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
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 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
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, 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
Discrete & Computational Geometry
Discrete & Computational Geometry is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry; the journal is indexed by Mathematical Reviews, Zentralblatt MATH, Science Citation Index, Current Contents/Engineering and Technology. Its 2009 impact factor is 0.935. Official website
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