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Icosahedral Pyramids
The icosahedral pyramid is a four-dimensional convex polytope, bounded by one icosahedron as its base and by 20 triangular pyramid cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ... which meet at its apex. Since an icosahedron's circumradius is less than its edge length,, circumradius sqrt 5+sqrt(5))/8 = 0.951057 the tetrahedral pyramids can be made with regular faces. Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an icosahedral bipyramid which is also a Blind Polytope. The regular 600-cell has icosahedral pyramids around every vertex. The dual to the icosahedral pyramid is the dodecahedral pyramid, seen as a dodecahedron, dodecahedral base, and 12 regular pentagonal pyramids meeting at an apex. : References External links * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Pyramid
The icosahedral pyramid is a four-dimensional convex polytope, bounded by one icosahedron as its base and by 20 triangular pyramid cells which meet at its apex. Since an icosahedron's circumradius is less than its edge length,, circumradius sqrt 5+sqrt(5))/8 = 0.951057 the tetrahedral pyramids can be made with regular faces. Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an icosahedral bipyramid which is also a Blind Polytope. The regular 600-cell has icosahedral pyramids around every vertex. The dual to the icosahedral pyramid is the dodecahedral pyramid, seen as a dodecahedron, dodecahedral base, and 12 regular pentagonal pyramid In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regu ...s meeting at an apex. : References External links * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blind Polytope
In geometry, a Blind polytope is a convex polytope composed of regular polytope facets. The category was named after the German couple Gerd and Roswitha Blind, who described them in a series of papers beginning in 1979. It generalizes the set of semiregular polyhedra and Johnson solids to higher dimensions. Uniform cases The set of convex uniform 4-polytopes (also called semiregular 4-polytopes) are completely known cases, nearly all grouped by their Wythoff constructions, sharing symmetries of the convex regular 4-polytopes and prismatic forms. Set of convex uniform 5-polytopes, uniform 6-polytopes, uniform 7-polytopes, etc are largely enumerated as Wythoff constructions, but not known to be complete. Other cases Pyramidal forms: (4D) # (''Tetrahedral pyramid'', ( ) ∨ , a tetrahedron base, and 4 tetrahedral sides, a lower symmetry name of regular 5-cell.) # Octahedral pyramid, ( ) ∨ , an octahedron base, and 8 tetrahedra sides meeting at an apex. # Icosahedral p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedral Pyramid
In 4-dimensional geometry, the dodecahedral pyramid is bounded by one dodecahedron on the base and 12 pentagonal pyramid cells which meet at the apex. Since a dodecahedron's circumradius is greater than its edge length, sqrt 9+3 sqrt(5))/8≒ 1.401259 the pentagonal pyramids require tall isosceles triangle faces. The dual to the dodecahedral pyramid is an icosahedral pyramid, seen as an icosahedral base, and 20 regular tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ... meeting at an apex. : References External links * * Richard KlitzingAxial-Symmetrical Edge Facetings of Uniform Polyhedra 4-polytopes {{Polychora-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells. The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4- dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell. Geometry The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell, as the 24-cell can be deconstructed into three overlapping insta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedral Bipyramid
In 4-dimensional geometry, the icosahedral bipyramid is the direct sum of a icosahedron and a segment, + . Each face of a central icosahedron is attached with two tetrahedra, creating 40 tetrahedral cells, 80 triangular faces, 54 edges, and 14 vertices.https://www.bendwavy.org/klitzing/incmats/ikedpy.htm An icosahedral bipyramid can be seen as two icosahedral pyramids augmented together at their bases. It is the dual of a dodecahedral prism, Coxeter-Dynkin diagram , so the bipyramid can be described as . Both have Coxeter notation symmetry ,3,5 order 240. Having all regular cells (tetrahedra), it is a Blind polytope. See also * Pentagonal bipyramid - A lower dimensional analogy * Tetrahedral bipyramid * ''Octahedral bipyramid'' - A lower symmetry form of the as 16-cell. * Cubic bipyramid * Dodecahedral bipyramid In 4-dimensional geometry, the dodecahedral bipyramid is the direct sum of a dodecahedron and a segment, + . Each face of a central dodecahedron is attached with tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell (mathematics)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedron Any convex polyhedron's surface has Euler characteristic :V - E + F = 2, where ''V'' is the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Pyramid
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another spher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlegel Diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes. Construction The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows: :A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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