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Concept
4 21 polytope
Summary
DISPLAYTITLE:4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.
Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .
The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421.
These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: .
The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 321 polytope. As its vertices represent the root vectors of the simple Lie group E8, this polytope is sometimes referred to as the E8 root polytope.
The vertices of this polytope can also be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop.
For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure. It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.
E. L. Elte named it V240 (for its 240 vertices) in his 1912 listing of semiregular polytopes.
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