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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] |
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] |
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4- simplex (Coxeter's \alpha_4 polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides. The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Bipyramid
In 4-dimensional geometry, the cubical bipyramid is the direct sum of a cube and a segment, + . Each face of a central cube is attached with two square pyramids, creating 12 square pyramidal cells, 30 triangular faces, 28 edges, and 10 vertices. A cubical bipyramid can be seen as two cubic pyramids augmented together at their base. It is the dual of a octahedral prism. Being convex and regular-faced, it is a CRF polytope. Coordinates It is a Hanner polytope with coordinates: * (0, 0, 0; ±1) * (±1, ±1, ±1; 0) See also * Tetrahedral 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 two pentagonal pyramids, creating 24 pentagonal pyramidal cells, 72 isosceles trian ... * Icosahedral bipyramid References External links Cubic tegum 4-polytopes {{Polychora-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Pyramid
In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one, the triangular pyramids can be made with regular faces (as regular tetrahedrons) by computing the appropriate height. Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an octahedral bipyramid which is also a Blind polytope. Occurrences of the octahedral pyramid The regular 16-cell has ''octahedral pyramids'' around every vertex, with the octahedron passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the 16-cell honeycomb. Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a 24-cell with octahedral bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism. Topologically, under its highest symmetry, ,3,3 there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron. The vertex figure of the ''rectified 5-cell'' is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends. Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual becau ... [...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] |
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid .Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68 It is a part of an infinite family of polytopes, called cross-polytopes or ''orthoplexes'', and is analogous to the octahedron in three dimensions. It is Coxeter's \beta_4 polytope. Conway's name for a cross-polytope is orthoplex, for ''orthant complex''. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices. Geometry The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). Each of its 4 successor convex regular 4-polytopes can be constructed as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Bipyramid
In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, + . Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices,.https://www.bendwavy.org/klitzing/incmats/tedpy.htm A tetrahedral bipyramid can be seen as two tetrahedral pyramids augmented together at their base. It is the dual of a tetrahedral prism, , so it can also be given a Coxeter-Dynkin diagram, , and both have Coxeter notation symmetry ,3,3 order 48. Being convex with all regular cells (tetrahedra) means that it is a Blind polytope. This bipyramid exists as the cells of the dual of the uniform rectified 5-simplex, and rectified 5-cube or the dual of any uniform 5-polytope with a tetrahedral prism vertex figure. And, as well, it exists as the cells of the dual to the rectified 24-cell honeycomb. See also * Triangular bipyramid - A lower dimensional analogy of the tetrahedral bipyra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , con ... [...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] |
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 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 sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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