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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] |
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] |
Hanner Polytope
In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956.. Construction The Hanner polytopes are constructed recursively by the following rules:. *A line segment is a one-dimensional Hanner polytope *The Cartesian product of every two Hanner polytopes is another Hanner polytope, whose dimension is the sum of the dimensions of the two given polytopes *The dual of a Hanner polytope is another Hanner polytope of the same dimension. They are exactly the polytopes that can be constructed using only these rules: that is, every Hanner polytope can be formed from line segments by a sequence of product and dual operations. Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products and direct sums, the dual of the Cartesian products. This direct sum operation combines two polytopes by placin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 triangular faces, 70 edges, and 22 vertices. A dodecahedral bipyramid can be seen as two dodecahedral pyramids augmented together at their base. It is the dual of a icosahedral prism. See also * Tetrahedral bipyramid * Cubic bipyramid * 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 ver ... References External links Dodecahedral tegum 4-polytopes {{Polychora-stub ... [...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] |
Octahedral Prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms. Alternative names *Octahedral dyadic prism ( Norman W. Johnson) *Ope (Jonathan Bowers, for octahedral prism) *Triangular antiprismatic prism *Triangular antiprismatic hyperprism Coordinates It is a Hanner polytope with vertex coordinates, permuting first 3 coordinates: :( ±1,0,0 ±1) Structure The octahedral prism consists of two octahedra connected to each other via 8 triangular prisms. The triangular prisms are joined to each other via their square faces. Projections The octahedron-first orthographic projection of the octahedral prism into 3D space has an octahedral envelope. The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces. The triangular-prism-first orthographic projection of the octahedral prism into 3D space has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Pyramid
In 4-dimensional geometry, the cubic pyramid is bounded by one cube on the base and 6 square pyramid cells which meet at the apex. Since a cube has a circumradius divided by edge length less than one, the square pyramids can be made with regular faces by computing the appropriate height. Images Related polytopes and honeycombs Exactly 8 regular cubic pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a tesseract with 8 cubical bounding cells, surrounding a central vertex with 16 edge-length long radii. The tesseract tessellates 4-dimensional space as the tesseractic honeycomb. The 4-dimensional content of a unit-edge-length tesseract is 1, so the content of the regular cubic pyramid is 1/8. The regular 24-cell has ''cubic pyramids'' around every vertex. Placing 8 cubic pyramids on the cubic bounding cells of a tesseract is Gosset's construction of the 24-cell. Thus the 24-cell is constructed from exactly 16 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, the Johnson solid General square pyramid A possibly oblique square pyramid with base length ''l'' and perpendicular height ''h'' has volume: :V=\frac l^2 h. Right square pyramid In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles. A right square pyramid with base length ''l'' and height ''h'' has surface area and volume: :A=l^2+l\sqrt, :V=\frac l^2 h. The lateral edge length is: :\sqrt; the slant height is: :\sqrt. The dihedral angles are: :*between the base and a side: :::\arctan \left(\right); :*between two sides: :::\arccos \left(\right). Equilateral square pyramid, Johnson solid J1 If all edges have the same length, then the sides are e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spa ... [...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] |
CRF 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] |
Cubic Bipyramid-ortho
Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system where the unit cell is in the shape of a cube * Cubic function, a polynomial function of degree three * Cubic equation, a polynomial equation (reducible to ''ax''3 + ''bx''2 + ''cx'' + ''d'' = 0) * Cubic form, a homogeneous polynomial of degree 3 * Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3 * Cubic plane curve (mathematics), a plane algebraic curve ''C'' defined by a cubic equation * Cubic reciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocity * Cubic surface, an algebraic surface in three-dimensional space * Cubic zirconia, in geology, a mineral that is widely synthesized for use as a diamond simulacra * CUBIC, a histology method Computing * Cubic IDE, a modular devel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
