justement, Whirlpool emploie désormais 1.200 personnes à la fabrication de cuisinières et de fours (1,2 million d'unités par an).« Whirlpool, c'est 800 millions de dollars de bénéfice, 21 milliards de chiffre d'affaire (...) mais ils fabriquent des chômeurs en France », expliquait ces dernières semaines François Gorlia, membre de la CGT.

Indemnités supra légales

Après la fermeture de Goodyear à Amiens Nord (1.143 salariés en 2014), et la perte de statut de capitale régionale, la fermeture de Whirlpool a été un nouveau coup dur pour la ville. Le bassin d'emploi qui compte plus de 13 % de chômage, a aussi perdu l'usine du fabricant de matelas Sapsa-bedding à Saleux (143 salariés en 2015).

Dans ce contexte, les négociations entre les salariés et la direction s'annoncent âpres, notamment concernant la question des indemnités supra-légales. A défaut d'empêcher sa fermeture, les salariés espèrent une reprise rapide du site. Une quinzaine de projets auraient déjà été déposés auprès de services de l'Etat, dont deux jugés très sérieux. Reste à savoir le nombre et le type d'emplois qu'ils permettront de maintenir.

Le caractère complexe du dossier s'explique aussi par sa situation géographique. L'usine est implantée dans une ville qui vient à peine de se remettre de la fermeture de Goodyear. Les syndicats accusent le groupe américain de délocaliser uniquement pour les profits.

AINSI, JE REVIENS DANS UN ASPECT DE SECOURISME POU DÉCRIRE L'ABANDON DES OUVRIERS

DE LA PART DE L’ÉTAT ET DE LA RÉGION INCARNE PAR MONSIEUR MACRON ET MADAME LE PEN MARINE...

Analysis

Animation of vortex street created by a cylindrical object; the flow on opposite sides of the object is given different colors, showing that the vortices are shed from alternating sides of the object

A look at the Kármán vortex street effect from ground level, as air flows quickly from the Pacific ocean eastward over Mojave desert mountains.

A vortex street will only form at a certain range of flow velocities, specified by a range of Reynolds numbers (Re), typically above a limiting Re value of about 90. The (global) Reynolds number for a flow is a measure of the ratio of inertial to viscous forces in the flow of a fluid around a body or in a channel, and may be defined as a nondimensional parameter of the global speed of the whole fluid flow:

R e L = U L ν 0 {\displaystyle \mathrm {Re} _{L}={\frac {UL}{\nu _{0}}}} {\displaystyle \mathrm {Re} _{L}={\frac {UL}{\nu _{0}}}}

where:

U {\displaystyle U} U = the free stream flow speed (i.e. the flow speed far from the fluid boundaries U ∞ {\displaystyle U_{\infty }} U_\infty like the body speed relative to the fluid at rest, or an inviscid flow speed, computed through the Bernoulli equation), which is the original global flow parameter, i.e. the target to be nondimensionalised.

L {\displaystyle L} L = a characteristic length parameter of the body or channel

ν 0 {\displaystyle \nu _{0}} \nu _{0} = the free stream kinematic viscosity parameter of the fluid, which in turn is the ratio:

ν = μ ρ {\displaystyle \nu ={\frac {\mu }{\rho }}} {\displaystyle \nu ={\frac {\mu }{\rho }}}

between:

ρ 0 {\displaystyle \rho _{0}} \rho _{0} = the reference fluid density.

μ 0 {\displaystyle \mu _{0}} \mu _{0} = the free stream fluid dynamic viscosity

For common flows (the ones which can usually be considered as incompressible ar isothermal), the kinematic viscosity is everywhere uniform over all the flow field and constant in time, so there is no choice on the viscosity parameter, which becomes naturally the kinematic viscosity of the fluid being considered at the temperature being considered. On the other hand, the reference length is always an arbitrary parameter, so particular attention should be put when comparing flows around different obstacles or in channels of different shapes: the global Reynolds numbers should be referred to the same reference length. This is actually the reason for which most precise sources for airfoil and channel flow data specify the reference length at a pedix to the Reynolds number. The reference length can vary depending on the analysis to be performed: for body with circle sections such as circular cyliners or spheres, one usually chooses the diameter; for an airfoil, a generic noncircular cylinder or a bluff body or a revolution body like a fuselage or a submarine, it is usually the profile chord or the profile thickness, or some other given widths that are in fact stable design inputs; for flow channels usually the hydraulic diameter) about which the fluid is flowing.

Interestingly, for an aerodynamic profile the reference length depends on the analysis. In fact, the profile chord is usually chosen as the reference length also for aerodynamic coefficient for wing sections and thin profiles in which the primary target is to maximize the lift coefficient or the lift/drag ratio (i.e. as usual in thin airfoil theory, one would employ the chord Reynolds as the flow speed parameter for comparing different profiles). On the other hand, for fairings and struts the given parameter is usually the dimension of internal structure to be streamlined (let us think for simplicity it is a beam with circular section), and the main target is to minimize the drag coefficient or the drag/lift ratio. The main design parameter which becomes naturally also a reference length is therefore the profile thickness (the profile dimension or area perpendicular to the flow direction), rather than the profile chord.

The range of Re values will vary with the size and shape of the body from which the eddies are being shed, as well as with the kinematic viscosity of the fluid. Over a large Red range (47<Red<105 for circular cylinders; reference length is d: diameter of the circular cylinder) eddies are shed continuously from each side of the circle boundary, forming rows of vortices in its wake. The alternation leads to the core of a vortex in one row being opposite the point midway between two vortex cores in the other row, giving rise to the distinctive pattern shown in the picture. Ultimately, the energy of the vortices is consumed by viscosity as they move further down stream, and the regular pattern disappears.

When a single vortex is shed, an asymmetrical flow pattern forms around the body and changes the pressure distribution. This means that the alternate shedding of vortices can create periodic lateral (sideways) forces on the body in question, causing it to vibrate. If the vortex shedding frequency is similar to the natural frequency of a body or structure, it causes resonance. It is this forced vibration that, at the correct frequency, causes suspended telephone or power lines to "sing" and the antenna on a car to vibrate more strongly at certain speeds.

In meteorology

Kármán vortex street caused by wind flowing around the Juan Fernández Islands off the Chilean coast

The flow of atmospheric air over obstacles such as islands or isolated mountains sometimes gives birth to von Kármán vortex streets. When a cloud layer is present at the relevant altitude, the streets become visible. Such cloud layer vortex streets have been photographed from satellites.[2]

Engineering problems

File:Karman Vortex Street Off Cylinder.ogvPlay media

Simulated vortex street around a no-slip cylindrical obstruction

File:Vortex.fin.small.ogvPlay media

The same cylinder, now with a fin, suppressing the vortex street by reducing the region in which the side eddies can interact

Chimneys with spirals outside to break up vortices

In low turbulence, tall buildings can produce a Kármán street so long as the structure is uniform along its height. In urban areas where there are many other tall structures nearby, the turbulence produced by these prevents the formation of coherent vortices.[3] Periodic crosswind forces set up by vortices along object's sides can be highly undesirable, and hence it is important for engineers to account for the possible effects of vortex shedding when designing a wide range of structures, from submarine periscopes to industrial chimneys and skyscrapers.

In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, provided it is longer than the diameter of the cylinder, will prevent the eddies from interacting, and consequently they remain attached. Obviously, for a tall building or mast, the relative wind could come from any direction. For this reason, helical projections that look like large screw threads are sometimes placed at the top, which effectively create asymmetric three-dimensional flow, thereby discouraging the alternate shedding of vortices; this is also found in some car antennas. Another countermeasure with tall buildings is using variation in the diameter with height, such as tapering - that prevents the entire building being driven at the same frequency.

Even more serious instability can be created in concrete cooling towers, for example, especially when built together in clusters. Vortex shedding caused the collapse of three towers at Ferrybridge Power Station C in 1965 during high winds.

The failure of the original Tacoma Narrows Bridge was originally attributed to excessive vibration due to vortex shedding, but was actually caused by aeroelastic flutter.

Kármán turbulence is also a problem for airplanes, especially at landing.[4][5]

Formula

This formula will generally hold true for the range 250 < Red < 2 × 105:

S t = 0.198 ( 1 − 19.7 R e d ) {\displaystyle St=0.198\left(1-{\frac {19.7}{Re_{d}}}\right)\ } {\displaystyle St=0.198\left(1-{\frac {19.7}{Re_{d}}}\right)\ }

where:

S t = f d U {\displaystyle St={\frac {fd}{U}}} {\displaystyle St={\frac {fd}{U}}}

f = vortex shedding frequency.

d = diameter of the cylinder

U = flow velocity.

This dimensionless parameter St is known as the Strouhal number and is named after the Czech physicist, Vincenc Strouhal (1850–1922) who first investigated the steady humming or singing of telegraph wires in 1878.

History

Although named after Theodore von Kármán,[6][7] he acknowledged[8] that the vortex street had been studied earlier by Mallock[9] and Bénard.[10]

See also

Eddy (fluid dynamics)

Kelvin–Helmholtz instability

Reynolds number

Vortex shedding

Vortex-induced vibration

Coandă effect

References

Theodore von Kármán, Aerodynamics. McGraw-Hill (1963): ISBN 978-0-07-067602-2. Dover (1994): ISBN 978-0-486-43485-8.

"Rapid Response - LANCE - Terra/MODIS 2010/226 14:55 UTC". Rapidfire.sci.gsfc.nasa.gov. Retrieved 2013-12-20.

Irwin, Peter A. (September 2010). "Vortices and tall buildings: A recipe for resonance". Physics Today. American Institute of Physics. 63 (9): 68–69. Bibcode:2010PhT....63i..68I. doi:10.1063/1.3490510. ISSN 0031-9228.

Wake turbulence

AIRPORT OPENING CEREMONY POSTPONED

T. von Kármán: Nachr. Ges. Wissenschaft. Göttingen Math. Phys. Klasse pp. 509–517 (1911) and pp. 547–556 (1912).

T. von Kármán: and H. Rubach, 1912: Phys. Z.", vol. 13, pp. 49–59.

T. Kármán, 1954. Aerodynamics: Selected Topics in the Light of Their Historical Development (Cornell University Press, Ithaca), pp. 68–69.

A. Mallock, 1907: On the resistance of air. Proc. Royal Soc., A79, pp. 262–265.

H. Bénard, 1908: Comptes rendus de l'Académie des Sciences (Paris), vol. 147, pp. 839–842, 970–972.

External links

Wikimedia Commons has media related to Von Kármán vortex streets.

Encyclopedia of Mathematics article on von Karman vortex shedding

Kármán vortex street formula calculator

3D animation of the Vortex Flow Measuring Principle

Vortex streets and Strouhal instability

How Insects Fly (which can produce von Kármán vortices)

YouTube — Flow visualisation of the vortex shedding mechanism on circular cylinder using hydrogen bubbles illuminated by a laser sheet in a water channel

Various Views of von Karman Vortices, NASA page

Authority control

GND: 4163337-4

[hide]

v t e

Patterns in nature

Patterns

Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry

in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern

Yemen Chameleon (cropped).jpg

Causes

Pattern formation Biology

Natural selection Camouflage Mimicry Sexual selection Mathematics

Chaos theory Fractal Logarithmic spiral Physics

Crystal Fluid dynamics Plateau's laws Self-organization

People

Plato Pythagoras Empedocles Leonardo Fibonacci

Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson

On Growth and Form Alan Turing

The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

Related

Emergence Mathematics and art

Y'becca et le Programme Spatial avec SPACE X, LA N.A.S.A, L'E.S.A et d'autres citoyens Morales et

Physiques ont des solutions qui se sont concrétisés durant les accord de la COP 21...

HELAS, MADAME LE PEN MARINE, MONSIEUR MACRON EMMANUEL sont trop querelleux

envers MADAME AUBRY MARTINE et MONSIEUR BERTRAND XAVIER...

RAPPORT DU

CITOYEN TIGNARD YANIS

In fluid dynamics, a Kármán vortex street (or a von Kármán vortex sheet) is a repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies. It is named after the engineer and fluid dynamicist Theodore von Kármán,[1] and is responsible for such phenomena as the "singing" of suspended telephone or power lines, and the vibration of a car antenna at certain speeds

DONC

Le groupe électroménager a décidé de délocaliser la production en Pologne et de fermer l'usine de sèche-linges en juin 2018. 40 millions d’euros d’investissements ont été réalisés sur le site

Duel surprise Macron-Le Pen ce mercredi sur le site de Whirlpool . Comme il s'y était engagé, Emmanuel Macron s'est rendu à Amiens pour rencontrer les salariés de l'usine d'électroménager, dont la production de sèche-linge sera délocalisée en Pologne en juin 2018. Mais pendant que le candidat d'En Marche s'entretenait en ville avec une délégation d'une dizaine de représentants des salariés à la Chambre de commerce et d'industrie, Marine Le Pen, son adversaire du FN, a effectué une visite surprise sur le site même de l'usine, là où les ouvriers tiennent un piquet de grève depuis deux jours.

