7-simplex honeycomb

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7-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[8]}
Coxeter diagram CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
6-face types {36} 7-simplex t0.svg, t1{36} 7-simplex t1.svg
t2{36} 7-simplex t2.svg, t3{36} 7-simplex t3.svg
6-face types {35} 6-simplex t0.svg, t1{35} 6-simplex t1.svg
t2{35} 6-simplex t2.svg
5-face types {34} 5-simplex t0.svg, t1{34} 5-simplex t1.svg
t2{34} 5-simplex t2.svg
4-face types {33} 4-simplex t0.svg, t1{33} 4-simplex t1.svg
Cell types {3,3} 3-simplex t0.svg, t1{3,3} 3-simplex t1.svg
Face types {3} 2-simplex t0.svg
Vertex figure t0,6{36} 7-simplex t06.svg
Symmetry {\tilde{A}}_7×2, <[3[8]]>
Properties vertex-transitive

In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

A7 lattice[edit]

This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the {\tilde{A}}_7 Coxeter group.[1] It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle.

{\tilde{E}}_7 contains {\tilde{A}}_7 as a subgroup of index 144.[2] Both {\tilde{E}}_7 and {\tilde{A}}_7 can be seen as affine extensions from A_7 from different nodes: Affine A7 E7 relations.png

The A2
7
lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png = CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png.

The A4
7
lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E2
7
).

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png.

The A*
7
lattice (also called A8
7
) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel split2.pngCDel node.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png.

Related polytopes and honeycombs[edit]

This honeycomb is one of 29 unique uniform honeycombs[3] constructed by the {\tilde{A}}_7 Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

Octagon
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycombs
a1 [3[8]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png ×1

CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png

d2 <[3[8]]> CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel nodeab c4.pngCDel split2.pngCDel node c5.png ×2

CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png1 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png

p2 [[3[8]]] CDel branch c1.pngCDel 3ab.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel branch c4.png ×2

CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png 2 CDel branch.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png

d4 <2[3[8]]> CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c3.pngCDel 3ab.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png ×4

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node.png

p4 [2[3[8]]] CDel branch c1.pngCDel 3ab.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c2.pngCDel 3ab.pngCDel branch c1.png ×4

CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch 11.png CDel branch.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch.png

d8 [4[3[8]]] CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png ×8 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png
r16 [8[3[8]]] CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel 3ab.pngCDel nodeab c1.pngCDel split2.pngCDel node c1.png ×16 CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png3

Projection by folding[edit]

The 7-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

{\tilde{A}}_7 CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
{\tilde{C}}_4 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

See also[edit]

Regular and uniform honeycombs in 7-space:

Notes[edit]

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html
  2. ^ N.W. Johnson: Geometries and Transformations, (2015) Chapter 12: Euclidean symmetry groups, p 177
  3. ^ Weisstein, Eric W., "Necklace", MathWorld., A000029 30-1 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]