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6-polytope
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. Definition A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron. A 4-face is a polychoron, and a 5-face is a 5-polytope. Furthermore, the following requirements must be met: * Each 4-face must join exactly two 5-faces (facets). * Adjacent facets are not in the same five-dimensional hyperplane. * The figure is not a compound of other figures which meet the requirements. Characteristics The topology of any given 6-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most 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-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex , the 6-cube (hexeract) , and the 6-orthoplex (hexacross) . 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expanded 6-simplex
In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the ''pentisteriruncicantitruncated 6-simplex'', is called an ''omnitruncated 6-simplex'' with all of the nodes ringed. Pentellated 6-simplex Alternate names * Expanded 6-simplex * Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers) Coordinates The vertices of the ''pentellated 6-simplex'' can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on facets of the pentellated 7-orthoplex. A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosset 1 22 Polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 (mathematics), E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices). Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, constructed by positions points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122. These polytopes are from a family of 39 convex uniform 6-polytope, uniform polytopes in 6-dimensions, made of uniform 5-polytope, uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . 122 polytope The 122 polytope contains 72 vertices, and 54 5-demicube, 5-demicubic facets. It has a birectified 5-simplex vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosset 2 21 Polytope
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 (mathematics), E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an Semiregular polytope, 6-ic semi-regular figure. It is also called the Schläfli polytope. Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221. The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the triangle face centers of the 221, and is the same as the rectified 122. These polytopes are a part of family of 39 convex uniform 6-polytope, uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-simplex T05
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alternate names It can also be called a heptapeton, or hepta-6-tope, as a 7- facetted polytope in 6-dimensions. The name ''heptapeton'' is derived from ''hepta'' for seven facets in Greek and ''-peta'' for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym hop. As a configuration This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin7 & 6 & 15 & 20 & 15 & 6 \\ 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 6-orthoplex
In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a Rectification (geometry), rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex. Rectified 6-orthoplex The ''rectified 6-orthoplex'' is the vertex figure for the demihexeractic honeycomb. : or Alternate names * rectified hexacross * rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers) Construction There are two Coxeter groups associated with the ''rectified hexacross'', one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group. Cartesian coordinates Cartesian coordinates for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-simplex T0
In geometry, a 6-simplex is a Duality (mathematics), self-dual Regular polytope, regular 6-polytope. It has 7 vertex (geometry), vertices, 21 Edge (geometry), edges, 35 triangle Face (geometry), faces, 35 Tetrahedron, tetrahedral Cell (mathematics), cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alternate names It can also be called a heptapeton, or hepta-6-tope, as a 7-facet (geometry), facetted polytope in 6-dimensions. The 5-polytope#A note on generality of terms for n-polytopes and elements, name ''heptapeton'' is derived from ''hepta'' for seven Facet (mathematics), facets in Greek language, Greek and Peta-, ''-peta'' for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym hop. As a configuration This Regular 4-polytope#As configurations, configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-face ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alternate names It can also be called a heptapeton, or hepta-6-tope, as a 7- facetted polytope in 6-dimensions. The name ''heptapeton'' is derived from ''hepta'' for seven facets in Greek and ''-peta'' for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym hop. As a configuration This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin7 & 6 & 15 & 20 & 15 & 6 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular with Schläfli symbol , and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or Coxeter symbol 311. It is a part of an infinite family of polytopes, called cross-polytopes or ''orthoplexes''. The dual polytope is the 6-hypercube, or hexeract. Alternate names *Hexacross, derived from combining the family name cross polytope with ''hex'' for six (dimensions) in Greek. * Hexacontitetrapeton as a 64- facetted 6-polytope. As a configuration This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the colu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Six-dimensional Space
Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed. Mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T5
In geometry, a 6-cube is a six-dimensional hypercube with 64 Vertex (geometry), vertices, 192 Edge (geometry), edges, 240 square Face (geometry), faces, 160 cubic Cell (mathematics), cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek language, Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-polytope, 6-dimensional polytope constructed from 12 regular Facet (geometry), facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The Dual polytope, dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an ''Alternation (geometry), alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T0
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an '' alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets. As a configuration This configuration matrix represents the 6- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
