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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.

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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.

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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)

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