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Heptagonal Tiling
In geometry, the heptagonal tiling is a regular hyperbolic tiling, regular tiling of the Hyperbolic geometry, hyperbolic plane. It is represented by Schläfli symbol of , having three regular heptagons around each vertex. Images Related polyhedra and tilings This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol . From a Wythoff construction there are eight hyperbolic Uniform tilings in hyperbolic plane, uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms. 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 al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hurwitz's Automorphisms Theorem
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms cannot exceed 84(''g'' − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms. The theorem is named after Adolf Hurwitz, who proved it in . Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isogonal Tilings
Isogonal, a mathematical term meaning "having similar angles", may refer to: *Isogonal figure or polygon, polyhedron, polytope or tiling * Isogonal trajectory, in curve theory *Isogonal conjugate __NOTOC__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ..., in triangle geometry See also * Isogonic line, in the study of Earth's magnetic field, a line of constant magnetic declination {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hyperbolic Tilings
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 Face (geometry), faces and is vertex-transitive (Transitive group action, transitive on its vertex (geometry), vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are Congruence (geometry), congruent, and the tessellation, 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 polyhedron, regular (if also face- and edge-transitive), quasi- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Heptagonal Tilings
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ''Wikt:hepta-, hepta-'', a Greek language, Greek-derived numerical prefix (both are cognate), together with the suffix ''-gon'' for , meaning angle. Regular heptagon A regular polygon, regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degree (angle), degrees). Its Schläfli symbol is . Area The area (''A'') of a regular heptagon of side length ''a'' is given by: :A = \fraca^2 \cot \frac \simeq 3.634 a^2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with Vertex (geometry), vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of \pi/7 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was 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 Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
List Of Regular Polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
List Of Uniform Planar 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 called 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 (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 two main problems relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hexagonal Tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a Truncation (geometry), truncated triangular tiling). English mathematician John Horton Conway, John Conway called it a hextille. The internal angle of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of List of regular polytopes#Euclidean tilings, three regular tilings of the plane. The other two are the triangular tiling and the square tiling. Structure and properties The hexagonal tiling has a structure consisting of a regular hexagon only as its prototile, sharing two vertices with other identical ones, an example of monohedral tiling. Each vertex at the tiling is surrounded by three regular hexagons, denoted as 6.6.6 by vertex configuration. The dual of a hexagonal tiling is triangular tiling, because the center of each hexagonal tiling ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polygon Triangulation
In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is . Triangulations may be viewed as special cases of planar straight-line graphs. When there are no holes or added points, triangulations form maximal outerplanar graphs. Polygon triangulation without extra vertices Over time, a number of algorithms have been proposed to triangulate a polygon. Special cases It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex ''n''-gon by non-intersecting diagonals is the (''n''−2)nd Catalan number, which equals :\frac, a formula found by Leonhard Euler. A monotone polygon can be triangulated in linear time with either the algorithm of A. Fournier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
