In seven-dimensional

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a runcinated 7-simplex is a convex

uniform 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...

with 3rd order truncations (

runcination In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers. It is a higher order truncatio ...

) 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−1(1/ ...

. 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 A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

of the

runcinated 8-orthoplex In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers. It is a higher order truncatio ...

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

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

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

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

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

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

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

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,

*** (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