text: tessellation 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 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 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 . history tessellations were used by the sumerians ( about 4000 bc ) in building wall decorations formed by patterns of clay tiles . decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity , sometimes displaying geometric patterns . 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 . some two hundred years later in 1891 , the russian crystallographer fyodorov 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 and belov ( 1964 ) , and heinrich heesch and otto ( 1963 ) . etymology in latin , is a small cubical piece of clay , stone or glass used to make mosaics . the word " " means " small square " ( from , square , which in turn is from the greek word for four ) . it corresponds with the everyday term tiling , which refers to applications of tessellations , often made of glazed clay . overview 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 bonded brickwork do not obey this rule . among those that do , a regular tessellation has both identical regular tiles 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 triangle , square , and regular hexagon . any one of these three shapes can be duplicated infinitely to fill a plane 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 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 . 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 . 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 . 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 . 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 types of convex pentagon that can tile the plane remains an unsolved problem . mathematically , tessellations can be extended to spaces other than the euclidean plane . the swiss geometer ludwig schläfli pioneered this by defining , which mathematicians nowadays call . 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 . for example , the schläfli symbol for an equilateral triangle is { 3 } , while that for a square is { 4 } . 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 } . 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 . in mathematics introduction to tessellations 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 . the fundamental region 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 four squares at every vertex . 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 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 . 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 in 1936 ; the tiling has a unit tile that is a . 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 , / 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 and forms 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 . 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 ) . 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 ) . many non - edge - to - edge tilings of the euclidean plane are possible , including the family of pythagorean tilings , tessellations that use two ( ) sizes of square , each square touching four squares of the other size . wallpaper groups with translational symmetry in two independent directions can be categorized by wallpaper groups , of which 17 exist . it has been claimed that all seventeen of these groups are represented in the alhambra palace in granada , spain . 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 wallpaper group and one is in . in 2d with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns . notation can be used to describe wallpaper groups of the euclidean plane . tilings 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 . pinwheel tilings are non - periodic , using a rep - tile construction ; 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 . 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 . a substitution rule , such as can be used to generate some penrose patterns using assemblies of tiles called rhombs , illustrates scaling symmetry . a fibonacci word can be used to build an aperiodic tiling , and to study quasicrystals , which are structures with aperiodic order . 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 . tiles are square tiles decorated with patterns so they do not have rotational symmetry ; in 1704 , sébastien used a square tile split into two triangles of contrasting colours . these can tile the plane either periodically or randomly . tessellations and colour 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 . tessellations with 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 . if only one shape of tile is allowed , tilings exists with convex n - 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 § 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 . ) 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 . voronoi tilings with randomly placed points can be used to construct random tilings of the plane . tessellations in higher dimensions 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 regular polyhedron to do so ) , the rhombic dodecahedron , and the truncated octahedron . naturally occurring rhombic dodecahedra are found as crystals of ( a kind of garnet ) and fluorite . a schwarz triangle is a spherical triangle that can be used to tile a sphere . 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 honeycomb , which has eight tetrahedra and six octahedra at each polyhedron vertex . however , there are many possible semiregular honeycombs in three dimensions . uniform polyhedra can be constructed using the construction . the schmitt - conway biprism is a convex polyhedron with the property of tiling space only aperiodically . tessellations in non - euclidean geometries it is possible to tessellate in non - euclidean geometries such as hyperbolic geometry . a uniform tiling in the hyperbolic plane ( which may be regular , 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 ) . 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 constructions , and represented by permutations of rings of the coxeter diagrams for each family . in art in architecture , tessellations have been used to create decorative motifs since ancient times . mosaic tilings often had geometric patterns . 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 and la mezquita . 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 . escher made four " circle limit " drawings of tilings that use hyperbolic geometry . for his woodcut " 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 . " 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 . 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 . 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 . in nature the honeycomb 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 fritillary and some species of are characteristically tessellate . many patterns in nature are formed by cracks in sheets of materials . these patterns can be described by gilbert tessellations , also known as random crack networks . the gilbert tessellation is a mathematical model for the formation of , 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 . 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 . pavement , a characteristic example of which is found at neck on the tasman peninsula of tasmania , is a rare sedimentary rock formation where the rock has fractured into rectangular blocks . 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 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 . in puzzles and recreational mathematics tessellations have given rise to many types of tiling puzzle , from traditional jigsaw puzzles ( with irregular pieces of wood or cardboard ) and the to more modern puzzles which often have a mathematical basis . for example , and polyominoes are figures of regular triangles and squares , often used in tiling puzzles . authors such as henry and martin gardner have made many uses of tessellation in recreational mathematics . for example , invented the hinged dissection , while gardner wrote about the rep - tile , a shape that can be dissected into smaller copies of the same shape . inspired by gardner 's articles in scientific american , the amateur mathematician marjorie rice found four new tessellations with pentagons . squaring the square 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 without repetitions ; james and frederick henle proved that this was possible . examples