1 32 polytope

DISPLAYTITLE:1 32 polytope In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences. The rectified 132 is constructed by points at the mid-edges of the 132. These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E7* lattice. Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes. Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers) It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-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 6-demicube, 131, Removing the node on the end of the 3-length branch leaves the 122, The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 032, Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134. The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.