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 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, 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
In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations. Let T be the right triangle with side length 1, 2 and 5. Conway noticed that T can be divided in five isometric copies of its image by the dilation of factor 1 / 5. By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of T; the union of all these triangles yields a tiling of the whole plane by isometric copies of T. In this tiling, isometric copies of T appears in infinitely many orientations. Despite this, all the vertices have rational coordinates. Radin relied on the above construction of Conway to define pinwheel tilings. Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of T, in which a tile may intersect another tile only either on a whole side or on half the length 2 side, such that the following property holds.
Given any pinwheel tiling P, there is a pinwheel tiling P ′ which, once each tile is divided in five following the Conway construction and the result is dilated by a factor 5, is equal to P. In other words, the tiles of any pinwheel tilings can be grouped in sets of five into homothetic tiles, so that these homothetic tiles form a new pinwheel tiling; the tiling constructed by Conway is a pinwheel tiling, but there are uncountably many other different pinwheel tiling. They are all locally undistinguishable, they all share with the Conway tiling the property that tiles appear in infinitely many orientations. The main result proven by Radin is that there is a finite set of so-called prototiles, with each being obtained by coloring the sides of T, so that the pinwheel tilings are the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point. In terms of symbolic dynamics, this means. Radin and Conway proposed a three-dimensional analogue, dubbed the quaquaversal tiling.
There are other generalizations of the original idea. One gets a fractal by iteratively dividing T in five isometrics copies, following the Conway construction, discarding the middle triangle; this "pinwheel fractal" has Hausdorff dimension d = ln 4 ln 5 ≈ 1.7227. Federation Square, a building complex in Melbourne, features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub-framing for the facades, allowing for the facades to be fabricated off-site, in a factory and erected to form the facades; the pinwheel tiling system was based on the single triangular element, composed of zinc, perforated zinc, sandstone or glass, joined to 4 other similar tiles on an aluminum frame, to form a "panel". Five panels were affixed to a galvanized steel frame, forming a "mega-panel", which were hoisted onto support frames for the facade; the rotational positioning of the tiles gives the facades a more random, uncertain compositional quality though the process of its construction is based on pre-fabrication and repetition.
The same pinwheel tiling system is used in the development of the structural frame and glazing for the "Atrium" at Federation Square, although in this instance, the pin-wheel grid has been made "3-dimensional" to form a portal frame structure. Pinwheel at the Tilings Encyclopedia Dynamic Pinwheel made in GeoGebra
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
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..
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