From Wikipedia, the free encyclopedia
"Tessellate" redirects here. For the song by Alt-J, see . For the computer graphics technique, see .
tiles in , forming edge-to-edge, regular and other tessellations
A wall sculpture at
celebrating the artistic tessellations of
A tessellation of a flat surface is the tiling of a
using one or more , called tiles, with no overlaps and no gaps. In , tessellations can be generalized to higher dimensions and a variety of geometries.
A periodic tiling has a repeating pattern. Some special kinds include
tiles all of the same shape, and
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 . A tiling that lacks a repeating pattern is called "non-periodic". An
uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or
is also called a tessellation of space.
A real physical tessellation is a tiling made of materials such as
squares or hexagons. Such tilings may be decorative , or may have functions such as providing durable and water-resistant , floor or wall coverings. Historically, tessellations were used in
such as in the decorative tiling of the
palace. In the twentieth century, the work of
often made use of tessellations, both in ordinary
and in , for artistic effect. Tessellations are sometimes employed for decorative effect in . Tessellations form a class of , for example in the arrays of
found in .
A temple mosaic from the ancient Sumerian city of
IV ( BC), showing a tessellation pattern in coloured tiles
Tessellations were used by the
(about 4000 BC) in building wall decorations formed by patterns of clay tiles.
Decorative
tilings made of small squared blocks called
were widely employed in , sometimes displaying geometric patterns.
made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his ; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes.
geometric mosaic
Some two hundred years later in 1891, the Russian crystallographer
proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov (1964), and
and Otto Kienzle (1963).
In Latin, tessella is a small cubical piece of ,
used to make mosaics. The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τ?σσερα for four). It corresponds with the everyday term tiling, which refers to applications of tessellations, often made of
A : tiled floor of a church in , 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. The tessellations created by
do not obey this rule. Among those that do, a
has both identical
and identical regular corners or vertices, having the same angle between adjacent edges for every tile. There are only three shapes that can form such regular tessellations: the equilateral , , and regular . Any one of these three shapes can be duplicated infinitely to fill a
with no gaps.
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. Irregular tessellations can also be made from other shapes such as ,
and in fact almost any kind of geometric shape. The artist
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 decorate physical surfaces such as church floors.
Elaborate and colourful tessellations of glazed tiles at the
More formally, a tessellation or tiling is a
of the Euclidean plane by a
number of closed sets, called tiles, such that the tiles intersect only on their . These tiles may be polygons or any other shapes. Many tessellations are formed from a finite number of
in which all tiles in the tessellation are
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
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. 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. For example,
the plane remains an unsolved problem.
Mathematically, tessellations can be extended to spaces other than the Euclidean plane. The
pioneered this by defining polyschemes, which mathematicians nowadays call . These are the analogues to polygons and
in spaces with more dimensions. He further defined the
notation to make it easy to describe polytopes. For example, the Schl?fli symbol for an equilateral triangle is {3}, while that for a square is {4}. The Schl?fli 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 Schl?fli symbol is {6,3}.
Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the , 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.
Further information: ,
Mathematicians use some technical terms when discussing tilings. An
is the intersection between it is often a straight line. A
is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or
tiling is a tiling where every verte that is, the arrangement of
about each vertex is the same. The
is a shape such as a rectangle that is repeated to form the tessellation. For example, a regular tessellation of the plane with squares has a meeting of .
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.
A normal tiling is a tessellation for which every tile is
equivalent to a , the intersection of any two tiles is a single
or the , and all tiles are . This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles the condition disallows tiles that are pathologically long or thin.
The 15th convex monohedral , discovered in 2015
A monohedral tiling is a tessellation in
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
has a unit tile that is a nonconvex . The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a
using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the
of a regular pentagon, 3π/5, is not a divisor of 2π.
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. If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms .
is a highly , edge-to-edge tiling made up of , all of the same shape. There are only three regular tessellations: those made up of , , or regular . All three of these tilings are isogonal and monohedral.
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). These can be 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). Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of , tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size.
This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.
Main article:
Tilings with
in two independent directions can be categorized by , of which 17 exist. It has been claimed that all seventeen of these groups are represented in the
palace in , . Though this is disputed, the variety and sophistication of the Alhambra tilings have surprised modern researchers. 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 .
can be used to describe wallpaper groups of the Euclidean plane.
Main articles:
A , with several symmetries but no periodic repetitions.
, 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 , which use tiles that cannot tessellate periodically. The
is a method of generating aperiodic tilings. One class that can be generated these tilings have surprising
properties.
are non-periodic, using a rep- the tiles appear in infinitely many orientations. It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in , 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. A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry. A
can be used to build an aperiodic tiling, and to study , which are structures with aperiodic order.