Du coup, le candidat d'En Marche, qui avait justifié son choix de ne pas aller à l'usine, en expliquant ne pas vouloir exploiter à son profit la détresse des salariés, est allé lui aussi aller à la rencontre des salariés grévistes . «Il se moque de nous. Nous lui avons écrit il y a trois mois. Aujourd'hui, il vient dans l'entre-deux-tours. De toute façon, la plupart des ouvriers ont voté Marine », avait commenté Frédéric Chantrelle, délégué CFDT de l'usine.

Délocalisation en Pologne

Le dossier Whirlpool d'Amiens est industriellement complexe et s'est imposé comme le symbole de la course à la compétitivité que se livrent les pays de l'Union Européenne. C'est en Pologne, à Lodz, que le géant américain de l'électroménager veut, d'ici juin 2018, délocaliser sa production de sèche-linge réalisée dans son usine picarde. Cette fermeture entraînera la suppression de 290 emplois directs, auxquels s'ajoutent des dizaines d'intérimaires, ainsi qu'une soixantaine de postes du sous-traitant in situ Prima. En janvier, le groupe a expliqué que cette fermeture était nécessaire au maintien de sa compétitivité à long terme. Il a indiqué qu'il mettrait tout en oeuvre pour trouver un repreneur.

Une usine moderne

L'usine est en très bon état. Ces cinq dernières années, Whirlpool a investi pas moins de 40 millions d'euros dans sa modernisation. Il a imaginé en faire, un temps, un pôle européen d'excellence de la production de sèche-linge. En contre-partie de cette promesse, les salariés avaient accepté de renégocier leur temps de travail. Le projet s'était finalement soldé par un échec, le modèle innovant lancé par Whirlpool ne répondant pas aux normes de qualité énergétique voulues. En 2012, le site comptait plus de 1300 personnes.

A ce flop industriel, est venu s'ajouter, en 2015, le rachat par l'Américain de l'Italien Indesit et de sa quinzaine d'usine, réparties sur le Vieux Continent : en Italie, mais aussi en Turquie, en Russie et en Pologne. A Lodz justement, Whirlpool emploie désormais 1.200 personnes à la fabrication de cuisinières et de fours (1,2 million d'unités par an).« Whirlpool, c'est 800 millions de dollars de bénéfice, 21 milliards de chiffre d'affaire (...) mais ils fabriquent des chômeurs en France », expliquait ces dernières semaines François Gorlia, membre de la CGT.

Indemnités supra légales

Après la fermeture de Goodyear à Amiens Nord (1.143 salariés en 2014), et la perte de statut de capitale régionale, la fermeture de Whirlpool a été un nouveau coup dur pour la ville. Le bassin d'emploi qui compte plus de 13 % de chômage, a aussi perdu l'usine du fabricant de matelas Sapsa-bedding à Saleux (143 salariés en 2015).

Dans ce contexte, les négociations entre les salariés et la direction s'annoncent âpres, notamment concernant la question des indemnités supra-légales. A défaut d'empêcher sa fermeture, les salariés espèrent une reprise rapide du site. Une quinzaine de projets auraient déjà été déposés auprès de services de l'Etat, dont deux jugés très sérieux. Reste à savoir le nombre et le type d'emplois qu'ils permettront de maintenir.

Le caractère complexe du dossier s'explique aussi par sa situation géographique. L'usine est implantée dans une ville qui vient à peine de se remettre de la fermeture de Goodyear. Les syndicats accusent le groupe américain de délocaliser uniquement pour les profits.

AINSI, JE REVIENS DANS UN ASPECT DE SECOURISME POU DÉCRIRE L'ABANDON DES OUVRIERS

DE LA PART DE L’ÉTAT ET DE LA RÉGION INCARNE PAR MONSIEUR MACRON ET MADAME LE PEN MARINE...

Analysis

Animation of vortex street created by a cylindrical object; the flow on opposite sides of the object is given different colors, showing that the vortices are shed from alternating sides of the object

A look at the Kármán vortex street effect from ground level, as air flows quickly from the Pacific ocean eastward over Mojave desert mountains.

A vortex street will only form at a certain range of flow velocities, specified by a range of Reynolds numbers (Re), typically above a limiting Re value of about 90. The (global) Reynolds number for a flow is a measure of the ratio of inertial to viscous forces in the flow of a fluid around a body or in a channel, and may be defined as a nondimensional parameter of the global speed of the whole fluid flow:

R e L = U L ν 0 {\displaystyle \mathrm {Re} _{L}={\frac {UL}{\nu _{0}}}} {\displaystyle \mathrm {Re} _{L}={\frac {UL}{\nu _{0}}}}

where:

U {\displaystyle U} U = the free stream flow speed (i.e. the flow speed far from the fluid boundaries U ∞ {\displaystyle U_{\infty }} U_\infty like the body speed relative to the fluid at rest, or an inviscid flow speed, computed through the Bernoulli equation), which is the original global flow parameter, i.e. the target to be nondimensionalised.

L {\displaystyle L} L = a characteristic length parameter of the body or channel

ν 0 {\displaystyle \nu _{0}} \nu _{0} = the free stream kinematic viscosity parameter of the fluid, which in turn is the ratio:

ν = μ ρ {\displaystyle \nu ={\frac {\mu }{\rho }}} {\displaystyle \nu ={\frac {\mu }{\rho }}}

between:

ρ 0 {\displaystyle \rho _{0}} \rho _{0} = the reference fluid density.

μ 0 {\displaystyle \mu _{0}} \mu _{0} = the free stream fluid dynamic viscosity

For common flows (the ones which can usually be considered as incompressible ar isothermal), the kinematic viscosity is everywhere uniform over all the flow field and constant in time, so there is no choice on the viscosity parameter, which becomes naturally the kinematic viscosity of the fluid being considered at the temperature being considered. On the other hand, the reference length is always an arbitrary parameter, so particular attention should be put when comparing flows around different obstacles or in channels of different shapes: the global Reynolds numbers should be referred to the same reference length. This is actually the reason for which most precise sources for airfoil and channel flow data specify the reference length at a pedix to the Reynolds number. The reference length can vary depending on the analysis to be performed: for body with circle sections such as circular cyliners or spheres, one usually chooses the diameter; for an airfoil, a generic noncircular cylinder or a bluff body or a revolution body like a fuselage or a submarine, it is usually the profile chord or the profile thickness, or some other given widths that are in fact stable design inputs; for flow channels usually the hydraulic diameter) about which the fluid is flowing.

Interestingly, for an aerodynamic profile the reference length depends on the analysis. In fact, the profile chord is usually chosen as the reference length also for aerodynamic coefficient for wing sections and thin profiles in which the primary target is to maximize the lift coefficient or the lift/drag ratio (i.e. as usual in thin airfoil theory, one would employ the chord Reynolds as the flow speed parameter for comparing different profiles). On the other hand, for fairings and struts the given parameter is usually the dimension of internal structure to be streamlined (let us think for simplicity it is a beam with circular section), and the main target is to minimize the drag coefficient or the drag/lift ratio. The main design parameter which becomes naturally also a reference length is therefore the profile thickness (the profile dimension or area perpendicular to the flow direction), rather than the profile chord.

The range of Re values will vary with the size and shape of the body from which the eddies are being shed, as well as with the kinematic viscosity of the fluid. Over a large Red range (47<Red<105 for circular cylinders; reference length is d: diameter of the circular cylinder) eddies are shed continuously from each side of the circle boundary, forming rows of vortices in its wake. The alternation leads to the core of a vortex in one row being opposite the point midway between two vortex cores in the other row, giving rise to the distinctive pattern shown in the picture. Ultimately, the energy of the vortices is consumed by viscosity as they move further down stream, and the regular pattern disappears.

When a single vortex is shed, an asymmetrical flow pattern forms around the body and changes the pressure distribution. This means that the alternate shedding of vortices can create periodic lateral (sideways) forces on the body in question, causing it to vibrate. If the vortex shedding frequency is similar to the natural frequency of a body or structure, it causes resonance. It is this forced vibration that, at the correct frequency, causes suspended telephone or power lines to "sing" and the antenna on a car to vibrate more strongly at certain speeds.

In meteorology

Kármán vortex street caused by wind flowing around the Juan Fernández Islands off the Chilean coast

The flow of atmospheric air over obstacles such as islands or isolated mountains sometimes gives birth to von Kármán vortex streets. When a cloud layer is present at the relevant altitude, the streets become visible. Such cloud layer vortex streets have been photographed from satellites.[2]

Engineering problems

File:Karman Vortex Street Off Cylinder.ogvPlay media

Simulated vortex street around a no-slip cylindrical obstruction

File:Vortex.fin.small.ogvPlay media

The same cylinder, now with a fin, suppressing the vortex street by reducing the region in which the side eddies can interact

Chimneys with spirals outside to break up vortices

In low turbulence, tall buildings can produce a Kármán street so long as the structure is uniform along its height. In urban areas where there are many other tall structures nearby, the turbulence produced by these prevents the formation of coherent vortices.[3] Periodic crosswind forces set up by vortices along object's sides can be highly undesirable, and hence it is important for engineers to account for the possible effects of vortex shedding when designing a wide range of structures, from submarine periscopes to industrial chimneys and skyscrapers.

In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, provided it is longer than the diameter of the cylinder, will prevent the eddies from interacting, and consequently they remain attached. Obviously, for a tall building or mast, the relative wind could come from any direction. For this reason, helical projections that look like large screw threads are sometimes placed at the top, which effectively create asymmetric three-dimensional flow, thereby discouraging the alternate shedding of vortices; this is also found in some car antennas. Another countermeasure with tall buildings is using variation in the diameter with height, such as tapering - that prevents the entire building being driven at the same frequency.

Even more serious instability can be created in concrete cooling towers, for example, especially when built together in clusters. Vortex shedding caused the collapse of three towers at Ferrybridge Power Station C in 1965 during high winds.

The failure of the original Tacoma Narrows Bridge was originally attributed to excessive vibration due to vortex shedding, but was actually caused by aeroelastic flutter.

Kármán turbulence is also a problem for airplanes, especially at landing.[4][5]

Formula

This formula will generally hold true for the range 250 < Red < 2 × 105:

S t = 0.198 ( 1 − 19.7 R e d ) {\displaystyle St=0.198\left(1-{\frac {19.7}{Re_{d}}}\right)\ } {\displaystyle St=0.198\left(1-{\frac {19.7}{Re_{d}}}\right)\ }

where:

S t = f d U {\displaystyle St={\frac {fd}{U}}} {\displaystyle St={\frac {fd}{U}}}

f = vortex shedding frequency.

d = diameter of the cylinder

U = flow velocity.

This dimensionless parameter St is known as the Strouhal number and is named after the Czech physicist, Vincenc Strouhal (1850–1922) who first investigated the steady humming or singing of telegraph wires in 1878.

History

Although named after Theodore von Kármán,[6][7] he acknowledged[8] that the vortex street had been studied earlier by Mallock[9] and Bénard.[10]

See also

Eddy (fluid dynamics)

Kelvin–Helmholtz instability

Reynolds number

Vortex shedding

Vortex-induced vibration

Coandă effect

References

Theodore von Kármán, Aerodynamics. McGraw-Hill (1963): ISBN 978-0-07-067602-2. Dover (1994): ISBN 978-0-486-43485-8.

"Rapid Response - LANCE - Terra/MODIS 2010/226 14:55 UTC". Rapidfire.sci.gsfc.nasa.gov. Retrieved 2013-12-20.

Irwin, Peter A. (September 2010). "Vortices and tall buildings: A recipe for resonance". Physics Today. American Institute of Physics. 63 (9): 68–69. Bibcode:2010PhT....63i..68I. doi:10.1063/1.3490510. ISSN 0031-9228.

Wake turbulence

AIRPORT OPENING CEREMONY POSTPONED

T. von Kármán: Nachr. Ges. Wissenschaft. Göttingen Math. Phys. Klasse pp. 509–517 (1911) and pp. 547–556 (1912).

T. von Kármán: and H. Rubach, 1912: Phys. Z.", vol. 13, pp. 49–59.

T. Kármán, 1954. Aerodynamics: Selected Topics in the Light of Their Historical Development (Cornell University Press, Ithaca), pp. 68–69.

A. Mallock, 1907: On the resistance of air. Proc. Royal Soc., A79, pp. 262–265.

H. Bénard, 1908: Comptes rendus de l'Académie des Sciences (Paris), vol. 147, pp. 839–842, 970–972.

External links

Wikimedia Commons has media related to Von Kármán vortex streets.

