There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,35,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,34,31,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}7.
The [4,35,4], , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.
A quadritruncated 7-cubic honeycomb, , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D7* lattice. Facets can be identically colored from a doubled ×2, [[4,35,4]] symmetry, alternately colored from , [4,35,4] symmetry, three colors from , [4,34,31,1] symmetry, and 4 colors from , [31,1,33,31,1] symmetry.
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, ISBN978-0-471-01003-6[1]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
cubic, honeycomb, image, type, regular, honeycombuniform, honeycombfamily, hypercube, honeycombschläfli, symbol, 7coxeter, dynkin, diagrams7, face, type, face, type, face, type, face, type, cell, type, face, type, face, figure, octahedron, edge, figure, cell, . 7 cubic honeycomb no image Type Regular 7 honeycombUniform 7 honeycombFamily Hypercube honeycombSchlafli symbol 4 35 4 4 34 31 1 7Coxeter Dynkin diagrams7 face type 4 3 3 3 3 3 6 face type 4 3 3 3 3 5 face type 4 3 3 3 4 face type 4 3 3 Cell type 4 3 Face type 4 Face figure 4 3 octahedron Edge figure 8 4 3 3 16 cell Vertex figure 128 4 35 7 orthoplex Coxeter group 4 35 4 Dual self dualProperties vertex transitive edge transitive face transitive cell transitiveThe 7 cubic honeycomb or hepteractic honeycomb is the only regular space filling tessellation or honeycomb in Euclidean 7 space It is analogous to the square tiling of the plane and to the cubic honeycomb of 3 space There are many different Wythoff constructions of this honeycomb The most symmetric form is regular with Schlafli symbol 4 35 4 Another form has two alternating 7 cube facets like a checkerboard with Schlafli symbol 4 34 31 1 The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schlafli symbol 7 Contents 1 Related honeycombs 1 1 Quadritruncated 7 cubic honeycomb 2 See also 3 ReferencesRelated honeycombs EditThe 4 35 4 Coxeter group generates 255 permutations of uniform tessellations 135 with unique symmetry and 134 with unique geometry The expanded 7 cubic honeycomb is geometrically identical to the 7 cubic honeycomb The 7 cubic honeycomb can be alternated into the 7 demicubic honeycomb replacing the 7 cubes with 7 demicubes and the alternated gaps are filled by 7 orthoplex facets Quadritruncated 7 cubic honeycomb Edit A quadritruncated 7 cubic honeycomb contains all tritruncated 7 orthoplex facets and is the Voronoi tessellation of the D7 lattice Facets can be identically colored from a doubled C 7 displaystyle tilde C 7 2 4 35 4 symmetry alternately colored from C 7 displaystyle tilde C 7 4 35 4 symmetry three colors from B 7 displaystyle tilde B 7 4 34 31 1 symmetry and 4 colors from D 7 displaystyle tilde D 7 31 1 33 31 1 symmetry See also EditList of regular polytopesReferences EditCoxeter H S M Regular Polytopes 3rd edition 1973 Dover edition ISBN 0 486 61480 8 p 296 Table II Regular honeycombs 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 ISBN 978 0 471 01003 6 1 Paper 24 H S M Coxeter Regular and Semi Regular Polytopes III Math Zeit 200 1988 3 45 vteFundamental convex regular and uniform honeycombs in dimensions 2 9Space Family A n 1 displaystyle tilde A n 1 C n 1 displaystyle tilde C n 1 B n 1 displaystyle tilde B n 1 D n 1 displaystyle tilde D n 1 G 2 displaystyle tilde G 2 F 4 displaystyle tilde F 4 E n 1 displaystyle tilde E n 1 E2 Uniform tiling 3 3 d3 hd3 qd3 HexagonalE3 Uniform convex honeycomb 3 4 d4 hd4 qd4E4 Uniform 4 honeycomb 3 5 d5 hd5 qd5 24 cell honeycombE5 Uniform 5 honeycomb 3 6 d6 hd6 qd6E6 Uniform 6 honeycomb 3 7 d7 hd7 qd7 222E7 Uniform 7 honeycomb 3 8 d8 hd8 qd8 133 331E8 Uniform 8 honeycomb 3 9 d9 hd9 qd9 152 251 521E9 Uniform 9 honeycomb 3 10 d10 hd10 qd10E10 Uniform 10 honeycomb 3 11 d11 hd11 qd11En 1 Uniform n 1 honeycomb 3 n dn hdn qdn 1k2 2k1 k21 Retrieved from https en wikipedia org w index php title 7 cubic honeycomb amp oldid 1001439495, wikipedia, wiki, book, books, library,