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Omnitruncated 8-simplex honeycomb

April 12, 2024

Omnitruncated 8-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Omnitruncated simplectic honeycomb
Schläfli symbol {3[9]}
Coxeter–Dynkin diagrams
7-face types t01234567{3,3,3,3,3,3,3}
Vertex figure
Irr. 8-simplex
Symmetry ×18, [9[3[9]]]
Properties vertex-transitive

In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-simplex facets.

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

A*
8 lattice edit

The A*
8 lattice (also called A9
8) is the union of nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

          ∪           ∪           ∪           ∪           ∪           ∪           ∪           ∪           = dual of          .

Related polytopes and honeycombs edit

This honeycomb is one of 45 unique uniform honeycombs[1] constructed by the   Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs
Enneagon
symmetry 		Symmetry Extended
diagram 		Extended
group 		Honeycombs
a1 [3[9]]            

                                                                     

                                                                     

i2 [[3[9]]]            ×2

         1                               2

                                       

                                                           

                                                           

                             

                                       

         

i6 [3[3[9]]]            ×6                    
r18 [9[3[9]]]            ×18           3

See also edit

Regular and uniform honeycombs in 8-space:

Notes edit

  1. ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 46-1 cases, skipping one with zero marks

References edit

  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family           /   /  
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21

April 12, 2024
omnitruncated, simplex, honeycomb, image, type, uniform, honeycombfamily, omnitruncated, simplectic, honeycombschläfli, symbol, coxeter, dynkin, diagrams7, face, types, t01234567, vertex, figure, simplexsymmetry, displaystyle, tilde, properties, vertex, transi. Omnitruncated 8 simplex honeycomb No image Type Uniform honeycombFamily Omnitruncated simplectic honeycombSchlafli symbol 3 9 Coxeter Dynkin diagrams7 face types t01234567 3 3 3 3 3 3 3 Vertex figure Irr 8 simplexSymmetry A 9 displaystyle tilde A 9 18 9 3 9 Properties vertex transitiveIn eight dimensional Euclidean geometry the omnitruncated 8 simplex honeycomb is a space filling tessellation or honeycomb It is composed entirely of omnitruncated 8 simplex facets The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n 1 space with integral coordinates permutations of the whole numbers 0 1 n Contents 1 A 8 lattice 2 Related polytopes and honeycombs 3 See also 4 Notes 5 ReferencesA 8 lattice editThe A 8 lattice also called A98 is the union of nine A8 lattices and has the vertex arrangement of the dual honeycomb to the omnitruncated 8 simplex honeycomb and therefore the Voronoi cell of this lattice is an omnitruncated 8 simplex nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp dual of nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Related polytopes and honeycombs editThis honeycomb is one of 45 unique uniform honeycombs 1 constructed by the A 8 displaystyle tilde A 8 nbsp Coxeter group The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams A8 honeycombsEnneagonsymmetry Symmetry Extendeddiagram Extendedgroup Honeycombsa1 3 9 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp A 8 displaystyle tilde A 8 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp i2 3 9 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp A 8 displaystyle tilde A 8 nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp i6 3 3 9 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp A 8 displaystyle tilde A 8 nbsp 6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp r18 9 3 9 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp A 8 displaystyle tilde A 8 nbsp 18 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3See also editRegular and uniform honeycombs in 8 space 8 cubic honeycomb 8 demicubic honeycomb 8 simplex honeycomb Truncated 8 simplex honeycomb 521 honeycomb 251 honeycomb 152 honeycombNotes edit Weisstein Eric W Necklace MathWorld OEIS sequence A000029 46 1 cases skipping one with zero marksReferences editNorman Johnson Uniform Polytopes Manuscript 1991 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 22 H S M Coxeter Regular and Semi Regular Polytopes I Math Zeit 46 1940 380 407 MR 2 10 1 9 Uniform space fillings 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 nbsp C n 1 displaystyle tilde C n 1 nbsp B n 1 displaystyle tilde B n 1 nbsp D n 1 displaystyle tilde D n 1 nbsp G 2 displaystyle tilde G 2 nbsp F 4 displaystyle tilde F 4 nbsp E n 1 displaystyle tilde E n 1 nbsp 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 Omnitruncated 8 simplex honeycomb amp oldid 786603915, wikipedia, wiki, book, books, library,

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