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E6 Polytope
In 6-dimensional geometry, there are 39 uniform 6-polytope, uniform polytopes with E6 symmetry. The two simplest forms are the 2_21 polytope, 221 and 1_22 polytope, 122 polytopes, composed of 27 and 72 vertex (geometry), vertices respectively. They can be visualized as symmetric orthographic projections in Coxeter planes of the E6 Coxeter group, and other subgroups. Graphs Symmetric orthographic projections of these 39 polytopes can be made in the E6, D5, D4, D2, A5, A4, A3 Coxeter planes. Ak has ''k+1'' symmetry, Dk has ''2(k-1)'' symmetry, and E6 has ''12'' symmetry. Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E6 symmetry. The vertices and edges drawn with 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 Edition, Dover New York, 1973 * Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arth ... [...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] |
Rectified 2 21
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 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 polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . 2_21 polytope The 221 has 27 verti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
