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Heptakis Heptagonal Tiling
In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t. Hyperbolic soccerball (football) This tiling is called a hyperbolic soccerball (football) for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space. Dual tiling The dual tiling is called a ''heptakis heptagonal tiling'', named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point. : Related tilings This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and ,3Coxeter group symmetry. From a W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptakis Heptagonal Tiling
In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t. Hyperbolic soccerball (football) This tiling is called a hyperbolic soccerball (football) for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space. Dual tiling The dual tiling is called a ''heptakis heptagonal tiling'', named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point. : Related tilings This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and ,3Coxeter group symmetry. From a W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Tilings
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they manifest hyperbolas, not because something about them is exaggerated. * Hyperbolic angle, an unbounded variable referring to a hyperbola instead of a circle * Hyperbolic coordinates, location by geometric mean and hyperbolic angle in quadrant I *Hyperbolic distribution, a probability distribution characterized by the logarithm of the probability density function being a hyperbola * Hyperbolic equilibrium point, a fixed point that does not have any center manifolds * Hyperbolic function, an analog of an ordinary trigonometric or circular function * Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolic space * Hyperbolic geometry, a non-Euclidean geometry * Hyperbolic group, a finitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Camb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Uniform Tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway calls these uniform duals ''Catalan tilings'', in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example ''4.8.8'' means one square and two octagons on a vertex. These 11 uniform tilings have 32 different ''uniform colorings''. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are ''not'' color-uniform) In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using star ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tilings Of Regular Polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' ( Latin: ''The Harmony of the World'', 1619). Notation of Euclidean tilings Euclidean tilings are usually named after Cundy & Rollett’s notation. This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 36; 36; 34.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 36; 36 (both of different transitivity class), or (36)2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 34.6, 4 more contiguous equilateral triangles and a single regular hexagon. However, this notation has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-7 Triangular Tiling
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . Hurwitz surfaces The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces. The smallest of these is the Klein quartic, the most symmetric genus 3 surface, together with a tiling by 56 triangles, meeting at 24 vertices, with symmetry group the simple group of order 168, known as PSL(2,7). The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron. The dual order-3 heptagonal tiling has the same symme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HyperRogue
''HyperRogue'' is an independent video game developed by Zeno Rogue. It is a roguelike inspired by the puzzle game ''Deadly Rooms of Death'' and the art of M. C. Escher, taking place in the hyperbolic plane. Gameplay ''HyperRogue'' is a turn-based game in which the player controls one character exploring a world based on hyperbolic geometry, with cells arranged as a truncated order-7 triangular tiling by default (with a few exceptions). The player can also choose to play on some other tilings and honeycombs in two and three dimensions, in all eight Geometrization conjecture#The eight Thurston geometries, Thurston geometries, along with a variety of quotient space (topology), quotient spaces. It borrows procedural generation and permadeath from the roguelike genre, and puzzle-based combat from ''Deadly Rooms of Death''. The world consists of 72 lands, each of which has a different theme, treasure, enemies, mechanics, and a magical power; the lands are usually separated by hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Tilings In Hyperbolic Plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol . Uniform tilings may be regular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transitive). For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wythoff Construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process 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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
