In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker; because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings described in Tilings and Patterns; the Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings, most notably: They are nonperiodic, which means that they lack any translational symmetry. Their non-periodicity is implied by their hierarchical structure: the tilings are substitution tilings arising from substitution rules for growing larger and larger patches; this substitution structure implies that: Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, they are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction.
This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation." All of this infinite global structure is forced through local matching rules on a pair of tiles, among the simplest aperiodic sets of tiles found, Ammann's A5 set. Various methods to describe the tilings have been proposed: matching rules, substitutions and project schemes and coverings. In 1987 Wang and Kuo announced the discovery of a quasicrystal with octagonal symmetry. Amman's A and B tiles in his pair A5 a 45-135-degree rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic and quasiperiodic structures of each of the infinite number of individual Ammann-Beenker tilings. An alternate set of tiles discovered by Ammann, labelled "Ammann 4" in Grünbaum and Shephard, consists of two nonconvex right-angle-edged pieces. One consists of two squares overlapping on a smaller square, while the other consists of a large square attached to a smaller square.
The diagrams below show a portion of the tilings. This is the substitution rule for the alternate tileset; the relationship between the two tilesets. In addition to the edge arrows in the usual tileset, the matching rules for both tilesets can be expressed by drawing pieces of large arrows at the vertices, requiring them to piece together into full arrows. Katz has studied the additional tilings allowed by dropping the vertex constraints and imposing only the requirement that the edge arrows match. Since this requirement is itself preserved by the substitution rules, any new tiling has an infinite sequence of "enlarged" copies obtained by successive applications of the substitution rule; each tiling in the sequence is indistinguishable from a true Ammann–Beenker tiling on a successively larger scale. Since some of these tilings are periodic, it follows that no decoration of the tiles which does force aperiodicity can be determined by looking at any finite patch of the tiling; the orientation of the vertex arrows which force aperiodicity can only be deduced from the entire infinite tiling.
The tiling has an extremal property: among the tilings whose rhombuses alternate, the proportion of squares is found to be minimal in the Ammann–Beenker tilings. The Ammann–Beenker tilings are related to the silver ratio and the Pell numbers; the substitution scheme R → R r R. The eigenvalues of the substitution matrix are 1 + 2 and 1 − 2. In the alternate tileset, the long edges have 1 + 2 times longer sides than the short edges. One set of Conway worms, formed by the short and long diagonals of the rhombs, forms the above strings, with r as the short diagonal and R as the long diagonal. Therefore, the Ammann bars form Pell ordered grids; the Ammann bars for the usual tileset. If the bold outer lines are taken to have length 2 2, the bars split the edges into segments of length 1 + 2 and 2 − 1; the Ammann bars for the alternate tileset. Note that the bars for the asymmetric tile extend outside it; the tesseractic honeycomb has an eightfold rotational symmetry, corresponding to an eightfold rotational symmetry of the tesseract.
A rotation matrix representing this symmetry is: A = [ 0 0 0 − 1 1 0 0 0 0 − 1
Circle Limit III
Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and "fall back again whence they came", it is one of a series of four woodcuts by Escher depicting ideas from hyperbolic geometry. Dutch physicist and mathematician Bruno Ernst called it "the best of the four". Escher became interested in tessellations of the plane after a 1936 visit to the Alhambra in Granada and from the time of his 1937 artwork Metamorphosis I he had begun incorporating tessellated human and animal figures into his artworks. In a 1958 letter from Escher to H. S. M. Coxeter, Escher wrote that he was inspired to make his Circle Limit series by a figure in Coxeter's article "Crystal Symmetry and its Generalizations". Coxeter's figure depicts a tessellation of the hyperbolic plane by right triangles with angles of 30°, 45°, 90°; this tessellation may be interpreted as depicting the lines of reflection and fundamental domains of the triangle group.
An elementary analysis of Coxeter's figure, as Escher might have understood it, is given by Casselman. Escher seems to have believed that the white curves of his woodcut, which bisect the fish, represent hyperbolic lines in the Poincaré disk model of the hyperbolic plane, in which the whole hyperbolic plane is modeled as a disk in the Euclidean plane, hyperbolic lines are modeled as circular arcs perpendicular to the disk boundary. Indeed, Escher wrote that the fish move "perpendicularly to the boundary". However, as Coxeter demonstrated, there is no hyperbolic arrangement of lines whose faces are alternately squares and equilateral triangles, as the figure depicts. Rather, the white curves are hypercycles that meet the boundary circle at angles of cos−1 21⁄4 − 2−1⁄4/2 80°; the symmetry axes of the triangles and squares that lie between the white lines are true hyperbolic lines. The squares and triangles of the woodcut resemble the alternated octagonal tiling of the hyperbolic plane, which features squares and triangles meeting in the same incidence pattern.
