|
Heptagonal Tiling
In geometry, a heptagonal tiling is a regular tiling of the 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 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 all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving the ... [...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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Tilings
Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts: * Isogonal polygon, polyhedron, polytope or tiling. *Isogonal trajectory in curve theory. *Isogonal conjugate in triangle geometry. An Isogonal is also the name for a line connecting points at which the magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) and ... is the same. {{disambig Geometry ... [...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] |
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 "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle. Regular heptagon A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 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 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, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptag ... [...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 Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 sta ... [...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 two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 truncated triangular tiling). English mathematician 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 three regular tilings of the plane. The other two are the triangular tiling and the square tiling. Applications The hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is optimal. However, the less regular Weaire–Phel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 and D.Y. ... [...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] |
.svg "Click for more on -> Polygon Triangulation")