Encyclopedia of Mathematics article on von Karman vortex shedding

Kármán vortex street formula calculator

3D animation of the Vortex Flow Measuring Principle

Vortex streets and Strouhal instability

How Insects Fly (which can produce von Kármán vortices)

YouTube — Flow visualisation of the vortex shedding mechanism on circular cylinder using hydrogen bubbles illuminated by a laser sheet in a water channel

Various Views of von Karman Vortices, NASA page

Authority control

GND: 4163337-4

[hide]

v t e

Patterns in nature

Patterns

Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry

in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern

Yemen Chameleon (cropped).jpg

Causes

Pattern formation Biology

Natural selection Camouflage Mimicry Sexual selection Mathematics

Chaos theory Fractal Logarithmic spiral Physics

Crystal Fluid dynamics Plateau's laws Self-organization

People

Plato Pythagoras Empedocles Leonardo Fibonacci

Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson

On Growth and Form Alan Turing

The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

Related

Emergence Mathematics and art

Y'becca et le Programme Spatial avec SPACE X, LA N.A.S.A, L'E.S.A et d'autres citoyens Morales et

Physiques ont des solutions qui se sont concrétisés durant les accord de la COP 21...

HELAS, MADAME LE PEN MARINE, MONSIEUR MACRON EMMANUEL sont trop querelleux

envers MADAME AUBRY MARTINE et MONSIEUR BERTRAND XAVIER...

RAPPORT DU

CITOYEN TIGNARD YANIS

**yanis la chouette**- Messages : 575

Date d'inscription : 24/02/2017

## Re: Whirlpool, a maelstrom of emotion disambiguation et Y'becca

UN AVENIR FRANÇAIS ET INTERNATIONALE:

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.

Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier,[2] but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals.[3]

Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. Symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally, the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than two, three, four, or six. In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. Due to fear of the scientific community's reaction, it took him two years to publish the results[4][5] for which he was awarded the Nobel Prize in Chemistry in 2011.[6]

History

A Penrose tiling

In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called Wang tiles) that can tile the plane but not in a periodic fashion. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern.[7] Around the same time Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.

Mathematically, quasicrystals have been shown to be derivable from a general method that treats them as projections of a higher-dimensional lattice. Just as circles, ellipses, and hyperbolic curves in the plane can be obtained as sections from a three-dimensional double cone, so too various (aperiodic or periodic) arrangements in two and three dimensions can be obtained from postulated hyperlattices with four or more dimensions. Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984.[8] The tiling is formed by two tiles with rhombohedral shape.

Shechtman first observed ten-fold electron diffraction patterns in 1982, as described in his notebook. The observation was made during a routine investigation, by electron microscopy, of a rapidly cooled alloy of aluminium and manganese prepared at the US National Bureau of Standards (later NIST).

In the summer of the same year Shechtman visited Ilan Blech and related his observation to him. Blech responded that such diffractions had been seen before.[9][10] Around that time, Shechtman also related his finding to John Cahn of NIST who did not offer any explanation and challenged him to solve the observation. Shechtman quoted Cahn as saying: "Danny, this material is telling us something and I challenge you to find out what it is".

The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, the existence of twins, was ruled out by his experiments. Since periodicity and twins were ruled out, Blech, unaware of the two-dimensional tiling work, was looking for another possibility: a completely new structure containing cells connected to each other by defined angles and distances but without translational periodicity. Blech decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material without long-range translational order but still not random. He termed this new structure multiple polyhedral[disambiguation needed].

The idea of a new structure was the necessary paradigm shift to break the impasse. The “Eureka moment” came when the computer simulation showed sharp ten-fold diffraction patterns, similar to the observed ones, emanating from the three-dimensional structure devoid of periodicity. The multiple polyhedral structure was termed later by many researchers as icosahedral glass but in effect it embraces any arrangement of polyhedra connected with definite angles and distances (this general definition includes tiling, for example).

Shechtman accepted Blech's discovery of a new type of material and it gave him the courage to publish his experimental observation. Shechtman and Blech jointly wrote a paper entitled "The Microstructure of Rapidly Solidified Al6Mn" [11] and sent it for publication around June 1984 to the Journal of Applied Physics (JAP). The JAP editor promptly rejected the paper as being better fit for a metallurgical readership. As a result, the same paper was re-submitted for publication to the Metallurgical Transactions A, where it was accepted. Although not noted in the body of the published text, the published paper was slightly revised prior to publication.

Meanwhile, on seeing the draft of the Shechtman-Blech paper in the summer of 1984, John Cahn suggested that Shechtman's experimental results merit a fast publication in a more appropriate scientific journal. Shechtman agreed and, in hindsight, called this fast publication "a winning move”. This paper, published in the Physical Review Letters (PRL),[5] repeated Shechtman's observation and used the same illustrations as the original Shechtman-Blech paper in the Metallurgical Transactions A. The PRL paper, the first to appear in print, caused considerable excitement in the scientific community.

Next year Ishimasa et al. reported twelvefold symmetry in Ni-Cr particles.[12] Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys.[13] Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable—when heated, they formed regular crystals. However, in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and opening the door to potential applications. In 2009, following a 10-year systematic search, scientists reported the first natural quasicrystal, a mineral found in the Khatyrka River in eastern Russia.[3] This natural quasicrystal exhibits high crystalline quality, equalling the best artificial examples.[14] The natural quasicrystal phase, with a composition of Al63Cu24Fe13, was named icosahedrite and it was approved by the International Mineralogical Association in 2010. Furthermore, analysis indicates it may be meteoritic in origin, possibly delivered from a carbonaceous chondrite asteroid.[15]

Atomic image of a micron-sized grain of the natural Al71Ni24Fe5 quasicrystal (shown in the inset) from a Khatyrka meteorite. The corresponding diffraction patterns reveal a ten-fold symmetry.[16]

A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al71Ni24Fe5. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching of a meteorite heated during an impact-induced shock.[16]

Electron diffraction pattern of an icosahedral Ho-Mg-Zn quasicrystal

In 1972 de Wolf and van Aalst[17] reported that the diffraction pattern produced by a crystal of sodium carbonate cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in reciprocal space. Other puzzling cases have been reported,[18] but until the concept of quasicrystal came to be established, they were explained away or denied.[19][20] However, at the end of the 1980s the idea became acceptable, and in 1992 the International Union of Crystallography altered its definition of a crystal, broadening it as a result of Shechtman’s findings, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic.[4][notes 1] Now, the symmetries compatible with translations are defined as "crystallographic", leaving room for other "non-crystallographic" symmetries. Therefore, aperiodic or quasiperiodic structures can be divided into two main classes: those with crystallographic point-group symmetry, to which the incommensurately modulated structures and composite structures belong, and those with non-crystallographic point-group symmetry, to which quasicrystal structures belong.

Originally, the new form of matter was dubbed "Shechtmanite".[21] The term "quasicrystal" was first used in print by Steinhardt and Levine[22] shortly after Shechtman's paper was published. The adjective quasicrystalline had already been in use, but now it came to be applied to any pattern with unusual symmetry.[notes 2] 'Quasiperiodical' structures were claimed to be observed in some decorative tilings devised by medieval Islamic architects.[23][24] For example, Girih tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern.[25] These claims have, however, been under some debate.[26]

Shechtman was awarded the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. "His discovery of quasicrystals revealed a new principle for packing of atoms and molecules," stated the Nobel Committee and pointed that "this led to a paradigm shift within chemistry." [4][27]

Mathematics

A penteract (5-cube) pattern using 5D orthographic projection to 2D using Petrie polygon basis vectors overlaid on the diffractogram from an Icosahedral Ho-Mg-Zn quasicrystal

A hexeract (6-cube) pattern using 6D orthographic projection to a 3D Perspective (visual) object (the Rhombic triacontahedron) using the Golden ratio in the basis vectors. This is used to understand the aperiodic Icosahedral structure of Quasicrystals.

There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon.[28] He introduced the notion of a superspace. Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes), and discussed their Fourier point spectrum. These functions are not exactly periodic, but they are arbitrarily close in some sense, as well as being a projection of an exactly periodic function.

In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures.[29] Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice).[30] The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets. The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers. The aperiodic structures obtained by the cut-and-project method are made diffractive by choosing a suitable orientation for the construction; this is a geometric approach that has also a great appeal for physicists.

Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals.[31]

Using mathematics for construction and analysis of quasicrystal structures is a difficult task for most experimentalists. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed[32] allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns.

Interacting spins were also analyzed in quasicrystals: AKLT Model and 8-vertex model were solved in quasicrystals analytically.[33]

Study of quasicrystals may shed light on the most basic notions related to quantum critical point observed in heavy fermion metals. Experimental measurements on the gold-aluminium-ytterbium quasicrystal have revealed a quantum critical point defining the divergence of the magnetic susceptibility as temperature tends to zero.[34] It is suggested that the electronic system of some quasicrystals is located at quantum critical point without tuning, while quasicrystals exhibit the typical scaling behaviour of their thermodynamic properties and belong to the famous family of heavy-fermion metals.

Materials science

Tiling of a plane by regular pentagons is impossible but can be realized on a sphere in the form of pentagonal dodecahedron.

A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron, the dual of the icosahedron. Unlike the similar pyritohedron shape of some cubic-system crystals such as pyrite, the quasicrystal has faces that are true regular pentagons

TiMn quasicrystal approximant lattice.

Since the original discovery by Dan Shechtman, hundreds of quasicrystals have been reported and confirmed. Undoubtedly, the quasicrystals are no longer a unique form of solid; they exist universally in many metallic alloys and some polymers. Quasicrystals are found most often in aluminium alloys (Al-Li-Cu, Al-Mn-Si, Al-Ni-Co, Al-Pd-Mn, Al-Cu-Fe, Al-Cu-V, etc.), but numerous other compositions are also known (Cd-Yb, Ti-Zr-Ni, Zn-Mg-Ho, Zn-Mg-Sc, In-Ag-Yb, Pd-U-Si, etc.).[35]

Two types of quasicrystals are known.[32] The first type, polygonal (dihedral) quasicrystals, have an axis of 8, 10, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type, icosahedral quasicrystals, are aperiodic in all directions.

Quasicrystals fall into three groups of different thermal stability:[36]

Stable quasicrystals grown by slow cooling or casting with subsequent annealing,

Metastable quasicrystals prepared by melt spinning, and

Metastable quasicrystals formed by the crystallization of the amorphous phase.

Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by X-ray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions.

The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.

A nanoscale icosahedral phase was formed in Zr-, Cu- and Hf-based bulk metallic glasses alloyed with noble metals.[37]

Most quasicrystals have ceramic-like properties including high thermal and electrical resistance, hardness and brittleness, resistance to corrosion, and non-stick properties.[38] Many metallic quasicrystalline substances are impractical for most applications due to their thermal instability; the Al-Cu-Fe ternary system and the Al-Cu-Fe-Cr and Al-Co-Fe-Cr quaternary systems, thermally stable up to 700 °C, are notable exceptions.

Applications

Quasicrystalline substances have potential applications in several forms.

Metallic quasicrystalline coatings can be applied by plasma-coating or magnetron sputtering. A problem that must be resolved is the tendency for cracking due to the materials' extreme brittleness.[38] The cracking could be suppressed by reducing sample dimensions or coating thickness.[39] Recent studies show typically brittle quasicrystals can exhibit remarkable ductility of over 50% strains at room temperature and sub-micrometer scales (<500 nm).[39]

An application was the use of low-friction Al-Cu-Fe-Cr quasicrystals[40] as a coating for frying pans. Food did not stick to it as much as to stainless steel making the pan moderately non-stick and easy to clean; heat transfer and durability were better than PTFE non-stick cookware and the pan was free from perfluorooctanoic acid (PFOA); the surface was very hard, claimed to be ten times harder than stainless steel, and not harmed by metal utensils or cleaning in a dishwasher; and the pan could withstand temperatures of 1,000 °C (1,800 °F) without harm. However, cooking with a lot of salt would etch the quasicrystalline coating used, and the pans were eventually withdrawn from production. Shechtman had one of these pans.[41]

The Nobel citation said that quasicrystals, while brittle, could reinforce steel "like armor". When Shechtman was asked about potential applications of quasicrystals he said that a precipitation-hardened stainless steel is produced that is strengthened by small quasicrystalline particles. It does not corrode and is extremely strong, suitable for razor blades and surgery instruments. The small quasicrystalline particles impede the motion of dislocation in the material.[41]

Quasicrystals were also being used to develop heat insulation, LEDs, diesel engines, and new materials that convert heat to electricity. Shechtman suggested new applications taking advantage of the low coefficient of friction and the hardness of some quasicrystalline materials, for example embedding particles in plastic to make strong, hard-wearing, low-friction plastic gears. The low heat conductivity of some quasicrystals makes them good for heat insulating coatings.[41]

Other potential applications include selective solar absorbers for power conversion, broad-wavelength reflectors, and bone repair and prostheses applications where biocompatibility, low friction and corrosion resistance are required. Magnetron sputtering can be readily applied to other stable quasicrystalline alloys such as Al-Pd-Mn.[38]

While saying that the discovery of icosahedrite, the first quasicrystal found in nature, was important, Shechtman saw no practical applications.

See also

Archimedean solid

Disordered Hyperuniformity

Fibonacci quasicrystal

Phason

Tessellation

Icosahedral twins

Notes

The concept of aperiodic crystal was coined by Erwin Schrödinger in another context with a somewhat different meaning. In his popular book What is life? in 1944, Schrödinger sought to explain how hereditary information is stored: molecules were deemed too small, amorphous solids were plainly chaotic, so it had to be a kind of crystal; as a periodic structure could encode very little information, it had to be aperiodic. DNA was later discovered, and, although not crystalline, it possesses properties predicted by Schrödinger—it is a regular but aperiodic molecule.