However, the precise geometry of these shapes is not the same. In the alternated octagonal tiling tiling, the sides of the squares and triangles are hyperbolically straight line segments, which do not link up in smooth curves. In Escher's woodcut, the sides of the squares and triangles are formed by arcs of hypercycles, which are not straight in hyperbolic geometry, but which connect smoothly to each other without corners; the points at the centers of the squares, where four fish meet at their fins, form the vertices of an order-8 triangular tiling, while the points where three fish fins meet and the points where three white lines cross together form the vertices of its dual, the octagonal tiling. Similar tessellations by lines of fish may be constructed for other hyperbolic tilings formed by polygons other than triangles and squares, or with more than three white curves at each crossing. Euclidean coordinates of circles containing the three most prominent white curves in the woodcut may be obtained by calculations in the field of rational numbers extended by the square roots of two and three.
Viewed as a pattern, ignoring the colors of the fish, in the hyperbolic plane, the woodcut has three-fold and four-fold rotational symmetry at the centers of its triangles and squares and order-three dihedral symmetry at the points where the white curves cross. In John Conway's orbifold notation, this set of symmetries is denoted 433; each fish provides a fundamental region for this symmetry group. Contrary to appearances, the fish do not have bilateral symmetry: the white curves of the drawing are not axes of reflection symmetry. For example, the angle at the back of the right fin is 90°, but at the back of the much smaller left fin it is 120°; the fish in Circle Limit III are depicted in four colors, allowing each string of fish to have a single color and each two adjacent fish to have different colors. Together with the black ink used to outline the fish, the overall woodcut has five colors, it is printed from five wood blocks, each of which provides one of the colors within a quarter of the disk, for a total of 20 impressions.
The diameter of the outer circle, as printed, is 41.5 cm. As well as being included in the collection of the Escher Museum in The Hague, there is a copy of Circle Limit III in the collection of the National Gallery of Canada. Douglas Dunham Department of Computer Science University of Minnesota, Duluth Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation
A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s; the aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right. Penrose tiling is non-periodic, it is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through "inflation" and every finite patch from the tiling occurs infinitely many times, it is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order. Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules and project schemes and coverings.
Penrose tilings are simple examples of aperiodic tilings of the plane. A tiling is a covering of the plane by tiles with no gaps; the most familiar tilings are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period of the tiling. If a tiling has no periods it is said to be non-periodic. A set of prototiles is said to be aperiodic if it tiles the plane but every such tiling is non-periodic; the subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles, posed the "Domino Problem": to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges, he observed that if this problem were undecidable there would have to exist an aperiodic set of Wang dominoes.
At the time, this seemed implausible, so Wang conjectured no such set could exist. Wang's student Robert Berger proved that the Domino Problem was undecidable in his 1964 thesis, obtained an aperiodic set of 20426 Wang dominoes, he described a reduction to 104 such prototiles. The color matching required in a tiling by Wang dominoes can be achieved by modifying the edges of the tiles like jigsaw puzzle pieces so that they can fit together only as prescribed by the edge colorings. Raphael Robinson, in a 1971 paper which simplified Berger's techniques and undecidability proof, used this technique to obtain an aperiodic set of just six prototiles; the first Penrose tiling is an aperiodic set of six prototiles, introduced by Roger Penrose in a 1974 paper, but it is based on pentagons rather than squares. Any attempt to tile the plane with regular pentagons leaves gaps, but Johannes Kepler showed, in his 1619 work Harmonices Mundi, that these gaps can be filled using pentagrams and related shapes.
Traces of these ideas can be found in the work of Albrecht Dürer. Acknowledging inspiration from Kepler, Penrose found matching rules for these shapes, obtaining an aperiodic set, his tiling can be viewed as a completion of Kepler's finite Aa pattern. Penrose subsequently reduced the number of prototiles to two, discovering the kite and dart tiling and the rhombus tiling; the rhombus tiling was independently discovered by Robert Ammann in 1976. Penrose and John H. Conway investigated the properties of Penrose tilings, discovered that a substitution property explained their hierarchical nature. In 1981, De Bruijn explained a method to construct Penrose tilings from five families of parallel lines as well as a "cut and project method", in which Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure. In this approach, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges.
The three types of Penrose tiling, P1–P3, are described individually below. They have many common features: in each case, the tiles are constructed from shapes related to the pentagon, but the basic tile shapes need to be supplemented by matching rules in order to tile aperiodically. Penrose's first tiling uses pentagons and three other shapes: a five-pointed "star", a "boat" and a "diamond". To ensure that all tilings are non-periodic, there are matching rules that specify how tiles may meet each other, there are three different types of matching rule for the pentagonal tiles, it is co
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions, its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are constructed in ordinary Euclidean space, they may be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. There are infinitely many honeycombs, which have only been classified; the more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary space.
Another interesting family is the Hill tetrahedra and their generalizations, which can tile the space. A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, having all vertices the same. There are 28 convex examples in Euclidean 3-space called the Archimedean honeycombs. A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell; every regular honeycomb is automatically uniform. However, there is just the cubic honeycomb. Two are quasiregular: The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric.
