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List Of E7 Polytopes
In 7-dimensional geometry, there are 127 uniform 7-polytope, uniform polytopes with E7 symmetry. The three simplest forms are the 3_21 polytope, 321, 2_31 polytope, 231, and 1_32 polytope, 132 polytopes, composed of 56, 126, and 576 vertex (geometry), vertices respectively. They can be visualized as symmetric orthographic projections in Coxeter planes of the E7 Coxeter group, and other subgroups. Graphs Symmetric orthographic projections of these 127 polytopes can be made in the E7, E6, D6, D5, D4, D3, A6, A5, A4, A3, A2 Coxeter planes. Ak has ''k+1'' symmetry, Dk has ''2(k-1)'' symmetry, and E6 and E7 have ''12'', ''18'' symmetry respectively. For 10 of 127 polytopes (7 single rings, and 3 truncations), they are shown in these 9 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position. References * Harold Scott MacDonald Coxeter, H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthographic Projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, 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''. (''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosset 2 31 Polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch. The rectified 231 is constructed by points at the mid-edges of the 231. These polytopes are part of a family of 127 (or 27−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . 2_31 polytope The 231 is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces ( 3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E7. This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 331. Alternate names * E. L. Elte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