The use of the adjective 'quasicrystalline' for qualifying a structure can be traced back to the mid-1940-50s, e.g. in Kratky, O; Porod, G (1949). "Diffuse small-angle scattering of x-rays in colloid systems". Journal of Colloid Science. 4 (1): 35–70. doi:10.1016/0095-8522(49)90032-X. PMID 18110601.; Gunn, R (1955). "The statistical electrification of aerosols by ionic diffusion". Journal of Colloid Science. 10: 107–119. doi:10.1016/0095-8522(55)90081-7.

References

Ünal, B; V. Fournée; K.J. Schnitzenbaumer; C. Ghosh; C.J. Jenks; A.R. Ross; T.A. Lograsso; J.W. Evans; P.A. Thiel (2007). "Nucleation and growth of Ag islands on ﬁvefold Al-Pd-Mn quasicrystal surfaces: Dependence of island density on temperature and ﬂux". Physical Review B. 75 (6): 064205. Bibcode:2007PhRvB..75f4205U. doi:10.1103/PhysRevB.75.064205.

Steurer W. (2004). "Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals". Z. Kristallogr. 219 (7–2004): 391–446. Bibcode:2004ZK....219..391S. doi:10.1524/zkri.219.7.391.35643.

Bindi, L.; Steinhardt, P. J.; Yao, N.; Lu, P. J. (2009). "Natural Quasicrystals". Science. 324 (5932): 1306–9. Bibcode:2009Sci...324.1306B. doi:10.1126/science.1170827. PMID 19498165.

Andrea Gerlin (5 October 2011). "Technion's Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery". Bloomberg.

Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. (1984). "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry". Physical Review Letters. 53 (20): 1951–1953. Bibcode:1984PhRvL..53.1951S. doi:10.1103/PhysRevLett.53.1951.

"The Nobel Prize in Chemistry 2011". Nobelprize.org. Retrieved 2011-10-06.

Mackay, A.L. (1982). "Crystallography and the Penrose Pattern". Physica A. 114: 609–613. Bibcode:1982PhyA..114..609M. doi:10.1016/0378-4371(82)90359-4.

Kramer, P.; Neri, R. (1984). "On periodic and non-periodic space fillings of Em obtained by projection". Acta Crystallographica A. 40 (5): 580–587. doi:10.1107/S0108767384001203.

Yang, C. Y. (1979). "Crystallography of decahedral and icosahedral particles". J. Cryst. Growth. 47 (2): 274–282. Bibcode:1979JCrGr..47..274Y. doi:10.1016/0022-0248(79)90252-5.

Yang, C. Y.; Yacaman, M. J.; Heinemann, K. (1979). "Crystallography of decahedral and icosahedral particles". J. Cryst. Growth. 47 (2): 283–290. Bibcode:1979JCrGr..47..283Y. doi:10.1016/0022-0248(79)90253-7.

Shechtman, Dan; I. A. Blech (1985). "The Microstructure of Rapidly Solidified Al6Mn". Met. Trans. A. 16A (6): 1005–1012. Bibcode:1985MTA....16.1005S. doi:10.1007/BF02811670.

Ishimasa, T.; Nissen, H.-U.; Fukano, Y. (1985). "New ordered state between crystalline and amorphous in Ni-Cr particles". Physical Review Letters. 55 (5): 511–513. Bibcode:1985PhRvL..55..511I. doi:10.1103/PhysRevLett.55.511. PMID 10032372.

Wang, N.; Chen, H.; Kuo, K. (1987). "Two-dimensional quasicrystal with eightfold rotational symmetry". Physical Review Letters. 59 (9): 1010–1013. Bibcode:1987PhRvL..59.1010W. doi:10.1103/PhysRevLett.59.1010. PMID 10035936.

Steinhardt, Paul; Bindi, Luca (2010). "Once upon a time in Kamchatka: the search for natural quasicrystals". Philosophical Magazine. 91 (19–21): 1. Bibcode:2011PMag...91.2421S. doi:10.1080/14786435.2010.510457.

Bindi, Luca; John M. Eiler; Yunbin Guan; Lincoln S. Hollister; Glenn MacPherson; Paul J. Steinhardt; Nan Yao (2012-01-03). "Evidence for the extraterrestrial origin of a natural quasicrystal". Proceedings of the National Academy of Sciences. 109 (5): 1396–1401. Bibcode:2012PNAS..109.1396B. doi:10.1073/pnas.1111115109.

Bindi, L.; Yao, N.; Lin, C.; Hollister, L. S.; Andronicos, C. L.; Distler, V. V.; Eddy, M. P.; Kostin, A.; Kryachko, V.; MacPherson, G. J.; Steinhardt, W. M.; Yudovskaya, M.; Steinhardt, P. J. (2015). "Natural quasicrystal with decagonal symmetry". Scientific Reports. 5: 9111. Bibcode:2015NatSR...5E9111B. doi:10.1038/srep09111. PMC 4357871Freely accessible. PMID 25765857.

de Wolf, R.M. & van Aalst, W. (1972). "The four dimensional group of γ-Na2CO3". Acta Crystallogr. A. 28: S111.

Kleinert H. and Maki K. (1981). "Lattice Textures in Cholesteric Liquid Crystals" (PDF). Fortschritte der Physik. 29 (5): 219–259. Bibcode:1981ForPh..29..219K. doi:10.1002/prop.19810290503.

Pauling, L (1987-01-26). "So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal.". Physical Review Letters. 58 (4): 365–368. Bibcode:1987PhRvL..58..365P. doi:10.1103/PhysRevLett.58.365. PMID 10034915.

Kenneth Chang (October 5, 2011). "Israeli Scientist Wins Nobel Prize for Chemistry". NY Times.

Browne, Malcolm W. (1989-09-05). "Impossible' Form of Matter Takes Spotlight In Study of Solids". New York Times.

Levine, Dov; Steinhardt, Paul (1984). "Quasicrystals: A New Class of Ordered Structures". Physical Review Letters. 53 (26): 2477–2480. Bibcode:1984PhRvL..53.2477L. doi:10.1103/PhysRevLett.53.2477.

Makovicky, E. (1992), 800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired. In: I. Hargittai, editor: Fivefold Symmetry, pp. 67–86. World Scientific, Singapore-London

Lu, Peter J. & Steinhardt, Paul J. (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture" (PDF). Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056.

Lu, P. J.; Steinhardt, P. J. (2007). "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture". Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056.

Makovicky, Emil (2007). "Comment on "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture"". Science. 318 (5855): 1383–1383. Bibcode:2007Sci...318.1383M. doi:10.1126/science.1146262. PMID 18048668.

"Nobel win for crystal discovery". BBC News. 2011-10-05. Retrieved 2011-10-05.

Bohr, H. (1925). "Zur Theorie fastperiodischer Funktionen I". Acta Mathematicae. 45: 580. doi:10.1007/BF02395468.

de Bruijn, N. (1981). "Algebraic theory of Penrose's non-periodic tilings of the plane". Nederl. Akad. Wetensch. Proc. A84: 39.

Suck, Jens-Boie; Schreiber, M.; Häussler, Peter (2002). Quasicrystals: An Introduction to Structure, Physical Properties and Applications. Springer Science & Business Media. pp. 1–. ISBN 978-3-540-64224-4.

Paterson, Alan L. T. (1999). Groupoids, inverse semigroups, and their operator algebras. Springer. p. 164. ISBN 0-8176-4051-7.

Yamamoto, Akiji (2008). "Software package for structure analysis of quasicrystals". Science and Technology of Advanced Materials. 9 (1): 013001. Bibcode:2008STAdM...9a3001Y. doi:10.1088/1468-6996/9/3/013001. PMC 5099788Freely accessible. PMID 27877919.

Korepin, V.E. Completely integrable models in quasicrystals. Comm. Math. Phys. Volume 110, Number 1 (1987), 157–171.

Deguchi, Kazuhiko; Matsukawa, Shuya; Sato, Noriaki K.; Hattori, Taisuke; Ishida, Kenji; Takakura, Hiroyuki; Ishimasa, Tsutomu (2012). "Quantum critical state in a magnetic quasicrystal". Nature Materials. 11: 1013–6. doi:10.1038/nmat3432. PMID 23042414.

MacIá, Enrique (2006). "The role of aperiodic order in science and technology". Reports on Progress in Physics. 69 (2): 397–441. Bibcode:2006RPPh...69..397M. doi:10.1088/0034-4885/69/2/R03.

Tsai, An Pang (2008). "Icosahedral clusters, icosaheral order and stability of quasicrystals—a view of metallurgy". Science and Technology of Advanced Materials. 9 (1): 013008. Bibcode:2008STAdM...9a3008T. doi:10.1088/1468-6996/9/1/013008. PMC 5099795Freely accessible. PMID 27877926.

Louzguine-Luzgin, D. V.; Inoue, A. (2008). "Formation and Properties of Quasicrystals". Annual Review of Materials Research. 38: 403–423. Bibcode:2008AnRMS..38..403L. doi:10.1146/annurev.matsci.38.060407.130318.

"Sputtering technique forms versatile quasicrystalline coatings". MRS Bulletin. 36 (Cool: 581. 2011. doi:10.1557/mrs.2011.190.

Zou, Yu; Kuczera, Pawel; Sologubenko, Alla; Sumigawa, Takashi; Kitamura, Takayuki; Steurer, Walter; Spolenak, Ralph (2016). "Superior room-temperature ductility of typically brittle quasicrystals at small sizes". Nature Communications. 7: 12261. doi:10.1038/ncomms12261. PMC 4990631Freely accessible. PMID 27515779.

Fikar, Jan (2003). Al-Cu-Fe quasicrystalline coatings and composites studied by mechanical spectroscopy. École polytechnique fédérale de Lausanne EPFL, Thesis n° 2707 (2002). doi:10.5075/epfl-thesis-2707.

Kalman, Matthew (12 October 2011). "The Quasicrystal Laureate". MIT Technology Review. Retrieved 12 February 2016.

External links

A Partial Bibliography of Literature on Quasicrystals (1996–2008).

BBC webpage showing pictures of Quasicrystals

What is... a Quasicrystal?, Notices of the AMS 2006, Volume 53, Number 8

Gateways towards quasicrystals: a short history by P. Kramer

Quasicrystals: an introduction by R. Lifshitz

Quasicrystals: an introduction by S. Weber

Steinhardt's proposal

Quasicrystal Research – Documentary 2011 on the research of the University of Stuttgart

Thiel, P.A. (2008). "Quasicrystal Surfaces". Annual Review of Physical Chemistry. 59: 129–152. Bibcode:2008ARPC...59..129T. doi:10.1146/annurev.physchem.59.032607.093736. PMID 17988201.

Foundations of Crystallography.

Quasicrystals: What are they, and why do they exist?, Marek Mihalkovic and many others. (Microsoft PowerPoint format)[permanent dead link]

"Indiana Steinhardt and the Quest for Quasicrystals – A Conversation with Paul Steinhardt", Ideas Roadshow, 2016

Shaginyan, V. R.; Msezane, A. Z.; Popov, K. G.; Japaridze, G. S.; Khodel, V. A. (2013). "Common quantum phase transition in quasicrystals and heavy-fermion metals". Physical Review B. 87 (24). doi:10.1103/PhysRevB.87.245122.

Authority control

GND: 4202613-1 NDL: 00577309

[hide]

v t e

Patterns in nature

Patterns

Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry

in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern

Yemen Chameleon (cropped).jpg

Causes

Pattern formation Biology

Natural selection Camouflage Mimicry Sexual selection Mathematics

Chaos theory Fractal Logarithmic spiral Physics

Crystal Fluid dynamics Plateau's laws Self-organization

People

Plato Pythagoras Empedocles Leonardo Fibonacci

Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson

On Growth and Form Alan Turing

The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

Related

Emergence Mathematics and art

RAPPORT DE

Y'BECCA,

PRÊTE A INVESTIR SUR :

Whirlpool à Amiens

Tessellation

From Wikipedia, the free encyclopedia

"Tessellate" redirects here. For the song by Alt-J, see Tessellate (song). For the computer graphics technique, see Tessellation (computer graphics).

Zellige terracotta tiles in Marrakech, forming edge-to-edge, regular and other tessellations

A wall sculpture at Leeuwarden celebrating the artistic tessellations of M. C. Escher

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and Semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space.

A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.

Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.[1]

Decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity,[2] sometimes displaying geometric patterns.[3][4]

In 1619 Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes.[5][6][7]

Roman geometric mosaic

Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.[8][9] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1964),[10] and Heinrich Heesch and Otto Kienzle (1963).[11]

Etymology

In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.[12] The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τέσσερα for four). It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay.

