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Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams. History of discovery *Regular polytopes: (convex faces) **1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. *Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) **1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions''. *Convex uniform polytopes: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 5-cube
In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex Cantellated 5-cube Alternate names * Small rhombated penteract (Acronym: sirn) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a ''cantellated 5-cube'' having edge length 2 are all permutations of: :\left(\pm1,\ \pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right) Images Bicantellated 5-cube In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope. Alternate names * Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross * Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a ''bicantellated 5-cube'' having edge length 2 are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 5-cube
In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed. Stericated 5-cube Alternate names * Stericated penteract / Stericated 5-orthoplex / Stericated pentacross * Expanded penteract / Expanded 5-orthoplex / Expanded pentacross * Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a ''stericated 5-cube'' having edge length 2 are all permutations of: :\left(\pm1,\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 5-orthoplex
In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex. There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube. Runcinated 5-orthoplex Alternate names * Runcinated pentacross * Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers) Coordinates The vertices of the can be made in 5-space, as permutations and sign combinations of: : (0,1,1,1,2) Images Runcitruncated 5-orthoplex Alternate names * Runcitruncated pentacross * Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers) Coordinates Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of : (±3,±2,±1,±1,0) Images Runcicantellated 5-orthoplex Alternate names * Runcican ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 5-orthoplex
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex. There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex. Rectified 5-orthoplex Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups. E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 5-simplex
In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex. There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex. Truncated 5-simplex The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells). Alternate names * Truncated hexateron (Acronym: tix) (Jonathan Bowers) Coordinates The vertices of the ''truncated 5-simplex'' can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) ''or'' of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively. Images Bitruncated 5-simplex Alternate names * Bitruncated hexateron (Acronym: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 5-simplex
In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations ( sterication) of the regular 5-simplex. There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the ''steriruncicantitruncated 5-simplex'' is more simply called an omnitruncated 5-simplex with all of the nodes ringed. Stericated 5-simplex A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duopris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, ,3or , it is represented by Coxeter diagram . The regular polytopes are grouped by dimension and subgrouped by convex, nonconve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T3
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclidean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T0
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Eu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T04
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclidean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 5-cube
In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube. There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex. Runcinated 5-cube Alternate names * Small prismated penteract (Acronym: span) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a ''runcinated 5-cube'' having edge length 2 are all permutations of: :\left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right) Images Runcitruncated 5-cube Alternate names * Runcitruncated penteract * Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers) Construction and coordinates The Cartesian coordinates of the vertices of a ''runcitruncated 5-cube'' having edge length 2 are all permutations of: :\left(\pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+2\sqrt),\ \pm(1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T03
In Five-dimensional space, five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 Vertex (geometry), vertices, 80 Edge (geometry), edges, 80 square Face (geometry), faces, 40 cubic Cell (mathematics), cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic Ridge (geometry), ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-polytope, 5-dimensional polytope constructed from 10 regular Facet (geometry), facets. Related polytopes It is a part of an infinite hypercube family. The Dual polytope, dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an ''Alternation (geometry), alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
