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List Of A7 Polytopes
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices. Each can be visualized as symmetric orthographic projections in Coxeter planes of the A7 Coxeter group, and other subgroups. __TOC__ Graphs Symmetric orthographic projections of these 71 polytopes can be made in the A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has '' +1' symmetry. For even ''k'' and symmetrically ringed-diagrams, symmetry doubles to '' (k+1)'. These 71 polytopes are each shown in these 6 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position. References * H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...: ** H.S.M. Coxeter, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthographic Projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are ''not'' orthogonal to the projection plane. The term ''orthographic'' sometimes means a technique in multiview projection in which principal axes or the planes of the subject are also parallel with the projection plane to create the ''primary views''. If the principal planes or axes of an object in an orthographic projection are ''not'' parallel with the projection plane, the depiction is called ''axonometric'' or an ''auxiliary views''. (''Axonometric projection'' is synonymous with ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-simplex T1
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°. Alternate names It can also be called an octaexon, or octa-7-tope, as an 8- facetted polytope in 7-dimensions. The name ''octaexon'' is derived from ''octa'' for eight facets in Greek and ''-ex'' for having six-dimensional facets, and ''-on''. Jonathan Bowers gives an octaexon the acronym oca. As a configuration This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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. \beg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trirectified 7-simplex
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the ''rectified 7-simplex'' are located at the edge-centers of the ''7-simplex''. Vertices of the ''birectified 7-simplex'' are located in the triangular face centers of the ''7-simplex''. Vertices of the ''trirectified 7-simplex'' are located in the tetrahedral cell centers of the ''7-simplex''. Rectified 7-simplex The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as . E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. Alternate names * Rectified octaexon (Acronym: roc) (Jonathan Bowers) Coordinates The vertices of the ''rectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-simplex T2 A2
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°. Alternate names It can also be called an octaexon, or octa-7-tope, as an 8- facetted polytope in 7-dimensions. The name ''octaexon'' is derived from ''octa'' for eight facets in Greek and ''-ex'' for having six-dimensional facets, and ''-on''. Jonathan Bowers gives an octaexon the acronym oca. As a configuration This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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. \be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-simplex T2
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°. Alternate names It can also be called an octaexon, or octa-7-tope, as an 8- facetted polytope in 7-dimensions. The name A name is a term used for identification by an external observer. They can identify a class or category of things, or a single thing, either uniquely, or within a given context. The entity identified by a name is called its referent. A personal ... ''octaexon'' is derived from ''octa'' for eight facets in Greek and ''-ex'' for having six-dimensional facets, and ''-on''. Jonathan Bowers gives an octaexon the acronym oca. As a configuration This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birectified 7-simplex
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the ''rectified 7-simplex'' are located at the edge-centers of the ''7-simplex''. Vertices of the ''birectified 7-simplex'' are located in the triangular face centers of the ''7-simplex''. Vertices of the ''trirectified 7-simplex'' are located in the tetrahedral cell centers of the ''7-simplex''. Rectified 7-simplex The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as . E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. Alternate names * Rectified octaexon (Acronym: roc) (Jonathan Bowers) Coordinates The vertices of the ''rectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-simplex T1 A2
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°. Alternate names It can also be called an octaexon, or octa-7-tope, as an 8- facetted polytope in 7-dimensions. The name ''octaexon'' is derived from ''octa'' for eight facets in Greek and ''-ex'' for having six-dimensional facets, and ''-on''. Jonathan Bowers gives an octaexon the acronym oca. As a configuration This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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. \be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
