4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
[1] Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .
The birectified 421 is constructed by points at the triangle face 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 vertices of this polytope can also be obtained by taking the 240 integral octonions of norm 1.
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).
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The 240 vertices of the 421 polytope can be constructed in two sets: 112 (22 × 8C2) with coordinates obtained from
by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be a multiple of 4).
Another decomposition gives the 240 points in 9-dimensions as an expanded 8-simplex, and two opposite birectified 8-simplexes, and .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
[4] These graphs represent orthographic projections in the E8, E7, E6, and B8, D8, D7, D6, D5, D4, D3, A7, A5 Coxeter planes.
The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).
Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.
240 321 17280 t1{36} 2160 t1{35,4} The rectified 421 can be seen as a rectification of the 421 polytope, creating new vertices on the center of edges of the 421.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 421.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
Removing the node on the short branch leaves the birectified 7-simplex.
Removing the node on the end of the 2-length branch leaves the birectified 7-orthoplex in its alternated form.
Removing the node on the end of the 4-length branch leaves the rectified 321.
These graphs represent orthographic projections in the E8, E7, E6, and B8, D8, D7, D6, D5, D4, D3, A7, A5 Coxeter planes.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
These graphs represent orthographic projections in the E7, E6, B8, D8, D7, D6, D5, D4, D3, A7, and A5 Coxeter planes.
