1 42 polytope

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, .