# Tessellation

Ceramic 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 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.

Some special kinds of tessellations include regular, with tiles all of the same shape; semi-regular, with tiles of more than one shape; and aperiodic tilings, which use tiles that cannot form a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups.

In computer graphics, the term "tessellation" is used to describe the organization of information needed to render to give the appearance of the surfaces of realistic three-dimensional objects.

In the real world, a tessellation is a tiling made of physical 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 tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations 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.

## History

A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC) showing a tessellation pattern in the tile colors.

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

In 1619 Johannes Kepler made one of the first documented studies of tessellations when he wrote about regular and semiregular tessellation, which are coverings of a plane with regular polygons, in his Harmonices Mundi.2 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.34 Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1951); and Heinrich Heesch and Otto Kienzle (1963).

### Etymology

In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.5 The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word "τέσσερα" for "four"). It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay.

## Overview

A semi-regular tessellation: tiled floor of a church in Seville, Spain, using square, triangle and hexagon prototiles

Tessellation or tiling is the branch of mathematics that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps. There are only three "regular" tessellations using exactly one kind of identical regular polygons arranged edge-to-edge, but many other types of tessellations are possible, differing in the constraints that are chosen to apply. For example, there are nine types of tessellations made with more than one kind of regular polygon, but having the same arrangement of polygons at every corner. These were described by the Swiss geometer Ludwig Schläfli in the 1850s. Tessellations can also be made from other shapes such as rectangles, 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. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to form physical surfaces such as church floors.6

More formally, a tessellation or tiling is a partition of the Euclidean plane into 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.a Many tessellations are formed from a finite number of prototiles; all tiles in the tessellation are congruent to one of 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. Mathematicians have found no general rule for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.7 For example, the types of convex pentagons that can tile the plane remains an unsolved problem.

Mathematically, tessellations can be extended to spaces other than the Euclidean plane.8 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}.9 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}.10

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 66.7:59

## In mathematics

### Kinds of tessellations

The 3.4.6.4 semi-regular tessellation is made with three prototiles: a triangle, a square and a hexagon. Every vertex has a triangle, square, hexagon, square around it, in that order.

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. For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex.7

The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tessellation 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 tessellation, 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.7

A normal tiling is a tessellation for which (1) every tile is topologically equivalent to a disk, (2) the intersection of any two tiles is a single connected set or the empty set, and (3) all tiles are uniformly bounded.11:172 A uniformly bounded tile is one in which a finite circle can be circumscribed around the tile and a finite circle can be inscribed within the tile; the condition disallows tiles that are pathologically long or thin.

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, with the Voderberg tiling having a unit tile that is a nonconvex enneagon.1 The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, has a unit tile that is an irregular pentagon.1213

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.11:175 If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is call 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.14

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).15 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).16

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, though they have surprising self-replicating properties using the recursive process of substitution tiling.17

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.)1819 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.20

This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3m1.

### Wallpaper groups

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist.21 It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed,2223 the variety and sophistication of the Alhambra tilings have surprised modern researchers.24 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.25

### Tessellations and colour

If this parallelogram pattern is coloured before tiling it over a plane, seven colours are required to ensure each complete parallelogram has a consistent colour that is distinct from that of adjacent areas. (This tiling can be compared to the surface of a torus.) Colouring after tiling, only four colours are needed.

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. Note that the colouring guaranteed by the four-colour theorem will not in general respect the symmetries of the tessellation. To produce a colouring which does, as many as seven colours may be needed, as in the picture at right.26

### Tessellations with triangles and quadrilaterals

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 2-fold rotational centres at the midpoints of all sides, and translational symmetry whose basis vectors are the diagonal of the quadrilateral or, equivalently, one of these and the sum or difference of the two. 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.27

### Tessellations in higher dimensions

Tessellating three-dimensional space: the rhombic dodecahedron is one of the solids that can be stacked to fill space exactly.
Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile.

Tessellation can be extended to three or more dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only regular polyhedron to do so); the rhombic dodecahedron; and the truncated octahedron.28 Some crystals including Andradite (a kind of Garnet) and Fluorite can take the form of rhombic dodecahedra.2930

The Schmitt-Conway biprism is a convex polyhedron which has the property of tiling space only aperiodically. John Horton Conway discovered it in 1993.31

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 quasiregularb honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However there are many possible semiregular honeycombs in three dimensions.32

## In computer graphics

In computer graphics, tessellation has a variety of usages. It is used to manage datasets of polygons (sometimes called vertex sets) presenting objects in a scene and divide them into suitable structures for rendering. Especially for real-time rendering, data are tessellated into triangles, for example in DirectX 11 and OpenGL.3334

## In 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 the broad geometric border, two squares and two rhombuses meet at every vertex of the tessellation.

Mosaic tilings were used by the Romans, often with geometric patterns.35

Tessellations frequently appeared in the art of M. C. Escher, who was inspired by studying the Moorish use of symmetry in the Moorish tilings of the Alhambra and the Córdoba, Andalusia mosque of La Mezquita during a visit to Spain in 1936.36

In the context of quilting, tessellation refers to regular37 and semiregular38 of tessellation of either patch shapes or the overall design. Tessellation patterns have been used to design interlocking motifs of patch shapes.3940 The repeating motif is sometimes called a block design.37

## In nature

Tessellate pattern in a Colchicum flower

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.

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.

