Pentellated 6-cube
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In six-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 6-cube is a convex
uniform 6-polytope 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, bu ...
with 5th order truncations of the regular
6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
. There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an
expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...
operation applied to the regular
6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
. The highest form, the ''pentisteriruncicantitruncated 6-cube'', is called an ''omnitruncated 6-cube'' with all of the nodes ringed. Six of them are better constructed from the
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular with ...
given at pentellated 6-orthoplex.


Pentellated 6-cube


Alternate names

* Pentellated 6-orthoplex * Expanded 6-cube, expanded 6-orthoplex * Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)


Images


Pentitruncated 6-cube


Alternate names

* Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers)


Images


Penticantellated 6-cube


Alternate names

* Terirhombated hexeract (Acronym: topag) (Jonathan Bowers)


Images


Penticantitruncated 6-cube


Alternate names

* Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers)


Images


Pentiruncitruncated 6-cube


Alternate names

* Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers)


Images


Pentiruncicantellated 6-cube


Alternate names

* Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)


Images


Pentiruncicantitruncated 6-cube


Alternate names

* Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers)


Images


Pentisteritruncated 6-cube


Alternate names

* Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)


Images


Pentistericantitruncated 6-cube


Alternate names

* Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers)


Images


Omnitruncated 6-cube

The omnitruncated 6-cube has 5040 vertices, 15120
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
s, 16800
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
(4200
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...
s and 1260
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
), 8400
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35
uniform 6-polytope 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, bu ...
s generated from the regular
6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
.


Alternate names

* Pentisteriruncicantituncated 6-cube or 6-orthoplex (
omnitruncation In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortc ...
for 6-polytopes) * Omnitruncated hexeract * Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)Klitzing, (x4x3x3x3x3x - gotaxog)


Images


Full snub 6-cube

The full snub 6-cube or omnisnub 6-cube, defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram and
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
,3,3,3,3sup>+, and constructed from 12 snub 5-cubes, 64 snub 5-simplexes, 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr duoantiprisms, 240 4-s duoantiprisms, and 23040 irregular
5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The 5-s ...
es filling the gaps at the deleted vertices.


Related polytopes

These polytopes are from a set of 63
uniform 6-polytope 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, bu ...
s generated from the B6
Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
, including the regular
6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...
or
6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular with ...
.


Notes


References

*
H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
: ** 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. * x4o3o3o3o3x - stoxog, x4x3o3o3o3x - tacog, x4o3x3o3o3x - topag, x4x3x3o3o3x - togrix, x4x3o3x3o3x - tocrag, x4o3x3x3o3x - tiprixog, x4x3x3o3x3x - tagpox, x4x3o3o3x3x - tactaxog, x4x3x3o3x3x - tocagrax, x4x3x3x3x3x - gotaxog


External links

*
Polytopes of Various Dimensions


{{Polytopes 6-polytopes