In six-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 6-orthoplex is a convex
uniform 6-polytope, constructed as a
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 ...
(4th order truncation) of the regular
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
.
There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the
stericated 6-cubes.
Stericated 6-orthoplex
Alternate names
* Small cellated hexacontatetrapeton (Acronym: scag) (Jonathan Bowers)
Images
Steritruncated 6-orthoplex
Alternate names
* Cellitruncated hexacontatetrapeton (Acronym: catog) (Jonathan Bowers)
Images
Stericantellated 6-orthoplex
Alternate names
* Cellirhombated hexacontatetrapeton (Acronym: crag) (Jonathan Bowers)
Images
Stericantitruncated 6-orthoplex
Alternate names
* Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (Jonathan Bowers)
Images
Steriruncinated 6-orthoplex
Alternate names
* Celliprismated hexacontatetrapeton (Acronym: copog) (Jonathan Bowers)
Images
Steriruncitruncated 6-orthoplex
Alternate names
* Celliprismatotruncated hexacontatetrapeton (Acronym: captog) (Jonathan Bowers)
Images
Steriruncicantellated 6-orthoplex
Alternate names
* Celliprismatorhombated hexacontatetrapeton (Acronym: coprag) (Jonathan Bowers)
Images
Steriruncicantitruncated 6-orthoplex
Alternate names
* Great cellated hexacontatetrapeton (Acronym: gocog) (Jonathan Bowers)
Images
Snub 6-demicube
The snub 6-demicube defined as an
alternation of the omnitruncated 6-demicube is not uniform, but it can be given Coxeter diagram or and
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
2,1,1,1sup>+ or
,(3,3,3,3)+ and constructed from 12
snub 5-demicubes, 64
snub 5-simplexes, 60 snub 24-cell antiprisms, 160 3-s duoantiprisms, 240 2-sr duoantiprisms, and 11520 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 ...
es filling the gaps at the deleted vertices.
Related polytopes
These polytopes are from a set of 63
uniform 6-polytopes generated from the B
6 Coxeter plane
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...
, including the regular
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
or
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
.
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.
*
External links
Polytopes of Various Dimensions
{{Polytopes
6-polytopes