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Apeirogonal Hosohedron
In geometry, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol Related tilings and polyhedra The apeirogonal hosohedron is the arithmetic limit of the family of hosohedra , as ''p'' tends to infinity, thereby turning the hosohedron into a Euclidean tiling. All the vertices have then receded to infinity and the digonal faces are no longer defined by closed circuits of finite edges. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling In geometry, an apeirogonal tiling is a tessellation of the ... [...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] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r or \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Tilings
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] |
Isohedral Tilings
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Euclidean Tilings
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher dimensional generalizations *Euclidean geometry, the study of the properties of Euclidean spaces *Non-Euclidean geometry, systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines *Euclidean distance, the distance between pairs of points in Euclidean spaces * Euclidean ball, the set of points within some fixed distance from a center point Number theory *Euclidean division, the division which produces a quotient and a remainder *Euclidean algorithm, a method for finding greatest common divisors *Extended Euclidean algorithm, a method for solving the Diophantine equation ''ax'' + ''by'' = ''d'' where ''d'' is the greatest common divisor of ''a'' and ''b'' *Euclid's lemma: if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogonal Tilings
In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include: *Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces *Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around a vertex *Order-4 apeirogonal tiling, hyperbolic tiling with 4 apeirogons around a vertex *Order-5 apeirogonal tiling, hyperbolic tiling with 5 apeirogons around a vertex *Infinite-order apeirogonal tiling, hyperbolic tiling with an infinite number of apeirogons around a vertex The vertices of an order-k apeirogonal tiling form a Bethe lattice, a regular infinite tree. See also *Apeirogonal antiprism *Apeirogonal prism *Apeirohedron References {{set index article, mathematics Apeirogonal tilings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogonal Antiprism
In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane. If the sides are equilateral triangles, it is a uniform tiling. In general, it can have two sets of alternating congruent isosceles triangles, surrounded by two half-planes. Related tilings and polyhedra The apeirogonal antiprism is the arithmetic limit of the family of antiprisms sr or ''p''.3.3.3, as ''p'' tends to infinity, thereby turning the antiprism into a Euclidean tiling. File:Infinite prism.svg, The apeirogonal antiprism can be constructed by applying an alternation operation to an apeirogonal prism. File:Apeirogonal trapezohedron.svg, The dual tiling of an apeirogonal antiprism is an ''apeirogonal deltohedron''. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogonal Prism
In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.Conway (2008), p.263 Thorold Gosset called it a ''2-dimensional semi-check'', like a single row of a checkerboard. If the sides are squares, it is a uniform tiling. If colored with two sets of alternating squares it is still uniform. File:Infinite prism alternating.svg, Uniform variant with alternate colored square faces. File:Infinite_bipyramid.svg, Its dual tiling is an ''apeirogonal bipyramid''. Related tilings and polyhedra The apeirogonal tiling is the arithmetic limit of the family of prisms t or ''p''.4.4, as ''p'' tends to infinity, thereby turning the prism into a Euclidean tiling. An alternation operation can create an apeirogonal antiprism composed of three triangles and one apeirogon at each vertex. : Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-2 Apeirogonal Tiling
In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedronConway (2008), p. 263 is a tessellation (gap-free filling with repeated shapes) of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol Two apeirogons joined along all their edges can completely fill the entire plane, as an apeirogon is infinite in size and has an interior angle of 180°, which is half of a full 360°. Related tilings and polyhedra Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the apeirogonal antiprism. See also * Orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncation
In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual operation to barycentric subdivision. Because the barycentric subdivision of any polytope can be realized as another polytope, the same is true for the omnitruncation of any polytope. When omnitruncation is applied to a regular polytope (or honeycomb) it can be described geometrically as a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortcut'' term which has a different meaning in progressively-higher-dimensional polytopes: * Uniform polytope truncation operators ** For regular polygons: An ordinary truncation, t_\ = t\ = \. *** Coxeter-Dynkin diagram ** For uniform polyhedra (3-polytopes): A cantitruncation, t_\ = tr\. (Application ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellation (geometry)
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tilings and honeycombs. Cantellating a polyhedron is also rectifying its rectification. Cantellation (for polyhedra and tilings) is also called '' expansion'' by Alicia Boole Stott: it corresponds to moving the faces of the regular form away from the center, and filling in a new face in the gap for each opened edge and for each opened vertex. Notation A cantellated polytope is represented by an extended Schläfli symbol ''t''0,2 or ''r''\beginp\\q\\...\end or ''rr''. For polyhedra, a cantellation offers a direct sequence from a regular polyhedron to its dual. Example: cantellation sequence between cube and octahedron: Example: a cuboctahedron is a cantellated tetrahedron. For higher-dimensional polytopes, a cantellation offers a direc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