A set of 13
that tile the plane only
are squares coloured on each edge, and placed so that abutting edges of adjacent tiles
hence they are sometimes called Wang . A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any
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
is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable.
are square tiles decorated with patterns
used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly.
Further information:
If the colours of this tiling are to form a pattern by repeating this rectangle as the , at least seven
more generally, at least
are needed.
Sometimes the colour of a tile is understood a 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
states that for every tessellation of a normal , 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.
A , in which the cells are always convex polygons
Next to the various , tilings by other polygons have also been studied.
Any triangle or
(even ) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary
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 . As
we have the quadrilateral. Equivalently, we can construct a
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.
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
and for N = 6, see .
For results on tiling the plane with , see .
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.) The Voronoi cell for each defining point is a convex polygon. The
is a tessellation that is the
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. Voronoi tilings with randomly placed points can be used to construct random tilings of the plane.
Tessellating three-dimensional space: the
is one of the solids that can be stacked to .
Main article:
Illustration of a Schmitt-Conway biprism, also called a Schmitt–Conway–Danzer tile.
Tessellation can be extended to three dimensions. Certain
can be stacked in a regular
to fill (or tile) three-dimensional space, including the
(the only regular polyhedron to do so), the , and the . Naturally occurring rhombic dodecahedra are found as
(a kind of ) and .
that can be used to tile a .
Tessellations in three or more dimensions are called . 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 honeycomb, which has eight
at each polyhedron vertex. However, there are many possible
in three dimensions. Uniform polyhedra can be constructed using the .
The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically.
in hyperbolic plane, seen in
projection
The regular , one of four regular compact honeycombs in
It is possible to tessellate in
geometries such as . A
(which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, these are
( on its ), and isogonal (there is an
mapping any vertex onto any other).
is a uniform tessellation of
. In 3-dimensional hyperbolic space there are nine
families of compact , generated as , and represented by
for each family.
Further information:
A quilt showing a regular tessellation pattern.
floor panel of stone, tile and glass, from a villa near
in Roman Syria. 2nd century A.D.
In architecture, tessellations have been used to create decorative motifs since ancient times.
tilings often had geometric patterns. Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the
wall tilings of , using
tiles in buildings such as the
Tessellations frequently appeared in he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited
in 1936. Escher made four "" drawings of tilings that use hyperbolic geometry. For his
"Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry. 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."
Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking
of patch shapes in .
Tessellations are also a main genre in
(paper folding), where pleats are used to connect molecules such as twist folds together in a repeating fashion.
Tessellation is used in
to reduce the wastage of material (yield losses) such as
when cutting out shapes for objects like
Tessellate pattern in a
Main article:
provides a well-known example of tessellation in nature with its hexagonal cells.
In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the
and some species of Colchicum are characteristically tessellate.
are formed by cracks in sheets of materials. These patterns can be described by , also known as random crack networks. The Gilbert tessellation is a mathematical model for the formation of , needle-like , and similar structures. The model, named after , allows cracks to form starting from randomly scat 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.
often display
as a result of
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
in Northern Ireland. , a characteristic example of which is found at
of , is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.
Other natura these are packed according to , which require . Such foams present a problem in how to pack cells as tightly as possible: in 1887,
proposed a packing using only one solid, the
with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the , which uses less surface area to separate cells of equal volume than Kelvin's foam.
Traditional
Main articles:
Tessellations have given rise to many types of , from traditional
(with irregular pieces of wood or cardboard) and the
to more modern puzzles which often have a mathematical basis. For example,
are figures of regular triangles and squares, often used in tiling puzzles. Authors such as
have made many uses of tessellation in . For example, Dudeney invented the , while Gardner wrote about the , a shape that can be
into smaller copies of the same shape. Inspired by Gardner's articles in , the amateur mathematician
found four new tessellations with pentagons.
is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares. An extension is squaring the plane, tiling it by squares whose sizes are all natural numbers James and Frederick Henle proved that this was possible.
,one of the three
of the plane.
of the plane
, dual to a semiregular tiling and one of 15 monohedral .
is a natural tessellated structure.
The , a spiral, monohedral tiling made of .
is a uniform tiling of the .
The mathematical term for identical shapes is "congruent" - in mathematics, "identical" means they are the same tile.
The tiles are usually required to be
(topologically equivalent) to a , which means bizarre shapes with holes, dangling line segments or infinite areas are excluded.
In this context, quasiregular means that the cells are regular (solids), and the vertex figures are semiregular.
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Wikimedia Commons has media related to .
(good bibliography, drawings of regular, semiregular and demiregular tessellations)
(extensive information on substitution tilings, including drawings, people, and references)
(how-to guides, Escher tessellation gallery, galleries of tessellations by other artists, lesson plans, history)
. . (list of web resources including articles and galleries)
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