In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of the Platonic solids other than the cube. Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only, they are called parallelohedra: Cubic honeycomb Hexagonal prismatic honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb. Bitruncated cubic honeycomb or truncated octahedraOther known examples of space-filling polyhedra include: The Triangular prismatic honeycomb; the gyrated triangular prismatic honeycomb. The Voronoi cells of the carbon atoms in diamond are this shape; the trapezo-rhombic dodecahedral honeycomb Isohedral tilings. Sometimes, two or more different polyhedra may be combined to fill space. Besides many of the uniform honeycombs, another well known example is the Weaire–Phelan structure, adopted from the structure of clathrate hydrate crystals Weaire–Phelan structure Documented examples are rare.
Two classes can be distinguished: Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane. In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size; the regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora; the 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated. For every honeycomb there is a dual honeycomb, which may be obtained by exchanging: cells for vertices. Faces for edges; these are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems.
The more regular honeycombs dualise neatly: The cubic honeycomb is self-dual. That of octahedra and tetrahedra is dual to that of rhombic dodecahedra; the slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are. The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald. Honeycombs can be self-dual. All n-dimensional hypercubic honeycombs with Schläfli symbols, are self-dual. List of uniform tilings Regular honeycombs Infinite skew polyhedron Plesiohedron Coxeter, H. S. M.: Regular Polytopes. Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. pp. 164–199. ISBN 0-486-23729-X. Chapter 5: Polyhedra packing and space filling Critchlow, K.: Order in space. Pearce, P.: Structure in nature is a strategy for design. Goldberg, Michael Three Infinite Families of Tetrahedral Space-Fillers Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
Goldberg, Michael The space-filling pentahedra, Journal of Combinatorial Theory, Series A Volume 13, Issue 3, November 1972, Pages 437-443 [
John Horton Conway
John Horton Conway is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus. Conway was born in the son of Cyril Horton Conway and Agnes Boyce, he became interested in mathematics at a early age. By the age of eleven his ambition was to become a mathematician. After leaving sixth form, Conway entered Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert", he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport.
Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room, he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. Conway is known for the invention of the Game of Life, one of the early examples of a cellular automaton, his initial experiments in that field were done with pen and paper, long before personal computers existed. Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, articles, it is a staple of recreational mathematics.
There is an extensive wiki devoted to cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done; the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner; when Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, over the years Gardner had written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts and his angel and devil problem.
In the September 1976 column he reviewed Conway's book On Numbers and Games and introduced the public to Conway's surreal numbers. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, Conway himself has been a featured speaker at these events, discussing various aspects of recreational mathematics. Conway is known for his contributions to combinatorial game theory, a theory of partisan games; this he developed with Elwyn Berlekamp and Richard Guy, with them co-authored the book Winning Ways for your Mathematical Plays. He wrote the book On Numbers and Games which lays out the mathematical foundations of CGT, he is one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, Conway's soldiers, he came up with the angel problem, solved in 2006. He invented a new system of numbers, the surreal numbers, which are related to certain games and have been the subject of a mathematical novel by Donald Knuth.
He invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. In the mid-1960s with Michael Guy, son of Richard Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms, they discovered the grand antiprism in the only non-Wythoffian uniform polychoron. Conway has suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which describes rules for deciding if a prototile will tile the plane, he investigated lattices in higher dimensions, was the first to determine the symmetry group of the Leech lattice. In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.
Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while correcting a number of errors in the 19th century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. See Topology Proceedings 7 118, he was the primary author of the ATLAS of Finite Groups giving prope
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 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, 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. In the geometry of higher dimensions, a space-filling or honeycomb is 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.
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 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 in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf 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 Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, 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, 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 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 be made from other shapes such as pentagons, polyominoes and in fact 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, 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 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. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; 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. 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. Other methods exist for describing polygonal tilings; when the tessellation
Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of catoptric tessellation; the cubille is the only Platonic tessellation of 3-space, is self-dual. There are other uniform honeycombs constructed as prismatic stacks which are excluded from these categories; the pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed; these four symmetry groups are labeled as: Crystallography of Quasicrystals: Concepts and Structures by Walter Steurer, Sofia Deloudi, p.54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry Conway, John H..
"21. Naming Archimedean and Catalan Polyhedra and Tilings"; the Symmetries of Things. A K Peters, Ltd. pp. 292–298. ISBN 978-1-56881-220-5. Inchbald, Guy. "The Archimedean honeycomb duals". The Mathematical Gazette. Leicester: The Mathematical Association. 81: 213–219. JSTOR 3619198. Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4, 49 - 56. Norman Johnson Uniform Polytopes, Manuscript A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative, Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF George Olshevsky, Uniform Panoploid Tetracombs, Manuscript PDF Pearce, Peter. Structure in Nature is a Strategy for Design; the MIT Press. Pp. 41–47. ISBN 9780262660457. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter and Semi-Regular Polytopes III, See p318