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Great Icosihemidodecahedron
In geometry, the great icosihemidodecahedron (or great icosahemidodecahedron) is a nonconvex uniform polyhedron, indexed as U71. It has 26 faces (20 triangles and 6 decagrams), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. It is a hemipolyhedron with 6 decagrammic faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the triangular faces in common), and with the great dodecahemidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron 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 c ... References External links * Uniform polyhedra and duals Uniform polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosihemidodecahedron
In geometry, the great icosihemidodecahedron (or great icosahemidodecahedron) is a nonconvex uniform polyhedron, indexed as U71. It has 26 faces (20 triangles and 6 decagrams), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. It is a hemipolyhedron with 6 decagrammic faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the triangular faces in common), and with the great dodecahemidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron 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 c ... References External links * Uniform polyhedra and duals Uniform polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same ''vertex arrangement'' if they share the same 0-skeleton. A group of polytopes that shares a vertex arrangement is called an ''army''. Vertex arrangement The same set of vertices can be connected by edges in different ways. For example, the ''pentagon'' and ''pentagram'' have the same ''vertex arrangement'', while the second connects alternate vertices. A ''vertex arrangement'' is often described by the convex hull polytope which contains it. For example, the regular ''pentagram'' can be said to have a (regular) ''pentagonal vertex arrangement''. Infinite tilings can also share common ''vertex arrangements''. For example, this triangular lattice of points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosihemidodecahedron 2
Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People * List of people known as "the Great" *Artel Great (born 1981), American actor Other uses * ''Great'' (1975 film), a British animated short about Isambard Kingdom Brunel * ''Great'' (2013 film), a German short film * Great (supermarket), a supermarket in Hong Kong * GReAT, Graph Rewriting and Transformation, a Model Transformation Language * Gang Resistance Education and Training Gang Resistance Education And Training, abbreviated G.R.E.A.T., provides a school-based, police officer instructed program that includes classroom instruction and various learning activities. Their intention is to teach the students to avoid gang ..., or GREAT, a school-based and police officer-instructed program * Global Research and Analysis Team (GReAT), a cybersecurity team at Kaspersky Lab *'' Great!'', a 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron. Geometry An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron located at the midpoints of the edges of either. Its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, which belong among the Johnson solids. The icosidodecahedron can be considered a ''pentagonal gyrobirotunda'', as a combination of two rotundae (compare pentagonal orthobirotunda, one of the Johnson solids) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahemidodecahedron
In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. It has 18 faces (12 Pentagram, pentagrams and 6 Decagram (geometry), decagrams), 60 edges, and 30 vertices. Its vertex figure is a quadrilateral#More quadrilaterals, crossed quadrilateral. Aside from the regular small stellated dodecahedron and great stellated dodecahedron , it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons (star polygons), namely the star polygons pentagram, and decagram (geometry), . It is a hemipolyhedron with 6 Decagram (geometry), decagrammic faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the pentagrammic faces in common) and the great icosihemidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra References External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosidodecahedron
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and . Related polyhedra The name is constructed analogously as how a cube-octahedron creates a cuboctahedron, and how a dodecahedron-icosahedron creates a (small) icosidodecahedron. It shares the same vertex arrangement with the icosidodecahedron, its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead. It also shares its edge arrangement with the great icosihemidodecahedron (having the triangular faces in common), and with the great dodecahemidodecahedron (having the pentagrammic faces in common). This polyhedron can be considered a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahemidodecahedron
In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. It has 18 faces (12 Pentagram, pentagrams and 6 Decagram (geometry), decagrams), 60 edges, and 30 vertices. Its vertex figure is a quadrilateral#More quadrilaterals, crossed quadrilateral. Aside from the regular small stellated dodecahedron and great stellated dodecahedron , it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons (star polygons), namely the star polygons pentagram, and decagram (geometry), . It is a hemipolyhedron with 6 Decagram (geometry), decagrammic faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the pentagrammic faces in common) and the great icosihemidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra References External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosidodecahedron
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and . Related polyhedra The name is constructed analogously as how a cube-octahedron creates a cuboctahedron, and how a dodecahedron-icosahedron creates a (small) icosidodecahedron. It shares the same vertex arrangement with the icosidodecahedron, its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead. It also shares its edge arrangement with the great icosihemidodecahedron (having the triangular faces in common), and with the great dodecahemidodecahedron (having the pentagrammic faces in common). This polyhedron can be considered a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron. Geometry An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron located at the midpoints of the edges of either. Its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, which belong among the Johnson solids. The icosidodecahedron can be considered a ''pentagonal gyrobirotunda'', as a combination of two rotundae (compare pentagonal orthobirotunda, one of the Johnson solids) ... [...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] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemipolyhedron
In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other polyhedron – hence the "hemi" prefix. The prefix "hemi" is also used to refer to certain projective polyhedra, such as the hemicube (geometry), hemi-cube, which are the image of a 2 to 1 map of a spherical polyhedron with central symmetry. Wythoff symbol and vertex figure Their Wythoff symbols are of the form ''p''/(''p'' − ''q'') ''p''/''q'' , ''r''; their vertex figures are quadrilateral#More quadrilaterals, crossed quadrilaterals. They are thus related to the cantellation (geometry), cantellated polyhedra, which have similar Wythoff symbols. The vertex configuration is ''p''/''q''.2''r''.''p''/(''p'' − ''q'').2''r''. The 2''r''-gon faces pass through the center of the model: if represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