Overview

A rhombitrihexagonal tiling: tiled floor of a church in Seville, Spain, using square, triangle and hexagon prototiles

Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another.[13] The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical[a] regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.[14] There are only three shapes that can form such regular tessellations: the equilateral triangle, square, and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.[6]

Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner.[15] Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects.[16] If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors.[17]

Elaborate and colourful zellige tessellations of glazed tiles at the Alhambra in Spain

More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. These tiles may be polygons or any other shapes.[b] Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane.[19] No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.[18] For example, the types of convex pentagon that can tile the plane remains an unsolved problem.[20]

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.[6] The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions. He further defined the Schläfli symbol notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is {3}, while that for a square is {4}.[21] The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is {6,3}.[22]

Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 44. The tiling of regular hexagons is noted 6.6.6, or 63.[18]

In mathematics

Introduction to tessellations

Further information: Euclidean tilings of regular polygons, Uniform tiling, and List of convex uniform tilings

Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles; it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling is a tiling where every vertex point is identical; that is, the arrangement of polygons about each vertex is the same.[18] The fundamental region is a shape such as a rectangle that is repeated to form the tessellation.[23] For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.[18]

The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tiling is any polygonal tessellation where adjacent tiles only share one full side, i.e., no tile shares a partial side or more than one side with any other tile. In an edge-to-edge tiling, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks.[18]

A normal tiling is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a single connected set or the empty set, and all tiles are uniformly bounded. This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling; the condition disallows tiles that are pathologically long or thin.[24]

The 15th convex monohedral pentagonal tiling, discovered in 2015

A monohedral tiling is a tessellation in which all tiles are congruent; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936; the Voderberg tiling has a unit tile that is a nonconvex enneagon.[1] The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a pentagon tiling using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the internal angle of a regular pentagon, 3π/5, is not a divisor of 2π.[25][26][27]

A Pythagorean tiling

An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling.[24] If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms anisohedral tilings.

A regular tessellation is a highly symmetric, edge-to-edge tiling made up of regular polygons, all of the same shape. There are only three regular tessellations: those made up of equilateral triangles, squares, or regular hexagons. All three of these tilings are isogonal and monohedral.[28]

A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two).[29] These can be described by their vertex configuration; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.82 (each vertex has one square and two octagons).[30] Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of Pythagorean tilings, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.[31] An edge tessellation is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles.[32]

This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.

Wallpaper groups

Main article: Wallpaper group

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.[33] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,[34] the variety and sophistication of the Alhambra tilings have surprised modern researchers.[35] Of the three regular tilings two are in the p6m wallpaper group and one is in p4m. Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns.[36] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane.[37]

Aperiodic tilings

Main articles: Aperiodic tiling and List of aperiodic sets of tiles

A Penrose tiling, with several symmetries but no periodic repetitions

Penrose tilings, which use two different quadrilaterals, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings, which use tiles that cannot tessellate periodically. The recursive process of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is the rep-tiles; these tilings have surprising self-replicating properties.[38] Pinwheel tilings are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.[39] It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in translational symmetry, do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches.[40] A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.[41] A Fibonacci word can be used to build an aperiodic tiling, and to study quasicrystals, which are structures with aperiodic order.[42]

A set of 13 Wang tiles that tile the plane only aperiodically

Wang tiles are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wang dominoes. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any Turing machine can be represented as a set of Wang dominoes that tile the plane if and only if the Turing machine does not halt. Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.[43][44][45][46][47]

Random Truchet tiling

Truchet tiles are square tiles decorated with patterns so they do not have rotational symmetry; in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.[48][49]

Tessellations and colour

Further information: four colour theorem

If the colours of this tiling are to form a pattern by repeating this rectangle as the fundamental domain, at least seven colours are required; more generally, at least four colours are needed.

Sometimes the colour of a tile is understood as part of the tiling; at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity one needs to specify whether the colours are part of the tiling or just part of its illustration. This affects whether tiles with the same shape but different colours are considered identical, which in turn affects questions of symmetry. The four colour theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The colouring guaranteed by the four-colour theorem does not generally respect the symmetries of the tessellation. To produce a colouring which does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as in the picture at right.[50]

Tessellations with polygons

A Voronoi tiling, in which the cells are always convex polygons

Next to the various tilings by regular polygons, tilings by other polygons have also been studied.

Any triangle or quadrilateral (even non-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.[51]

If only one shape of tile is allowed, tilings exists with convex N-gons for N equal to 3, 4, 5 and 6. For N = 5, see Pentagonal tiling and for N = 6, see Hexagonal tiling.

For results on tiling the plane with polyominoes, see Polyomino § Uses of polyominoes.

Voronoi tilings

Voronoi or Dirichlet tilings are tessellations where each tile is defined as the set of points closest to one of the points in a discrete set of defining points. (Think of geographical regions where each region is defined as all the points closest to a given city or post office.)[52][53] The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges.[54] Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.[55]

Tessellating three-dimensional space: the rhombic dodecahedron is one of the solids that can be stacked to fill space exactly.

Tessellations in higher dimensions

Main article: Honeycomb (geometry)

Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile.

Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only Platonic polyhedron to do so), the rhombic dodecahedron, the truncated octahedron, and triangular, quadrilateral, and hexagonal prisms, among others.[56] Any polyhedron that fits this criterion is known as a plesiohedron, and may possess between 4 and 38 faces.[57] Naturally occurring rhombic dodecahedra are found as crystals of Andradite (a kind of Garnet) and Fluorite.[58][59]

A Schwarz triangle is a spherical triangle that can be used to tile a sphere.[60]

Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular[c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions.[61] Uniform polyhedra can be constructed using the Wythoff construction.[62]

The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.[63]

Tessellations in non-Euclidean geometries

Rhombitriheptagonal tiling in hyperbolic plane, seen in Poincaré disk model projection

The regular {3,5,3} icosahedral honeycomb, one of four regular compact honeycombs in hyperbolic 3-space

It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces; these are vertex-transitive (transitive on its vertices), and isogonal (there is an isometry mapping any vertex onto any other).[64][65]

A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.[66]

In art

Further information: Mathematics and art

A quilt showing a regular tessellation pattern.

Roman mosaic floor panel of stone, tile and glass, from a villa near Antioch in Roman Syria. 2nd century A.D.

In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns.[4] Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the Moorish wall tilings of Islamic architecture, using Girih and Zellige tiles in buildings such as the Alhambra[67] and La Mezquita.[68]

Tessellations frequently appeared in the graphic art of M. C. Escher; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 1936.[69] Escher made four "Circle Limit" drawings of tilings that use hyperbolic geometry.[70][71] For his woodcut "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.[72] Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line."[73]

Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts.[74][75]

Tessellations are also a main genre in origami (paper folding), where pleats are used to connect molecules such as twist folds together in a repeating fashion.[76]

In manufacturing

Tessellation is used in manufacturing industry to reduce the wastage of material (yield losses) such as sheet metal when cutting out shapes for objects like car doors or drinks cans.[77]

In nature

Tessellate pattern in a Colchicum flower

Main article: Patterns in nature

The honeycomb provides a well-known example of tessellation in nature with its hexagonal cells.[78]

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the Fritillary[79] and some species of Colchicum are characteristically tessellate.[80]

Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations,[81] also known as random crack networks.[82] The Gilbert tessellation is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. The model, named after Edgar Gilbert, allows cracks to form starting from randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons.[83] Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland.[84] Tessellated pavement, a characteristic example of which is found at Eaglehawk Neck on the Tasman Peninsula of Tasmania, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.[85]

Other natural patterns occur in foams; these are packed according to Plateau's laws, which require minimal surfaces. Such foams present a problem in how to pack cells as tightly as possible: in 1887, Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure, which uses less surface area to separate cells of equal volume than Kelvin's foam.[86]

In puzzles and recreational mathematics

Traditional tangram dissection puzzle

Main articles: Tiling puzzle and recreational mathematics

Tessellations have given rise to many types of tiling puzzle, from traditional jigsaw puzzles (with irregular pieces of wood or cardboard)[87] and the tangram[88] to more modern puzzles which often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles.[89][90] Authors such as Henry Dudeney and Martin Gardner have made many uses of tessellation in recreational mathematics. For example, Dudeney invented the hinged dissection,[91] while Gardner wrote about the rep-tile, a shape that can be dissected into smaller copies of the same shape.[92][93] Inspired by Gardner's articles in Scientific American, the amateur mathematician Marjorie Rice found four new tessellations with pentagons.[94][95] Squaring the square is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.[96][97] An extension is squaring the plane, tiling it by squares whose sizes are all natural numbers without repetitions; James and Frederick Henle proved that this was possible.[98]

Sources

Coxeter, H.S.M. (1973). Regular Polytopes, Section IV : Tessellations and Honeycombs. Dover. ISBN 0-486-61480-8.

Escher, M. C. (1974). J. L. Locher, ed. The World of M. C. Escher (New Concise NAL ed.). Abrams. ISBN 0-451-79961-5.

Gardner, Martin (1989). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.

Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1.

Gullberg, Jan (1997). Mathematics From the Birth of Numbers. Norton. ISBN 0-393-04002-X.

Magnus, Wilhelm (1974). Noneuclidean Tesselations and Their Groups. Academic Press. ISBN 978-0-12-465450-1.

Stewart, Ian (2001). What Shape is a Snowflake?. Weidenfeld and Nicolson. ISBN 0-297-60723-5.

External links

Wikimedia Commons has media related to Tessellation.

Wolfram MathWorld: Tessellation (good bibliography, drawings of regular, semiregular and demiregular tessellations)

Tilings Encyclopedia (extensive information on substitution tilings, including drawings, people, and references)

Tessellations.org (how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)

Eppstein, David. "The Geometry Junkyard: Hyperbolic Tiling". (list of web resources including articles and galleries)

[show]

v t e

Mathematics and art

RAPPORT DE

Y'BECCA,

PRÊTE A INVESTIR SUR :

Whirlpool à Amiens

et sur l'ensemble du territoire...

EN ACCORD AVEC LA COP 21

A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.

Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier,[2] but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals.[3]

Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. Symmetrical diffraction patterns result from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally, the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than two, three, four, or six. In 1982 materials scientist Dan Shechtman observed that certain aluminium-manganese alloys produced the unusual diffractograms which today are seen as revelatory of quasicrystal structures. Due to fear of the scientific community's reaction, it took him two years to publish the results[4][5] for which he was awarded the Nobel Prize in Chemistry in 2011.[6]

History

A Penrose tiling

In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called Wang tiles) that can tile the plane but not in a periodic fashion. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern.[7] Around the same time Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.

Mathematically, quasicrystals have been shown to be derivable from a general method that treats them as projections of a higher-dimensional lattice. Just as circles, ellipses, and hyperbolic curves in the plane can be obtained as sections from a three-dimensional double cone, so too various (aperiodic or periodic) arrangements in two and three dimensions can be obtained from postulated hyperlattices with four or more dimensions. Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984.[8] The tiling is formed by two tiles with rhombohedral shape.

Shechtman first observed ten-fold electron diffraction patterns in 1982, as described in his notebook. The observation was made during a routine investigation, by electron microscopy, of a rapidly cooled alloy of aluminium and manganese prepared at the US National Bureau of Standards (later NIST).

In the summer of the same year Shechtman visited Ilan Blech and related his observation to him. Blech responded that such diffractions had been seen before.[9][10] Around that time, Shechtman also related his finding to John Cahn of NIST who did not offer any explanation and challenged him to solve the observation. Shechtman quoted Cahn as saying: "Danny, this material is telling us something and I challenge you to find out what it is".

The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, the existence of twins, was ruled out by his experiments. Since periodicity and twins were ruled out, Blech, unaware of the two-dimensional tiling work, was looking for another possibility: a completely new structure containing cells connected to each other by defined angles and distances but without translational periodicity. Blech decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material without long-range translational order but still not random. He termed this new structure multiple polyhedral[disambiguation needed].

The idea of a new structure was the necessary paradigm shift to break the impasse. The “Eureka moment” came when the computer simulation showed sharp ten-fold diffraction patterns, similar to the observed ones, emanating from the three-dimensional structure devoid of periodicity. The multiple polyhedral structure was termed later by many researchers as icosahedral glass but in effect it embraces any arrangement of polyhedra connected with definite angles and distances (this general definition includes tiling, for example).

Shechtman accepted Blech's discovery of a new type of material and it gave him the courage to publish his experimental observation. Shechtman and Blech jointly wrote a paper entitled "The Microstructure of Rapidly Solidified Al6Mn" [11] and sent it for publication around June 1984 to the Journal of Applied Physics (JAP). The JAP editor promptly rejected the paper as being better fit for a metallurgical readership. As a result, the same paper was re-submitted for publication to the Metallurgical Transactions A, where it was accepted. Although not noted in the body of the published text, the published paper was slightly revised prior to publication.

Meanwhile, on seeing the draft of the Shechtman-Blech paper in the summer of 1984, John Cahn suggested that Shechtman's experimental results merit a fast publication in a more appropriate scientific journal. Shechtman agreed and, in hindsight, called this fast publication "a winning move”. This paper, published in the Physical Review Letters (PRL),[5] repeated Shechtman's observation and used the same illustrations as the original Shechtman-Blech paper in the Metallurgical Transactions A. The PRL paper, the first to appear in print, caused considerable excitement in the scientific community.

