1 33 honeycomb

DISPLAYTITLE:1 33 honeycomb In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of [[1 32 polytope|132]] facets. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing a node on the end of one of the 3-length branch leaves the 132, its only facet type. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 033. The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}. Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126. The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb. contains as a subgroup of index 144. Both and can be seen as affine extension from from different nodes: The E7* lattice (also called E72) has double the symmetry, represented by [[3,33,3]]. The Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb. The E7* lattice is constructed by 2 copies of the E7 lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A7* lattices, also called A74: ∪ = ∪ ∪ ∪ = dual of . The 133 is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The final is a noncompact hyperbolic honeycomb, 134. The rectified 133 or 0331, Coxeter diagram has facets and , and vertex figure . 8-polytope 331 honeycomb H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes) Kaleidoscopes: Selected Writings of H.

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3 31 honeycomb

DISPLAYTITLE:3 31 honeycomb In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram.

Gosset–Elte figures

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams. The Coxeter symbol for these figures has the form ki,j, where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a k length sequence of branches.

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: .