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 c ...

, a pentellated 7-orthoplex 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 5th order truncations (

pentellation In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but ...

) of the regular

7-orthoplex In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 Vertex (geometry), vertices, 84 Edge (geometry), edges, 280 triangle Face (geometry), faces, 560 tetrahedron Cell (mathematics), cells, 672 5-cells ''4-faces'', 448 ...

. There are 32 unique of the 7-orthoplex with permutations of truncations, cantellations, runcinations, and . 16 are more simply constructed relative to the

7-cube In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being c ...

. These polytopes are a part of a set of 127

s with B₇ symmetry.

Pentellated 7-orthoplex

Alternate names

* Small hecatonicosoctaexon (acronym: Staz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,1,1,1,2)

Images

Pentitruncated 7-orthoplex

Alternate names

* Teritruncated hecatonicosoctaexon (acronym: Tetaz) (Jonathan Bowers)

Images

Coordinates

Coordinates are permutations of (0,1,1,1,1,2,3).

Penticantellated 7-orthoplex

Alternate names

* Terirhombated hecatonicosoctaexon (acronym: Teroz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,1,2,2,3).

Images

Penticantitruncated 7-orthoplex

Alternate names

* Terigreatorhombated hecatonicosoctaexon (acronym: Tograz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,1,2,3,4).

Pentiruncinated 7-orthoplex

Alternate names

* Teriprismated hecatonicosoctaexon (acronym: Topaz) (Jonathan Bowers)

Coordinates

The coordinates are permutations of (0,1,1,2,2,2,3).

Images

Pentiruncitruncated 7-orthoplex

Alternate names

* Teriprismatotruncated hecatonicosoctaexon (acronym: Toptaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,2,2,3,4).

Images

Pentiruncicantellated 7-orthoplex

Alternate names

* Teriprismatorhombated hecatonicosoctaexon (acronym: Toparz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,2,3,3,4).

Images

Pentiruncicantitruncated 7-orthoplex

Alternate names

* Terigreatoprismated hecatonicosoctaexon (acronym: Tegopaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,2,3,4,5).

Images

Pentistericated 7-orthoplex

Alternate names

* Tericellated hecatonicosoctaexon (acronym: Tocaz) (Jonathan Bowers)

Images

Coordinates

Coordinates are permutations of (0,1,2,2,2,2,3).

Pentisteritruncated 7-orthoplex

Alternate names

* Tericellitruncated hecatonicosoctaexon (acronym: Tacotaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,2,2,3,4).

Images

Pentistericantellated 7-orthoplex

Alternate names

* Tericellirhombated hecatonicosoctaexon (acronym: Tocarz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,2,3,3,4).

Images

Pentistericantitruncated 7-orthoplex

Alternate names

* Tericelligreatorhombated hecatonicosoctaexon (acronym: Tecagraz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,2,3,4,5).

Images

Pentisteriruncinated 7-orthoplex

Alternate names

* Bipenticantitruncated 7-orthoplex as t_1,2,3,6 * Tericelliprismated hecatonicosoctaexon (acronym: Tecpaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,3,3,4).

Images

Pentisteriruncitruncated 7-orthoplex

Alternate names

* Tericelliprismatotruncated hecatonicosoctaexon (acronym: Tecpotaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,3,4,5).

Images

Pentisteriruncicantellated 7-orthoplex

Alternate names

* Bipentiruncicantitruncated 7-orthoplex as t_1,2,3,4,6 * Tericelliprismatorhombated hecatonicosoctaexon (acronym: Tacparez) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,4,4,5).

Images

Pentisteriruncicantitruncated 7-orthoplex

Alternate names

* Great hecatonicosoctaexon (acronym: Gotaz) (Jonathan Bowers)Klitzing, (x3x3x3x3x3x4o - )

Coordinates

Coordinates are permutations of (0,1,2,3,4,5,6).

Images

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. *

External links

Polytopes of Various Dimensions

{{Polytopes 7-polytopes