Next year Ishimasa et al. reported twelvefold symmetry in Ni-Cr particles.[12] Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys.[13] Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable—when heated, they formed regular crystals. However, in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and opening the door to potential applications. In 2009, following a 10-year systematic search, scientists reported the first natural quasicrystal, a mineral found in the Khatyrka River in eastern Russia.[3] This natural quasicrystal exhibits high crystalline quality, equalling the best artificial examples.[14] The natural quasicrystal phase, with a composition of Al63Cu24Fe13, was named icosahedrite and it was approved by the International Mineralogical Association in 2010. Furthermore, analysis indicates it may be meteoritic in origin, possibly delivered from a carbonaceous chondrite asteroid.[15]

Atomic image of a micron-sized grain of the natural Al71Ni24Fe5 quasicrystal (shown in the inset) from a Khatyrka meteorite. The corresponding diffraction patterns reveal a ten-fold symmetry.[16]

A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al71Ni24Fe5. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching of a meteorite heated during an impact-induced shock.[16]

Electron diffraction pattern of an icosahedral Ho-Mg-Zn quasicrystal

In 1972 de Wolf and van Aalst[17] reported that the diffraction pattern produced by a crystal of sodium carbonate cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in reciprocal space. Other puzzling cases have been reported,[18] but until the concept of quasicrystal came to be established, they were explained away or denied.[19][20] However, at the end of the 1980s the idea became acceptable, and in 1992 the International Union of Crystallography altered its definition of a crystal, broadening it as a result of Shechtman’s findings, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic.[4][notes 1] Now, the symmetries compatible with translations are defined as "crystallographic", leaving room for other "non-crystallographic" symmetries. Therefore, aperiodic or quasiperiodic structures can be divided into two main classes: those with crystallographic point-group symmetry, to which the incommensurately modulated structures and composite structures belong, and those with non-crystallographic point-group symmetry, to which quasicrystal structures belong.

Originally, the new form of matter was dubbed "Shechtmanite".[21] The term "quasicrystal" was first used in print by Steinhardt and Levine[22] shortly after Shechtman's paper was published. The adjective quasicrystalline had already been in use, but now it came to be applied to any pattern with unusual symmetry.[notes 2] 'Quasiperiodical' structures were claimed to be observed in some decorative tilings devised by medieval Islamic architects.[23][24] For example, Girih tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern.[25] These claims have, however, been under some debate.[26]

Shechtman was awarded the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. "His discovery of quasicrystals revealed a new principle for packing of atoms and molecules," stated the Nobel Committee and pointed that "this led to a paradigm shift within chemistry." [4][27]

Mathematics

A penteract (5-cube) pattern using 5D orthographic projection to 2D using Petrie polygon basis vectors overlaid on the diffractogram from an Icosahedral Ho-Mg-Zn quasicrystal

A hexeract (6-cube) pattern using 6D orthographic projection to a 3D Perspective (visual) object (the Rhombic triacontahedron) using the Golden ratio in the basis vectors. This is used to understand the aperiodic Icosahedral structure of Quasicrystals.

There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon.[28] He introduced the notion of a superspace. Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes), and discussed their Fourier point spectrum. These functions are not exactly periodic, but they are arbitrarily close in some sense, as well as being a projection of an exactly periodic function.

In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures.[29] Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice).[30] The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets. The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers. The aperiodic structures obtained by the cut-and-project method are made diffractive by choosing a suitable orientation for the construction; this is a geometric approach that has also a great appeal for physicists.

Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals.[31]

Using mathematics for construction and analysis of quasicrystal structures is a difficult task for most experimentalists. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed[32] allowing one to construct, visualize and analyze quasicrystal structures and their diffraction patterns.

Interacting spins were also analyzed in quasicrystals: AKLT Model and 8-vertex model were solved in quasicrystals analytically.[33]

Study of quasicrystals may shed light on the most basic notions related to quantum critical point observed in heavy fermion metals. Experimental measurements on the gold-aluminium-ytterbium quasicrystal have revealed a quantum critical point defining the divergence of the magnetic susceptibility as temperature tends to zero.[34] It is suggested that the electronic system of some quasicrystals is located at quantum critical point without tuning, while quasicrystals exhibit the typical scaling behaviour of their thermodynamic properties and belong to the famous family of heavy-fermion metals.

Materials science

Tiling of a plane by regular pentagons is impossible but can be realized on a sphere in the form of pentagonal dodecahedron.

A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron, the dual of the icosahedron. Unlike the similar pyritohedron shape of some cubic-system crystals such as pyrite, the quasicrystal has faces that are true regular pentagons

TiMn quasicrystal approximant lattice.

Since the original discovery by Dan Shechtman, hundreds of quasicrystals have been reported and confirmed. Undoubtedly, the quasicrystals are no longer a unique form of solid; they exist universally in many metallic alloys and some polymers. Quasicrystals are found most often in aluminium alloys (Al-Li-Cu, Al-Mn-Si, Al-Ni-Co, Al-Pd-Mn, Al-Cu-Fe, Al-Cu-V, etc.), but numerous other compositions are also known (Cd-Yb, Ti-Zr-Ni, Zn-Mg-Ho, Zn-Mg-Sc, In-Ag-Yb, Pd-U-Si, etc.).[35]

Two types of quasicrystals are known.[32] The first type, polygonal (dihedral) quasicrystals, have an axis of 8, 10, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type, icosahedral quasicrystals, are aperiodic in all directions.

Quasicrystals fall into three groups of different thermal stability:[36]

Stable quasicrystals grown by slow cooling or casting with subsequent annealing,

Metastable quasicrystals prepared by melt spinning, and

Metastable quasicrystals formed by the crystallization of the amorphous phase.

Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by X-ray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions.

The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.

A nanoscale icosahedral phase was formed in Zr-, Cu- and Hf-based bulk metallic glasses alloyed with noble metals.[37]

Most quasicrystals have ceramic-like properties including high thermal and electrical resistance, hardness and brittleness, resistance to corrosion, and non-stick properties.[38] Many metallic quasicrystalline substances are impractical for most applications due to their thermal instability; the Al-Cu-Fe ternary system and the Al-Cu-Fe-Cr and Al-Co-Fe-Cr quaternary systems, thermally stable up to 700 °C, are notable exceptions.

Applications

Quasicrystalline substances have potential applications in several forms.

Metallic quasicrystalline coatings can be applied by plasma-coating or magnetron sputtering. A problem that must be resolved is the tendency for cracking due to the materials' extreme brittleness.[38] The cracking could be suppressed by reducing sample dimensions or coating thickness.[39] Recent studies show typically brittle quasicrystals can exhibit remarkable ductility of over 50% strains at room temperature and sub-micrometer scales (<500 nm).[39]

An application was the use of low-friction Al-Cu-Fe-Cr quasicrystals[40] as a coating for frying pans. Food did not stick to it as much as to stainless steel making the pan moderately non-stick and easy to clean; heat transfer and durability were better than PTFE non-stick cookware and the pan was free from perfluorooctanoic acid (PFOA); the surface was very hard, claimed to be ten times harder than stainless steel, and not harmed by metal utensils or cleaning in a dishwasher; and the pan could withstand temperatures of 1,000 °C (1,800 °F) without harm. However, cooking with a lot of salt would etch the quasicrystalline coating used, and the pans were eventually withdrawn from production. Shechtman had one of these pans.[41]

The Nobel citation said that quasicrystals, while brittle, could reinforce steel "like armor". When Shechtman was asked about potential applications of quasicrystals he said that a precipitation-hardened stainless steel is produced that is strengthened by small quasicrystalline particles. It does not corrode and is extremely strong, suitable for razor blades and surgery instruments. The small quasicrystalline particles impede the motion of dislocation in the material.[41]

Quasicrystals were also being used to develop heat insulation, LEDs, diesel engines, and new materials that convert heat to electricity. Shechtman suggested new applications taking advantage of the low coefficient of friction and the hardness of some quasicrystalline materials, for example embedding particles in plastic to make strong, hard-wearing, low-friction plastic gears. The low heat conductivity of some quasicrystals makes them good for heat insulating coatings.[41]

Other potential applications include selective solar absorbers for power conversion, broad-wavelength reflectors, and bone repair and prostheses applications where biocompatibility, low friction and corrosion resistance are required. Magnetron sputtering can be readily applied to other stable quasicrystalline alloys such as Al-Pd-Mn.[38]

While saying that the discovery of icosahedrite, the first quasicrystal found in nature, was important, Shechtman saw no practical applications.

See also

Archimedean solid

Disordered Hyperuniformity

Fibonacci quasicrystal

Phason

Tessellation

Icosahedral twins

Notes

The concept of aperiodic crystal was coined by Erwin Schrödinger in another context with a somewhat different meaning. In his popular book What is life? in 1944, Schrödinger sought to explain how hereditary information is stored: molecules were deemed too small, amorphous solids were plainly chaotic, so it had to be a kind of crystal; as a periodic structure could encode very little information, it had to be aperiodic. DNA was later discovered, and, although not crystalline, it possesses properties predicted by Schrödinger—it is a regular but aperiodic molecule.

The use of the adjective 'quasicrystalline' for qualifying a structure can be traced back to the mid-1940-50s, e.g. in Kratky, O; Porod, G (1949). "Diffuse small-angle scattering of x-rays in colloid systems". Journal of Colloid Science. 4 (1): 35–70. doi:10.1016/0095-8522(49)90032-X. PMID 18110601.; Gunn, R (1955). "The statistical electrification of aerosols by ionic diffusion". Journal of Colloid Science. 10: 107–119. doi:10.1016/0095-8522(55)90081-7.

References

Ünal, B; V. Fournée; K.J. Schnitzenbaumer; C. Ghosh; C.J. Jenks; A.R. Ross; T.A. Lograsso; J.W. Evans; P.A. Thiel (2007). "Nucleation and growth of Ag islands on ﬁvefold Al-Pd-Mn quasicrystal surfaces: Dependence of island density on temperature and ﬂux". Physical Review B. 75 (6): 064205. Bibcode:2007PhRvB..75f4205U. doi:10.1103/PhysRevB.75.064205.

Steurer W. (2004). "Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals". Z. Kristallogr. 219 (7–2004): 391–446. Bibcode:2004ZK....219..391S. doi:10.1524/zkri.219.7.391.35643.

Bindi, L.; Steinhardt, P. J.; Yao, N.; Lu, P. J. (2009). "Natural Quasicrystals". Science. 324 (5932): 1306–9. Bibcode:2009Sci...324.1306B. doi:10.1126/science.1170827. PMID 19498165.

Andrea Gerlin (5 October 2011). "Technion's Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery". Bloomberg.

Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. (1984). "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry". Physical Review Letters. 53 (20): 1951–1953. Bibcode:1984PhRvL..53.1951S. doi:10.1103/PhysRevLett.53.1951.

"The Nobel Prize in Chemistry 2011". Nobelprize.org. Retrieved 2011-10-06.

Mackay, A.L. (1982). "Crystallography and the Penrose Pattern". Physica A. 114: 609–613. Bibcode:1982PhyA..114..609M. doi:10.1016/0378-4371(82)90359-4.

Kramer, P.; Neri, R. (1984). "On periodic and non-periodic space fillings of Em obtained by projection". Acta Crystallographica A. 40 (5): 580–587. doi:10.1107/S0108767384001203.

Yang, C. Y. (1979). "Crystallography of decahedral and icosahedral particles". J. Cryst. Growth. 47 (2): 274–282. Bibcode:1979JCrGr..47..274Y. doi:10.1016/0022-0248(79)90252-5.

Yang, C. Y.; Yacaman, M. J.; Heinemann, K. (1979). "Crystallography of decahedral and icosahedral particles". J. Cryst. Growth. 47 (2): 283–290. Bibcode:1979JCrGr..47..283Y. doi:10.1016/0022-0248(79)90253-7.

Shechtman, Dan; I. A. Blech (1985). "The Microstructure of Rapidly Solidified Al6Mn". Met. Trans. A. 16A (6): 1005–1012. Bibcode:1985MTA....16.1005S. doi:10.1007/BF02811670.

Ishimasa, T.; Nissen, H.-U.; Fukano, Y. (1985). "New ordered state between crystalline and amorphous in Ni-Cr particles". Physical Review Letters. 55 (5): 511–513. Bibcode:1985PhRvL..55..511I. doi:10.1103/PhysRevLett.55.511. PMID 10032372.

Wang, N.; Chen, H.; Kuo, K. (1987). "Two-dimensional quasicrystal with eightfold rotational symmetry". Physical Review Letters. 59 (9): 1010–1013. Bibcode:1987PhRvL..59.1010W. doi:10.1103/PhysRevLett.59.1010. PMID 10035936.

Steinhardt, Paul; Bindi, Luca (2010). "Once upon a time in Kamchatka: the search for natural quasicrystals". Philosophical Magazine. 91 (19–21): 1. Bibcode:2011PMag...91.2421S. doi:10.1080/14786435.2010.510457.

