7-simplex honeycomb
7-simplex honeycomb | |
---|---|
(No image) | |
Type | Uniform 7-honeycomb |
Family | Simplectic honeycomb |
Schläfli symbol | {3[8]} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-face types | {36} ![]() ![]() t2{36} ![]() ![]() |
6-face types | {35} ![]() ![]() t2{35} ![]() |
5-face types | {34} ![]() ![]() t2{34} ![]() |
4-face types | {33} ![]() ![]() |
Cell types | {3,3} ![]() ![]() |
Face types | {3} ![]() |
Vertex figure | t0,6{36} ![]() |
Symmetry | ×21, <[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
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 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.
contains as a subgroup of index 144.[2] Both and can be seen as affine extensions from from different nodes:
The A2
7 lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.
∪
=
.
The A4
7 lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E2
7).
∪
∪
∪
=
+
= dual of
.
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.
∪
∪
∪
∪
∪
∪
∪
= dual of
.
Related polytopes and honeycombs
This honeycomb is one of 29 unique uniform honeycombs[3] constructed by the Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
A7 honeycombs | ||||
---|---|---|---|---|
Octagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
a1![]() |
[3[8]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
| |
d2![]() |
<[3[8]]> | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×21 |
|
p2![]() |
[[3[8]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×22 | |
d4![]() |
<2[3[8]]> | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×41 |
|
p4![]() |
[2[3[8]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×42 |
|
d8![]() |
[4[3[8]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×8 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
r16![]() |
[8[3[8]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×16 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Projection by folding
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:
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
See also
Regular and uniform honeycombs in 7-space:
- 7-cubic honeycomb
- 7-demicubic honeycomb
- Truncated 7-simplex honeycomb
- Omnitruncated 7-simplex honeycomb
- E7 honeycomb
Notes
- ^ "The Lattice A7".
- ^ N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
- ^ Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 30-1 cases, skipping one with zero marks
References
- 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]
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |