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Skew Infinite Polygon
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all Collinearity, colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a Cylinder (geometry), cylinder. Regular infinite skew polygons exist in the Petrie polygons of the affine and hyperbolic Coxeter groups. They are constructed a single operator as the composite of all the reflections of the Coxeter group. Regular zig-zag skew apeirogons in two dimensions A regular zig-zag skew apeirogon has (2*∞), D∞d Frieze group symmetry. Regular zig-zag skew apeirogons exist as Petrie polygons of the three regular tilings of the plane: , , and . These regular zig-zag skew apeirogons have internal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings: Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Isogonal Apeirogon Skew-equal
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, in triangle geometry See also *Isogonic line A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensi ..., in the study of Earth's magnetic field, a line of constant magnetic declination {{disambig Geometry ... [...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 sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Tilings Of The Hyperbolic Plane
This article lists the regular polytopes in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, 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 (mathematician), Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the Edge (geometry), edges of polygons and other higher dimensional polytopes. It is used in the definition of prism (geometry), 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
