Runcinated 5-orthoplexes
Open-knowledge reference entry
In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.
There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.
Runcinated 5-orthoplex
Alternate names
Runcinated pentacross
Small prismated triacontaditeron (Acronym: spat) (Jonathan Bowers)
Coordinates
The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,1,1,1,2)
Images
Runcitruncated 5-orthoplex
Alternate names
Runcitruncated pentacross
Prismatotruncated triacontaditeron (Acronym: pattit) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±3,±2,±1,±1,0)
Images
Runcicantellated 5-orthoplex
Alternate names
Runcicantellated pentacross
Prismatorhombated triacontaditeron (Acronym: pirt) (Jonathan Bowers)
Coordinates
The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of:
(0,1,2,2,3)
Images
Runcicantitruncated 5-orthoplex
Alternate names
Runcicantitruncated pentacross
Great prismated triacontaditeron (gippit) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of √2 are given by all permutations of coordinates and sign of:
(
0
,
1
,
2
,
3
,
4
)
{\displaystyle \left(0,1,2,3,4\right)}
Images
Snub 5-demicube
The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.
Related polytopes
These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
(Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math.
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