In eight-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 truncated 8-orthoplex is a convex
uniform 8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.
A uniform 8-polytope is one which is vertex-transiti ...
, being a
truncation of the regular
8-orthoplex
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells ''4-faces'', 1792 ''5-faces'', 1024 ''6-faces'', and 256 ''7-faces''.
It has two constr ...
.
There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the
tetrahedral
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.
Truncated 8-orthoplex
Alternate names
* Truncated octacross (acronym tek) (Jonthan Bowers)
Construction
There are two
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s associated with the ''truncated 8-orthoplex'', one with the C
8 or
,3,3,3,3,3,3Coxeter group, and a lower symmetry with the D
8 or
5,1,1Coxeter group.
Coordinates
Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56)
permutations of
: (±2,±1,0,0,0,0,0,0)
Images
Bitruncated 8-orthoplex
Alternate names
* Bitruncated octacross (acronym batek) (Jonthan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate
permutations of
: (±2,±2,±1,0,0,0,0,0)
Images
Tritruncated 8-orthoplex
Alternate names
* Tritruncated octacross (acronym tatek) (Jonthan Bowers)
[Klitizing, (o3o3x3x3o3o3o4o - tatek)]
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate
permutations of
: (±2,±2,±2,±1,0,0,0,0)
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. (1966)
* x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek
External links
Polytopes of Various Dimensions
{{Polytopes
8-polytopes