1. A7 lattice

2. Related polytopes and honeycombs
     Projection by folding  

3. See also

4. Notes

5. References

7-simplex honeycomb
(No image)
Type	Uniform 7-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[8]}
Coxeter diagram	node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
6-face types	{36} , t1{36} t2{36} , t3{36}
6-face types	{35} , t1{35} t2{35}
5-face types	{34} , t1{34} t2{34}
4-face types	{33} , t1{33}
Cell types	{3,3} , t1{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 A{{sup sub|2|7}} lattice can be constructed as the union of two A₇ lattices, and is identical to the E7 lattice.

The A{{sup sub|4|7}} lattice is the union of four A₇ lattices, which is identical to the E7* lattice (or E{{sup sub|2|7}}).

The A{{sup sub|*|7}} lattice (also called A{{sup sub|8|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.

Related polytopes and honeycombs

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:

	node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
	node_1|4|node|3|node|3|node|4|node}}

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

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}}  
  • (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]

• Grinter Place State Historic Site
• Grinties
• Grintiesu Mare River
• Grintiesu Mic River
• Grintovec, Koper
• Grintovec pri Osilnici
• Grinvalds G-801 Orion
• Grinvalds G-802 Orion
• Grinvalds Orion
• Grinvalti
• Grinzola
• Grinăuţi (disambiguation)
• Grinăuţi Moldova
• Grinăuţi-Raia
• Grinăuţi Raia
• Grinăuți
• Grinăuți-Moldova
• Grinăuți Moldova
• Grinăuți Raia
• Grinăuți-Raia
• Grinăuți, Rîșcani
• Grințieș
• Grințieșu Mare River
• Grințieșu Mic River
• Griomasaigh