|
Uniform 4-honeycomb
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 Facet (geometry), 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 Coxeter groups, 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 Dimension ... [...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, nonconvex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 5-orthoplex
In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex. There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube. Truncated 5-orthoplex Alternate names * Truncated pentacross * Truncated triacontiditeron (Acronym: tot) (Jonathan Bowers) Coordinates Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of : (±2,±1,0,0,0) Images The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge. ... [...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 ... [...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] |
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 Euclidea ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T03
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 Euclidea ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T02
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 Euclidea ... [...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 * Runci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T14
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 Euclidea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 5-orthoplex
In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex. There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube. Cantellated 5-orthoplex Alternate names * Cantellated 5-orthoplex * Bicantellated 5-demicube * Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers) Coordinates The vertices of the can be made in 5-space, as permutations and sign combinations of: : (0,0,1,1,2) Images The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex. Cantitruncated 5-orthoplex Alternate names * Cantitruncated pentacross * Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)Klitizing, (x3x3x3o4o - gart) Coordinates Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
