1 33 honeycomb

{{DISPLAYTITLE:1 33 honeycomb}} 133 honeycomb(no image)TypeUniform tessellationSchläfli symbol {3,33,3}Coxeter symbol 133Coxeter-Dynkin diagram nodea 3a nodea 3a nodea 3a branch_01lr 3a nodea 3a nodea 3a nodeaor node_1 3 node split1 nodes 3ab nodes 3ab nodes7-face type132 30px6-face types122 30px131 30px5-face types121 25px{34} 25px4-face type111 25px{33} 25pxCell type101 25pxFace type{3} 25pxCell figureSquareFace figureTriangular duoprism 25pxEdge figureTetrahedral duoprismVertex figureTrirectified 7-simplex 25pxCoxeter group, [[3,33,3]]Propertiesvertex-transitive, facet-transitive In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets. It is also named pentacontahexa-hecatonicosihexa-exic heptacomb and Jonathan Bowers gives it acronym linohKlitzing (o3o3o3o3o3o3o *d3x - linoh) Construction 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. node_1 3 node split1 nodes 3ab nodes 3ab nodes Removing a node on the end of one of the 3-length branch leaves the 132, its only facet type. node_1 3 node split1 nodes 3ab nodes 3a nodea The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 033. node_1 split1 nodes 3ab nodes 3ab nodes 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}. nodes_11 3ab nodes 3ab nodes Kissing number 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. Geometric folding The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb. node_1 3 node split1 nodes 3ab nodes 3ab nodesnode_1 3 node 4 node 3 node 3 node{3,33,3}{3,3,4,3} E7* lattice contains as a subgroup of index 144.N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294 Both and can be seen as affine extension from from different nodes: File:Affine_A7_E7_relations.png The E7* lattice (also called E72)http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html The Lattice E7 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 Voronoi Cells of the E6* and E7* Lattices https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html 2016-01-30 , Edward Pervin 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: node 3 node split1 nodes 3ab nodes 3ab nodes_10l ∪ node 3 node split1 nodes 3ab nodes 3ab nodes_01l = node_1 split1 nodes 3ab nodes 3ab nodes split2 node ∪ node split1 nodes 3ab nodes_10lr 3ab nodes split2 node ∪ node split1 nodes 3ab nodes 3ab nodes split2 node_1 ∪ node split1 nodes 3ab nodes_01lr 3ab nodes split2 node = dual of node_1 3 node split1 nodes 3ab nodes 3ab nodes. Related polytopes and honeycombs 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. Rectified 133 honeycomb Rectified 133 honeycomb(no image)TypeUniform tessellationSchläfli symbol {33,3,1}Coxeter symbol 0331Coxeter-Dynkin diagram nodea 3a nodea 3a nodea 3a branch_10 3a nodea 3a nodea 3a nodeaor node 3 node_1 split1 nodes 3ab nodes 3ab nodes7-face typeTrirectified 7-simplexRectified 1326-face typesBirectified 6-simplexBirectified 6-cubeRectified 1225-face typesRectified 5-simplexBirectified 5-simplexBirectified 5-orthoplex4-face type5-cellRectified 5-cell24-cellCell type{3,3}{3,4}Face type{3}Vertex figure{}×{3,3}×{3,3}Coxeter group, [[3,33,3]]Propertiesvertex-transitive, facet-transitive The rectified 133 or 0331, Coxeter diagram node 3 node_1 split1 nodes 3ab nodes 3ab nodes has facets node_1 3 node split1 nodes 3ab nodes 3b nodeb and node_1 split1 nodes 3ab nodes 3ab nodes, and vertex figure node_1 2 nodes_11 3ab nodes 3ab nodes. Alternative names Pentacontahexa-hecatonicosihexa-pentacosiheptacontahexa-exic heptacomb Rectified pentacontahexa-hecatonicosihexa-exic heptacomb Acronym: lanquoh (Jonathan Bowers)Klitzing (o3o3o3x3o3o3o *d3o - lanquoh) See also 8-polytope 331 honeycomb Notes References H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, 0-486-61480-8 Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com, 978-0-471-01003-6 (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] flat.htm#7D 7D Heptacombs o3o3o3o3o3o3o *d3x - linoh, o3o3o3x3o3o3o *d3o - lanquoh CITEREFKlitzing Category:8-polytopes Category:Honeycombs (geometry)