Bindi, Luca; John M. Eiler; Yunbin Guan; Lincoln S. Hollister; Glenn MacPherson; Paul J. Steinhardt; Nan Yao (2012-01-03). "Evidence for the extraterrestrial origin of a natural quasicrystal". Proceedings of the National Academy of Sciences. 109 (5): 1396–1401. Bibcode:2012PNAS..109.1396B. doi:10.1073/pnas.1111115109.

Bindi, L.; Yao, N.; Lin, C.; Hollister, L. S.; Andronicos, C. L.; Distler, V. V.; Eddy, M. P.; Kostin, A.; Kryachko, V.; MacPherson, G. J.; Steinhardt, W. M.; Yudovskaya, M.; Steinhardt, P. J. (2015). "Natural quasicrystal with decagonal symmetry". Scientific Reports. 5: 9111. Bibcode:2015NatSR...5E9111B. doi:10.1038/srep09111. PMC 4357871Freely accessible. PMID 25765857.

de Wolf, R.M. & van Aalst, W. (1972). "The four dimensional group of γ-Na2CO3". Acta Crystallogr. A. 28: S111.

Kleinert H. and Maki K. (1981). "Lattice Textures in Cholesteric Liquid Crystals" (PDF). Fortschritte der Physik. 29 (5): 219–259. Bibcode:1981ForPh..29..219K. doi:10.1002/prop.19810290503.

Pauling, L (1987-01-26). "So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal.". Physical Review Letters. 58 (4): 365–368. Bibcode:1987PhRvL..58..365P. doi:10.1103/PhysRevLett.58.365. PMID 10034915.

Kenneth Chang (October 5, 2011). "Israeli Scientist Wins Nobel Prize for Chemistry". NY Times.

Browne, Malcolm W. (1989-09-05). "Impossible' Form of Matter Takes Spotlight In Study of Solids". New York Times.

Levine, Dov; Steinhardt, Paul (1984). "Quasicrystals: A New Class of Ordered Structures". Physical Review Letters. 53 (26): 2477–2480. Bibcode:1984PhRvL..53.2477L. doi:10.1103/PhysRevLett.53.2477.

Makovicky, E. (1992), 800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired. In: I. Hargittai, editor: Fivefold Symmetry, pp. 67–86. World Scientific, Singapore-London

Lu, Peter J. & Steinhardt, Paul J. (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture" (PDF). Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056.

Lu, P. J.; Steinhardt, P. J. (2007). "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture". Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056.

Makovicky, Emil (2007). "Comment on "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture"". Science. 318 (5855): 1383–1383. Bibcode:2007Sci...318.1383M. doi:10.1126/science.1146262. PMID 18048668.

"Nobel win for crystal discovery". BBC News. 2011-10-05. Retrieved 2011-10-05.

Bohr, H. (1925). "Zur Theorie fastperiodischer Funktionen I". Acta Mathematicae. 45: 580. doi:10.1007/BF02395468.

de Bruijn, N. (1981). "Algebraic theory of Penrose's non-periodic tilings of the plane". Nederl. Akad. Wetensch. Proc. A84: 39.

Suck, Jens-Boie; Schreiber, M.; Häussler, Peter (2002). Quasicrystals: An Introduction to Structure, Physical Properties and Applications. Springer Science & Business Media. pp. 1–. ISBN 978-3-540-64224-4.

Paterson, Alan L. T. (1999). Groupoids, inverse semigroups, and their operator algebras. Springer. p. 164. ISBN 0-8176-4051-7.

Yamamoto, Akiji (2008). "Software package for structure analysis of quasicrystals". Science and Technology of Advanced Materials. 9 (1): 013001. Bibcode:2008STAdM...9a3001Y. doi:10.1088/1468-6996/9/3/013001. PMC 5099788Freely accessible. PMID 27877919.

Korepin, V.E. Completely integrable models in quasicrystals. Comm. Math. Phys. Volume 110, Number 1 (1987), 157–171.

Deguchi, Kazuhiko; Matsukawa, Shuya; Sato, Noriaki K.; Hattori, Taisuke; Ishida, Kenji; Takakura, Hiroyuki; Ishimasa, Tsutomu (2012). "Quantum critical state in a magnetic quasicrystal". Nature Materials. 11: 1013–6. doi:10.1038/nmat3432. PMID 23042414.

MacIá, Enrique (2006). "The role of aperiodic order in science and technology". Reports on Progress in Physics. 69 (2): 397–441. Bibcode:2006RPPh...69..397M. doi:10.1088/0034-4885/69/2/R03.

Tsai, An Pang (2008). "Icosahedral clusters, icosaheral order and stability of quasicrystals—a view of metallurgy". Science and Technology of Advanced Materials. 9 (1): 013008. Bibcode:2008STAdM...9a3008T. doi:10.1088/1468-6996/9/1/013008. PMC 5099795Freely accessible. PMID 27877926.

Louzguine-Luzgin, D. V.; Inoue, A. (2008). "Formation and Properties of Quasicrystals". Annual Review of Materials Research. 38: 403–423. Bibcode:2008AnRMS..38..403L. doi:10.1146/annurev.matsci.38.060407.130318.

"Sputtering technique forms versatile quasicrystalline coatings". MRS Bulletin. 36 (Cool: 581. 2011. doi:10.1557/mrs.2011.190.

Zou, Yu; Kuczera, Pawel; Sologubenko, Alla; Sumigawa, Takashi; Kitamura, Takayuki; Steurer, Walter; Spolenak, Ralph (2016). "Superior room-temperature ductility of typically brittle quasicrystals at small sizes". Nature Communications. 7: 12261. doi:10.1038/ncomms12261. PMC 4990631Freely accessible. PMID 27515779.

Fikar, Jan (2003). Al-Cu-Fe quasicrystalline coatings and composites studied by mechanical spectroscopy. École polytechnique fédérale de Lausanne EPFL, Thesis n° 2707 (2002). doi:10.5075/epfl-thesis-2707.

Kalman, Matthew (12 October 2011). "The Quasicrystal Laureate". MIT Technology Review. Retrieved 12 February 2016.

External links

A Partial Bibliography of Literature on Quasicrystals (1996–2008).

BBC webpage showing pictures of Quasicrystals

What is... a Quasicrystal?, Notices of the AMS 2006, Volume 53, Number 8

Gateways towards quasicrystals: a short history by P. Kramer

Quasicrystals: an introduction by R. Lifshitz

Quasicrystals: an introduction by S. Weber

Steinhardt's proposal

Quasicrystal Research – Documentary 2011 on the research of the University of Stuttgart

Thiel, P.A. (2008). "Quasicrystal Surfaces". Annual Review of Physical Chemistry. 59: 129–152. Bibcode:2008ARPC...59..129T. doi:10.1146/annurev.physchem.59.032607.093736. PMID 17988201.

Foundations of Crystallography.

Quasicrystals: What are they, and why do they exist?, Marek Mihalkovic and many others. (Microsoft PowerPoint format)[permanent dead link]

"Indiana Steinhardt and the Quest for Quasicrystals – A Conversation with Paul Steinhardt", Ideas Roadshow, 2016

Shaginyan, V. R.; Msezane, A. Z.; Popov, K. G.; Japaridze, G. S.; Khodel, V. A. (2013). "Common quantum phase transition in quasicrystals and heavy-fermion metals". Physical Review B. 87 (24). doi:10.1103/PhysRevB.87.245122.

Authority control

GND: 4202613-1 NDL: 00577309

[hide]

v t e

Patterns in nature

Patterns

Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry

in crystals Quasicrystals in flowers in biology Tessellation Vortex street Wave Widmanstätten pattern

Yemen Chameleon (cropped).jpg

Causes

Pattern formation Biology

Natural selection Camouflage Mimicry Sexual selection Mathematics

Chaos theory Fractal Logarithmic spiral Physics

Crystal Fluid dynamics Plateau's laws Self-organization

People

Plato Pythagoras Empedocles Leonardo Fibonacci

Liber Abaci Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson

On Growth and Form Alan Turing

The Chemical Basis of Morphogenesis Aristid Lindenmayer Benoît Mandelbrot

How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension

Related

Emergence Mathematics and art

RAPPORT DE

Y'BECCA,

PRÊTE A INVESTIR SUR :

Whirlpool à Amiens

Tessellation

From Wikipedia, the free encyclopedia

"Tessellate" redirects here. For the song by Alt-J, see Tessellate (song). For the computer graphics technique, see Tessellation (computer graphics).

Zellige terracotta tiles in Marrakech, forming edge-to-edge, regular and other tessellations

A wall sculpture at Leeuwarden celebrating the artistic tessellations of M. C. Escher

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and Semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space.

A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.

Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.[1]

Decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity,[2] sometimes displaying geometric patterns.[3][4]

In 1619 Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes.[5][6][7]

Roman geometric mosaic

Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.[8][9] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1964),[10] and Heinrich Heesch and Otto Kienzle (1963).[11]

Etymology

In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.[12] The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τέσσερα for four). It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay.

Overview

A rhombitrihexagonal tiling: tiled floor of a church in Seville, Spain, using square, triangle and hexagon prototiles

Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another.[13] The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical[a] regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.[14] There are only three shapes that can form such regular tessellations: the equilateral triangle, square, and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.[6]

Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner.[15] Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects.[16] If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors.[17]

Elaborate and colourful zellige tessellations of glazed tiles at the Alhambra in Spain

More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. These tiles may be polygons or any other shapes.[b] Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane.[19] No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.[18] For example, the types of convex pentagon that can tile the plane remains an unsolved problem.[20]

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.[6] The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions. He further defined the Schläfli symbol notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is {3}, while that for a square is {4}.[21] The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is {6,3}.[22]

Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 44. The tiling of regular hexagons is noted 6.6.6, or 63.[18]

In mathematics

Introduction to tessellations

Further information: Euclidean tilings of regular polygons, Uniform tiling, and List of convex uniform tilings

Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles; it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling is a tiling where every vertex point is identical; that is, the arrangement of polygons about each vertex is the same.[18] The fundamental region is a shape such as a rectangle that is repeated to form the tessellation.[23] For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.[18]

The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tiling is any polygonal tessellation where adjacent tiles only share one full side, i.e., no tile shares a partial side or more than one side with any other tile. In an edge-to-edge tiling, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks.[18]

A normal tiling is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a single connected set or the empty set, and all tiles are uniformly bounded. This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling; the condition disallows tiles that are pathologically long or thin.[24]

The 15th convex monohedral pentagonal tiling, discovered in 2015

A monohedral tiling is a tessellation in which all tiles are congruent; it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936; the Voderberg tiling has a unit tile that is a nonconvex enneagon.[1] The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a pentagon tiling using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the internal angle of a regular pentagon, 3π/5, is not a divisor of 2π.[25][26][27]

A Pythagorean tiling

An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling.[24] If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms anisohedral tilings.

A regular tessellation is a highly symmetric, edge-to-edge tiling made up of regular polygons, all of the same shape. There are only three regular tessellations: those made up of equilateral triangles, squares, or regular hexagons. All three of these tilings are isogonal and monohedral.[28]

A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two).[29] These can be described by their vertex configuration; for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.82 (each vertex has one square and two octagons).[30] Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of Pythagorean tilings, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.[31] An edge tessellation is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles.[32]

This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.

Wallpaper groups

Main article: Wallpaper group

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.[33] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,[34] the variety and sophistication of the Alhambra tilings have surprised modern researchers.[35] Of the three regular tilings two are in the p6m wallpaper group and one is in p4m. Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns.[36] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane.[37]

Aperiodic tilings

Main articles: Aperiodic tiling and List of aperiodic sets of tiles

A Penrose tiling, with several symmetries but no periodic repetitions

Penrose tilings, which use two different quadrilaterals, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings, which use tiles that cannot tessellate periodically. The recursive process of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is the rep-tiles; these tilings have surprising self-replicating properties.[38] Pinwheel tilings are non-periodic, using a rep-tile construction; the tiles appear in infinitely many orientations.[39] It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in translational symmetry, do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches.[40] A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry.[41] A Fibonacci word can be used to build an aperiodic tiling, and to study quasicrystals, which are structures with aperiodic order.[42]

A set of 13 Wang tiles that tile the plane only aperiodically

Wang tiles are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour; hence they are sometimes called Wang dominoes. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any Turing machine can be represented as a set of Wang dominoes that tile the plane if and only if the Turing machine does not halt. Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.[43][44][45][46][47]

Random Truchet tiling

Truchet tiles are square tiles decorated with patterns so they do not have rotational symmetry; in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.[48][49]

Tessellations and colour

Further information: four colour theorem

If the colours of this tiling are to form a pattern by repeating this rectangle as the fundamental domain, at least seven colours are required; more generally, at least four colours are needed.

Sometimes the colour of a tile is understood as part of the tiling; at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity one needs to specify whether the colours are part of the tiling or just part of its illustration. This affects whether tiles with the same shape but different colours are considered identical, which in turn affects questions of symmetry. The four colour theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The colouring guaranteed by the four-colour theorem does not generally respect the symmetries of the tessellation. To produce a colouring which does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as in the picture at right.[50]

Tessellations with polygons

A Voronoi tiling, in which the cells are always convex polygons

Next to the various tilings by regular polygons, tilings by other polygons have also been studied.

