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Concept
1 42 polytope
Summary
DISPLAYTITLE:1 42 polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.
The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421.
These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex figures, defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram: .
The 142 is composed of 2400 facets: 240 132 polytopes, and 2160 7-demicubes (141). Its vertex figure is a birectified 7-simplex.
This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 152, and Coxeter-Dynkin diagram: .
E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V17280 for its 17280 vertices.
Coxeter named it 142 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
Diacositetracont-dischiliahectohexaconta-zetton (acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers)
The 17280 vertices can be defined as sign and location permutations of:
All sign combinations (32): (280×32=8960 vertices)
(4, 2, 2, 2, 2, 0, 0, 0)
Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)
(2, 2, 2, 2, 2, 2, 2, 2)
(5, 1, 1, 1, 1, 1, 1, 1)
(3, 3, 3, 1, 1, 1, 1, 1)
The edge length is 2 in this coordinate set, and the polytope radius is 4.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the end of the 2-length branch leaves the 7-demicube, 141, .
Removing the node on the end of the 4-length branch leaves the 132, .
Official source
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