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 runcinated 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 3rd order truncations ( runcination) 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 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.

Runcinated 7-simplex

Alternate names

* Small prismated octaexon (acronym: spo) (Jonathan Bowers)

Coordinates

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

Images

Biruncinated 7-simplex

Alternate names

* Small biprismated octaexon (sibpo) (Jonathan Bowers)

Coordinates

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

Images

Runcitruncated 7-simplex

Alternate names

* Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)

Coordinates

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

Images

Biruncitruncated 7-simplex

Alternate names

* Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)

Coordinates

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

Images

Runcicantellated 7-simplex

Alternate names

* Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)

Coordinates

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

Images

Biruncicantellated 7-simplex

Alternate names

* Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)

Coordinates

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

Images

Runcicantitruncated 7-simplex

Alternate names

* Great prismated octaexon (acronym: gapo) (Jonathan Bowers)

Coordinates

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

Images

Biruncicantitruncated 7-simplex

Alternate names

* Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)Klitzing, (o3x3x3x3x3o3o- gibpo)

Coordinates

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

Images

Related polytopes

These polytopes are among 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. * x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo

External links

Polytopes of Various Dimensions

{{Polytopes 7-polytopes