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

### Examples

 Types of tessellation Convex uniform honeycomb List of aperiodic sets of tiles List of uniform tilings Pinwheel tiling – non-periodic tilings using Conway Rep-tile - a type of substitution tiling Tilings of regular polygons Uniform coloring Uniform tessellation Mathematics Convex uniform honeycombs in hyperbolic space Coxeter groups – algebraic groups to find tessellations Girih tiles – set of 5 tiles used in Islamic architecture Triangulation (geometry) Uniform tiling Uniform tilings in hyperbolic plane Wang tiles Wythoff construction Related topics Jigsaw puzzle Mathematics and fiber arts Polyiamond and Polyomino — figures of regular triangles and squares, often in tiling puzzles Quilt block designs and quilt blocks Tiling puzzle Trianglepoint – needlepoint with polyiamonds (equilateral triangles)

## Footnotes

1. ^ The tiles are usually required to be topologically equivalent to a closed disk, which means bizarre shapes with holes, dangling line segments or infinite areas are excluded.7
2. ^ In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.

## References

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2. ^ Kepler, Johannes (1619). Harmonices Mundi (Harmony of the Worlds).
3. ^ Djidjev, Hristo; Potkonjak, Miodrag (2012). "Dynamic Coverage Problems in Sensor Networks". Los Alamos National Laboratory (USA). p. 2. Retrieved 6 April 2013.
4. ^ E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, volume 28, pages 245-291 (in Russian).
5. ^ tessellate, Merriam-Webster Online
6. ^ "Basilica di San Marco". Section dedicated to the tessellated floor. Basilica di San Marco, Venice, Italy. Retrieved 26 April 2013.
7. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman.
8. ^ Gullberg, 1997. p. 395
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10. ^ Weisstein, Eric W. (1999-2013). "Tessellation". Wolfram MathWorld. Retrieved 26 April 2013.
11. ^ a b Horne, Clare E. (2000). Geometric Symmetry in Patterns and Tilings. Woodhead Publishing. ISBN 9781855734920.
12. ^ Dutch, Steven (29 July 1999). "Some Special Radial and Spiral Tilings". University of Wisconsin. Retrieved 6 April 2013.
13. ^ Hirschhorn, M. D.; D. C. Hunt (1985). "Equilateral convex pentagons which tile the plane". Journal of Combinatorial Theory, Series A 39 (1): 1–18. doi:10.1016/0097-3165(85)90078-0. Retrieved 29 April 2013.
14. ^ MathWorld: Regular Tessellations
15. ^ Stewart, 2001. p. 75
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17. ^ Gardner, 1989.
18. ^ Franz Aurenhammer (1991). Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys, 23(3):345–405, 1991
19. ^ Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. 2nd edition. John Wiley, 2000. ISBN 0-471-98635-6
20. ^ Paul Louis George and Houman Borouchaki (1998). Delaunay Triangulation and Meshing: Application to Finite Elements. Paris: Hermes. pp. 34–35. ISBN 2-86601-692-0.
21. ^ Armstrong, M.A. (1988). Groups and Symmetry. New York: Springer-Verlag. ISBN 978-3-540-96675-3.
22. ^ Grünbaum, Branko (June/July 2006). Notices of the American Mathematical Society 53 (6): 670–673.
23. ^ Jaworski, J. "A mathematician’s guide to the Alhambra". Retrieved September 1, 2011.
24. ^ Lu, Peter J.; Steinhardt (23 February, 2007). Science 315: 1106.
25. ^ Weisstein, Eric W. ""Frieze Group."". MathWorld--A Wolfram Web Resource. Retrieved 29 April 2013.
26. ^ Hazewinkel, 2001.
27. ^ Jones, 1856.
28. ^ Weisstein, Eric W. (1999-2013). "Schmitt-Conway Biprism". Wolfram MathWorld. Retrieved 28 April 2013.
29. ^ "Rhodolite Garnet Gemstone Information". AJS Gems. Retrieved 28 April 2013.
30. ^ "The mineral Andradite". Amethyst Galleries. 1995-2013. Retrieved 28 April 2013.
31. ^ Weisstein, Eric W. (1999-2013). "Schmitt-Conway Biprism". Wolfram MathWorld. Retrieved 28 April 2013.
32. ^ Weisstein, Eric W. (1999-2013). "Schmitt-Conway Biprism". Wolfram MathWorld. Retrieved 28 April 2013.
33. ^ MSDN: Tessellation Overview
34. ^ The OpenGL® Graphics System: A Specification (Version 4.0 (Core Proﬁle) - March 11, 2010)
35. ^ Field, Robert (1988). Geometric Patterns from Roman Mosaics. Tarquin. ISBN 978-0-906-21263-9.
36. ^ Locher, J. L.; Escher, M. C. (1971). Locher, J. L., ed. The Work of M. C. Escher. Harry N. Abrams. p. 5.
37. ^ a b Beyer, Jinny. "Tessellations". Retrieved 28 April 2013.
38. ^ Swanson, Irena. "Quilting semi-regular tessellations". Retrieved 28 April 2013.
39. ^ Porter, Christine (2006). Tessellation Quilts: Sensational Designs From Interlocking Patterns. F+W Media. pp. 4–8. ISBN 9780715319413.
40. ^ Beyer, Jinny (1999). Designing tessellations: the secrets of interlocking patterns. Contemporary Books. pp. Ch. 7. ISBN 9780809228669.

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