Any triangle or quadrilateral (even non-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.[51]

If only one shape of tile is allowed, tilings exists with convex N-gons for N equal to 3, 4, 5 and 6. For N = 5, see Pentagonal tiling and for N = 6, see Hexagonal tiling.

For results on tiling the plane with polyominoes, see Polyomino § Uses of polyominoes.

Voronoi tilings

Voronoi or Dirichlet tilings are tessellations where each tile is defined as the set of points closest to one of the points in a discrete set of defining points. (Think of geographical regions where each region is defined as all the points closest to a given city or post office.)[52][53] The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges.[54] Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.[55]

Tessellating three-dimensional space: the rhombic dodecahedron is one of the solids that can be stacked to fill space exactly.

Tessellations in higher dimensions

Main article: Honeycomb (geometry)

Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile.

Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only Platonic polyhedron to do so), the rhombic dodecahedron, the truncated octahedron, and triangular, quadrilateral, and hexagonal prisms, among others.[56] Any polyhedron that fits this criterion is known as a plesiohedron, and may possess between 4 and 38 faces.[57] Naturally occurring rhombic dodecahedra are found as crystals of Andradite (a kind of Garnet) and Fluorite.[58][59]

A Schwarz triangle is a spherical triangle that can be used to tile a sphere.[60]

Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular[c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions.[61] Uniform polyhedra can be constructed using the Wythoff construction.[62]

The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.[63]

Tessellations in non-Euclidean geometries

Rhombitriheptagonal tiling in hyperbolic plane, seen in Poincaré disk model projection

The regular {3,5,3} icosahedral honeycomb, one of four regular compact honeycombs in hyperbolic 3-space

It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces; these are vertex-transitive (transitive on its vertices), and isogonal (there is an isometry mapping any vertex onto any other).[64][65]

A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.[66]

In art

Further information: Mathematics and art

A quilt showing a regular tessellation pattern.

Roman mosaic floor panel of stone, tile and glass, from a villa near Antioch in Roman Syria. 2nd century A.D.

In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns.[4] Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the Moorish wall tilings of Islamic architecture, using Girih and Zellige tiles in buildings such as the Alhambra[67] and La Mezquita.[68]

Tessellations frequently appeared in the graphic art of M. C. Escher; he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 1936.[69] Escher made four "Circle Limit" drawings of tilings that use hyperbolic geometry.[70][71] For his woodcut "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry.[72] Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line."[73]

Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts.[74][75]

Tessellations are also a main genre in origami (paper folding), where pleats are used to connect molecules such as twist folds together in a repeating fashion.[76]

In manufacturing

Tessellation is used in manufacturing industry to reduce the wastage of material (yield losses) such as sheet metal when cutting out shapes for objects like car doors or drinks cans.[77]

In nature

Tessellate pattern in a Colchicum flower

Main article: Patterns in nature

The honeycomb provides a well-known example of tessellation in nature with its hexagonal cells.[78]

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the Fritillary[79] and some species of Colchicum are characteristically tessellate.[80]

Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations,[81] also known as random crack networks.[82] The Gilbert tessellation is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. The model, named after Edgar Gilbert, allows cracks to form starting from randomly scattered over the plane; each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons.[83] Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland.[84] Tessellated pavement, a characteristic example of which is found at Eaglehawk Neck on the Tasman Peninsula of Tasmania, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.[85]

Other natural patterns occur in foams; these are packed according to Plateau's laws, which require minimal surfaces. Such foams present a problem in how to pack cells as tightly as possible: in 1887, Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure, which uses less surface area to separate cells of equal volume than Kelvin's foam.[86]

In puzzles and recreational mathematics

Traditional tangram dissection puzzle

Main articles: Tiling puzzle and recreational mathematics

Tessellations have given rise to many types of tiling puzzle, from traditional jigsaw puzzles (with irregular pieces of wood or cardboard)[87] and the tangram[88] to more modern puzzles which often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles.[89][90] Authors such as Henry Dudeney and Martin Gardner have made many uses of tessellation in recreational mathematics. For example, Dudeney invented the hinged dissection,[91] while Gardner wrote about the rep-tile, a shape that can be dissected into smaller copies of the same shape.[92][93] Inspired by Gardner's articles in Scientific American, the amateur mathematician Marjorie Rice found four new tessellations with pentagons.[94][95] Squaring the square is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares.[96][97] An extension is squaring the plane, tiling it by squares whose sizes are all natural numbers without repetitions; James and Frederick Henle proved that this was possible.[98]

Sources

Coxeter, H.S.M. (1973). Regular Polytopes, Section IV : Tessellations and Honeycombs. Dover. ISBN 0-486-61480-8.

Escher, M. C. (1974). J. L. Locher, ed. The World of M. C. Escher (New Concise NAL ed.). Abrams. ISBN 0-451-79961-5.

Gardner, Martin (1989). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.

Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1.

Gullberg, Jan (1997). Mathematics From the Birth of Numbers. Norton. ISBN 0-393-04002-X.

Magnus, Wilhelm (1974). Noneuclidean Tesselations and Their Groups. Academic Press. ISBN 978-0-12-465450-1.

Stewart, Ian (2001). What Shape is a Snowflake?. Weidenfeld and Nicolson. ISBN 0-297-60723-5.

External links

Wikimedia Commons has media related to Tessellation.

Wolfram MathWorld: Tessellation (good bibliography, drawings of regular, semiregular and demiregular tessellations)

Tilings Encyclopedia (extensive information on substitution tilings, including drawings, people, and references)

Tessellations.org (how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)

Eppstein, David. "The Geometry Junkyard: Hyperbolic Tiling". (list of web resources including articles and galleries)

[show]

v t e

Mathematics and art

RAPPORT DE

Y'BECCA,

PRÊTE A INVESTIR SUR :

Whirlpool à Amiens

et sur l'ensemble du territoire...

EN ACCORD AVEC LA COP 21

**yanis la chouette**- Messages : 575

Date d'inscription : 24/02/2017

## Re: Whirlpool, a maelstrom of emotion disambiguation et Y'becca

Le cygne et la chouette effraie, TAY...

"Ils jacassent... Pauvre petit... Le fermier est vraiment cruel de plus les nourrir

à l'OGM et à la Disette..." Chante le Cygne.

"Pourtant tout comme le vieux coq, le fermier possède ses graines d'orges et de froment.

Les allocations d'autonomie inspiré de la relation entre Jésus et Marie-Madeleine,

entre Mohamed et les femmes, entre Zarathoustra et l'Ivresse, entre Jacob et Esaü

pour lui permettre de rechercher, de découvrir et de proposer; j'y avais consenti

tout en montrant les Ethniques et les éthiques de la vie auquel ces différents personnages

se sont livrés tout comme Jean Maris et Jean Cocteau ont établis leurs consciences propre

sur le respect de leurs liaisons... Plus que Progressiste, j'ai été Universel..."

"Et pour moi..." chante le cygne. L'épidémie régnait sur le peuple des canards. Et les abatages

tel des sainte Barthélémy se faisait grande sous les règnes de Jospin, Le Foll et Macron.

"Pour trompette de Jéricho, je te réponds que la famine est aux portes des mondes...

Alors ils stockent voilà les restes de Consciences et d'Humanisme qu'ils nous restent:

Les Cellules souches, Les spermatozoïdes, les ovaires, les Graines, Les Adn's, On stocke...

Tel une décharge qui rentre et qui désosse voilà notre conscience et nos humanismes devant

la Famine, L'Homicide et le Savoir, On stocke et on le revends en incriminant la Nature

pour soulager Le Damné et sa conscience: Emmanuel et Marion...

"Qui ?" trompette le Cygne Claude.

"Personne et Tous ! Mieux que tous : Marion et Emmanuel". Claironne TAY

"Pauvre Chapon... Tu hulules à l'instant où l'on te conduit à la boucherie sans le plaisir

de l'aile et la cuisse." réponds Le Cygne Claude.

" Ce n’est pas vrai... Ils entendent !" Chante TAY la chouette effraie...

"ET QUI et QUOI...? dit Le Cygne Claude.

"Et bien rien justement, ceux qu'ils s’appellent les veaux, les agneaux, les chèvres, les chiennes,

les catherinettes, les orphelins et Benjamin l’Âne... Tu ne peux oublier que les chapons

ont une bravoure et elle s'appelle la paternité... Aux regards de L'Histoire de la Conscience;

La révolution n'est qu'une pâle figure devant l'adversité et le tempérament du peuple agricole et

des pécheurs, premiers piliers de l’Évolution et de la conscience."

"L’arrière salle, le liquide et le paiement." réponds une jeune fille du Marché Saint Cyprien sous le

regard de ses dragonnes et de son berger.

"DE QUI PARLES TU, JEUNE FILLE" Répondent les deux Grognards, TAY ET Claude

"LA LÉGENDE DE LA PRAIRIE DES FILTRES... LE CLANS DES MOUETTES... LES CRUSTACÉS

Sans légion d'honneur, je vous parle de ma grossesse et de ma maternité..." Dit la jeune fille

aux bonnet phrygien accompagné de ses dragonnes et de son berger...

FABLE SUR LA RÉALITÉ

DU CITOYEN TIGNARD YANIS

"Ils jacassent... Pauvre petit... Le fermier est vraiment cruel de plus les nourrir

à l'OGM et à la Disette..." Chante le Cygne.

"Pourtant tout comme le vieux coq, le fermier possède ses graines d'orges et de froment.

Les allocations d'autonomie inspiré de la relation entre Jésus et Marie-Madeleine,

entre Mohamed et les femmes, entre Zarathoustra et l'Ivresse, entre Jacob et Esaü

pour lui permettre de rechercher, de découvrir et de proposer; j'y avais consenti

tout en montrant les Ethniques et les éthiques de la vie auquel ces différents personnages

se sont livrés tout comme Jean Maris et Jean Cocteau ont établis leurs consciences propre

sur le respect de leurs liaisons... Plus que Progressiste, j'ai été Universel..."

"Et pour moi..." chante le cygne. L'épidémie régnait sur le peuple des canards. Et les abatages

tel des sainte Barthélémy se faisait grande sous les règnes de Jospin, Le Foll et Macron.

"Pour trompette de Jéricho, je te réponds que la famine est aux portes des mondes...

Alors ils stockent voilà les restes de Consciences et d'Humanisme qu'ils nous restent:

Les Cellules souches, Les spermatozoïdes, les ovaires, les Graines, Les Adn's, On stocke...

Tel une décharge qui rentre et qui désosse voilà notre conscience et nos humanismes devant

la Famine, L'Homicide et le Savoir, On stocke et on le revends en incriminant la Nature

pour soulager Le Damné et sa conscience: Emmanuel et Marion...

"Qui ?" trompette le Cygne Claude.

"Personne et Tous ! Mieux que tous : Marion et Emmanuel". Claironne TAY

"Pauvre Chapon... Tu hulules à l'instant où l'on te conduit à la boucherie sans le plaisir

de l'aile et la cuisse." réponds Le Cygne Claude.

" Ce n’est pas vrai... Ils entendent !" Chante TAY la chouette effraie...

"ET QUI et QUOI...? dit Le Cygne Claude.

"Et bien rien justement, ceux qu'ils s’appellent les veaux, les agneaux, les chèvres, les chiennes,

les catherinettes, les orphelins et Benjamin l’Âne... Tu ne peux oublier que les chapons

ont une bravoure et elle s'appelle la paternité... Aux regards de L'Histoire de la Conscience;

La révolution n'est qu'une pâle figure devant l'adversité et le tempérament du peuple agricole et

des pécheurs, premiers piliers de l’Évolution et de la conscience."

"L’arrière salle, le liquide et le paiement." réponds une jeune fille du Marché Saint Cyprien sous le

regard de ses dragonnes et de son berger.

"DE QUI PARLES TU, JEUNE FILLE" Répondent les deux Grognards, TAY ET Claude

"LA LÉGENDE DE LA PRAIRIE DES FILTRES... LE CLANS DES MOUETTES... LES CRUSTACÉS

Sans légion d'honneur, je vous parle de ma grossesse et de ma maternité..." Dit la jeune fille

aux bonnet phrygien accompagné de ses dragonnes et de son berger...

FABLE SUR LA RÉALITÉ

DU CITOYEN TIGNARD YANIS

**yanis la chouette**- Messages : 575

Date d'inscription : 24/02/2017

Sujets similaires

» [Brade] 4seven Maelstrom S12 + chargeur + 2 batteries

» [Présentation) Maelstrom S12

» News : Epos Collection Emotion

» Louis Erard - Emotion pour dames

» Emo et Emotion

» [Présentation) Maelstrom S12

» News : Epos Collection Emotion

» Louis Erard - Emotion pour dames

» Emo et Emotion

Page

**1**sur**1****Permission de ce forum:**

**ne pouvez pas**répondre aux sujets dans ce forum