In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is referred to as Wythoff's kaleidoscopic construction; the method is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope. However, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point, they therefore enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that point produce a uniform polyhedron. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron: A vertex is placed at the point A.
This produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C; this produces a polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC; this produces a polyhedron with Wythoff symbol a b c|. The vertex is at a point such that, when it is rotated around any of the triangle's corners by twice the angle at that point, it is displaced by the same distance for every angle. Only even-numbered reflections of the original vertex are used; the polyhedron has the Wythoff symbol |a b c. The process in general applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes. Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian, they can be derived from Wythoffian forms either by alternation or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry.
Sometimes snub forms are considered Wythoffian though they can only be constructed by the alternation of omnitruncated forms. Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra and uniform tilings. Coxeter-Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs. Coxeter Regular Polytopes, Third edition, Dover edition, ISBN 0-486-61480-8 Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 Har'El, Z. Uniform Solution for Uniform Polyhedra. Geometriae Dedicata 47, 57-110, 1993. W. A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences, 20 966–970. Greg Egan's applet to display uniform polyhedra using Wythoff's construction method A Shadertoy renderization of Wythoff's construction methodJenn, software that generates views of polyhedra and polychora from symmetry groups
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
Branko Grünbaum was a Yugoslavian-born mathematician of Jewish descent and a professor emeritus at the University of Washington in Seattle. He received his Ph. D. in 1957 from Hebrew University of Jerusalem in Israel. He authored over 200 papers in discrete geometry, an area in which he is known for various classification theorems, he wrote on the theory of abstract polyhedra. His paper on line arrangements may have inspired a paper by N. G. de Bruijn on quasiperiodic tilings. This paper is cited by the authors of a monograph on hyperplane arrangements as having inspired their research. Grünbaum devised a multi-set generalisation of Venn diagrams, he was a frequent contributor to Geombinatorics. Grünbaum's classic monograph Convex polytopes, first published in 1967, became the main textbook on the subject, his monograph Tilings and Patterns, coauthored with G. C. Shephard, helped to rejuvenate interest in this classic field, has proved popular with nonmathematical audiences, as well as with mathematicians.
In 1976 Grünbaum won a Lester R. Ford Award for his expository article Venn diagrams and independent families of sets. In 2004, Gil Kalai and Victor Klee edited a special issue of Discrete and Computational Geometry in his honor, the "Grünbaum Festschrift". In 2005, Grünbaum was awarded the Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society, he was a Guggenheim Fellow, a Fellow of the AAAS and in 2012 he became a fellow of the American Mathematical Society. Grünbaum supervised 19 Ph. D.s and has at least 200 mathematical "descendants". Grünbaum, Kaibel, Volker. Convex Polytopes, Graduate Texts in Mathematics, 221, Springer-Verlag, ISBN 0-387-00424-6. Grünbaum, Branko. Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Pentagram map Orlik, Peter.
The Socolar–Taylor tile is a single tile, aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane, with rotations and reflections of the tile allowed. It is the first known example of a single aperiodic tile, or "einstein"; the basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed. This rule cannot be geometrically implemented in two dimensions while keeping the tile a connected set; this is, possible in three dimensions, in their original paper Socolar and Taylor suggest a three-dimensional analogue to the monotile. Taylor and Socolar remark; however the tile does allow tilings with a period, shifting one two dimensional layer to the next, so the tile is only ″weakly aperiodic″. Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space. Previewable digital models of the three-dimensional tile, suitable for 3D printing, at Thingiverse Original diagrams and further information on Joan Taylor's personal website
In geometry, the rhombille tiling known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 120 ° angles. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles; the rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling, it can be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals of each rhomb are in the ratio 1:√3; this is the dual tiling of the trihexagonal kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, in the face configuration for monohedral tilings it is denoted, it is one of 56 possible isohedral tilings by quadrilaterals, one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, more such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube; the rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion. In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. In another of his works, Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements and an upstairs patio tiled with the rhombille tiling.
A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so. These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more and includes a depiction of the reversible cubes illusion on a flag within the scene; the rhombille tiling is used as a design for parquetry and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. It appears in ancient Greek floor mosaics from Delos and from Italian floor tilings from the 11th century, although the tiles with this pattern in Siena Cathedral are of a more recent vintage. In quilting, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation; as a quilting pattern it has many other names including cubework, heavenly stairs, Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape.
See Quilts of the Underground Railroad. In these decorative applications, the rhombi may appear in multiple colors, but are given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms; the rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field.
The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers. In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice, it is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals, it has been studied in percolation theory. The rhombille tiling has *632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a *333 symmetry; the rhombille tiling is the dual of the trihexagonal tiling, as such is part of a set of uniform dual tilings. It is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry, starting from the cube, which can be seen as a rhombic hexahedron where the rhombi are squares; the nth element in this sequence has a face configuration of V3.n.3.n. The rhombille tiling is one of many different ways of tiling the plane by congruent rhombi.
Others include a diagonally flattened variation of the square tiling, the tiling used by the Miura-ori folding pattern, the Penrose tiling which
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur in architecture and decorative art in textiles and tiles as well as wallpaper. A proof that there were only 17 distinct groups of possible patterns was first carried out by Evgraf Fedorov in 1891 and derived independently by George Pólya in 1924; the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are different in style, scale or orientation may belong to the same group. Consider the following examples: Examples A and B have the same wallpaper group.
Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe; the pattern is unchanged. Speaking, a true symmetry only exists in patterns that repeat and continue indefinitely. A set of only, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, classification is applied to finite patterns, small imperfections may be ignored. Sometimes two categorizations are meaningful, one based on shapes alone and one including colors.
When colors are ignored there may be more symmetry. In black and white there are 17 wallpaper groups; the types of transformations that are relevant here are called Euclidean plane isometries. For example: If we shift example B one unit to the right, so that each square covers the square, adjacent to it the resulting pattern is the same as the pattern we started with; this type of symmetry is called a translation. Examples A and C are similar. If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain the same pattern; this is called a rotation. Examples A and C have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can flip example B across a horizontal axis that runs across the middle of the image; this is called a reflection. Example B has reflections across a vertical axis, across two diagonal axes; the same can be said for A. However, example C is different, it only has reflections in vertical directions, not across diagonal axes.
If we flip across a diagonal line, we do not get the same pattern back. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C. Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same type if they are the same up to an affine transformation of the plane, thus e.g. a translation of the plane does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry. Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation, it follows from the Bieberbach theorem that all wallpaper groups are different as abstract groups.
2D patterns with double translational symmetry can be categorized according to their symmetry group type. Isometries of the Euclidean plane fall into four categories. Translations, denoted by Tv, where v is a vector in R2; this has the effect of shifting the plane applying displacement vector v. Rotations, denoted by Rc,θ, where c is a point in the plane, θ is the angle of rotation. Reflections, or mirror isometries, denoted by FL, where L is a line in R2.. This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror. Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance; this is a combination of a reflection in the line L and a translation along L by a distance d. The condition