In seven-dimensional

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a stericated 7-simplex is a convex

uniform 7-polytope In seven-dimensional space, seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope Ridge (geometry), ridge being shared by exactly two 6-polytope Facet (mathematics), facets. A uniform 7-polytope is ...

with 4th order truncations (

sterication In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...

) of the regular

7-simplex In 7-dimensional geometry, a 7- simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos ...

. There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations.

Stericated 7-simplex

Alternate names

* Small cellated octaexon (acronym: sco) (Jonathan Bowers)

Coordinates

The vertices of the ''stericated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 8-orthoplex.

Images

Bistericated 7-simplex

Alternate names

* Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)

Coordinates

The vertices of the ''bistericated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 8-orthoplex.

Images

Steritruncated 7-simplex

Alternate names

* Cellitruncated octaexon (acronym: cato) (Jonathan Bowers)

Coordinates

The vertices of the ''steritruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 8-orthoplex.

Images

Bisteritruncated 7-simplex

Alternate names

* Bicellitruncated octaexon (acronym: bacto) (Jonathan Bowers)

Coordinates

The vertices of the ''bisteritruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,3). This construction is based on facets of the bisteritruncated 8-orthoplex.

Images

Stericantellated 7-simplex

Alternate names

* Cellirhombated octaexon (acronym: caro) (Jonathan Bowers)

Coordinates

The vertices of the ''stericantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 8-orthoplex.

Images

Bistericantellated 7-simplex

Alternate names

* Bicellirhombihexadecaexon (acronym: bacroh) (Jonathan Bowers)

Coordinates

The vertices of the ''bistericantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,3). This construction is based on facets of the stericantellated 8-orthoplex.

Images

Stericantitruncated 7-simplex

Alternate names

* Celligreatorhombated octaexon (acronym: cagro) (Jonathan Bowers)

Coordinates

The vertices of the ''stericantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,4). This construction is based on facets of the stericantitruncated 8-orthoplex.

Images

Bistericantitruncated 7-simplex

Alternate names

* Bicelligreatorhombated octaexon (acronym: bacogro) (Jonathan Bowers)

Coordinates

The vertices of the ''bistericantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,4). This construction is based on facets of the bistericantitruncated 8-orthoplex.

Images

Steriruncinated 7-simplex

Alternate names

* Celliprismated octaexon (acronym: cepo) (Jonathan Bowers)

Coordinates

The vertices of the ''steriruncinated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the steriruncinated 8-orthoplex.

Images

Steriruncitruncated 7-simplex

Alternate names

* Celliprismatotruncated octaexon (acronym: capto) (Jonathan Bowers)

Coordinates

The vertices of the ''steriruncitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,4). This construction is based on facets of the steriruncitruncated 8-orthoplex.

Images

Steriruncicantellated 7-simplex

Alternate names

* Celliprismatorhombated octaexon (acronym: capro) (Jonathan Bowers)

Coordinates

The vertices of the ''steriruncicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,3,4). This construction is based on facets of the steriruncicantellated 8-orthoplex.

Images

Bisteriruncitruncated 7-simplex

Alternate names

* Bicelliprismatotruncated hexadecaexon (acronym: bicpath) (Jonathan Bowers)

Coordinates

The vertices of the ''bisteriruncitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the bisteriruncitruncated 8-orthoplex.

Images

Steriruncicantitruncated 7-simplex

Alternate names

* Great cellated octaexon (acronym: gecco) (Jonathan Bowers)

Coordinates

The vertices of the ''steriruncicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 8-orthoplex.

Images

Bisteriruncicantitruncated 7-simplex

Alternate names

* Great bicellated hexadecaexon (gabach) (Jonathan Bowers) Klitizing, (o3x3x3x3x3x3o - gabach)

Coordinates

The vertices of the ''bisteriruncicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,5). This construction is based on facets of the bisteriruncicantitruncated 8-orthoplex.

Images

Related polytopes

This polytope is one of 71

s with A₇ symmetry.

Notes

References

* H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 ** 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
wiley.com
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380–407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. * x3o3o3o3x3o3o - sco, o3x3o3o3o3x3o - sabach, x3x3o3o3x3o3o - cato, o3x3x3o3o3x3o - bacto, x3o3x3o3x3o3o - caro, o3x3o3x3o3x3o - bacroh, x3x3x3o3x3o3o - cagro, o3x3x3x3o3x3o - bacogro, x3o3o3x3x3o3o - cepo, x3x3x3o3x3o3o - capto, x3o3x3x3x3o3o - capro, o3x3x3o3x3x3o - bicpath, x3x3x3x3x3o3o - gecco, o3x3x3x3x3x3o - gabach

External links

Polytopes of Various Dimensions

{{Polytopes 7-polytopes