Runcinated 5-orthoplexes

5-orthoplex	Runcinated 5-orthoplex	Runcinated 5-cube
Runcitruncated 5-orthoplex	Runcicantellated 5-orthoplex	Runcicantitruncated 5-orthoplex
Runcitruncated 5-cube	Runcicantellated 5-cube	Runcicantitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

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

Runcinated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t0,3{3,3,3,4}
Coxeter-Dynkin diagram
4-faces	162
Cells	1200
Faces	2160
Edges	1440
Vertices	320
Vertex figure
Coxeter group	B5 [4,3,3,3] D5 [32,1,1]
Properties	convex

Alternate names

• Runcinated pentacross
• Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,1,1,1,2)

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Runcitruncated 5-orthoplex

Runcitruncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	t0,1,3{3,3,3,4} t0,1,3{3,31,1}
Coxeter-Dynkin diagrams
4-faces	162
Cells	1440
Faces	3680
Edges	3360
Vertices	960
Vertex figure
Coxeter groups	B5, [3,3,3,4] D5, [32,1,1]
Properties	convex

Alternate names

• Runcitruncated pentacross
• Prismatotruncated triacontiditeron (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

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Runcicantellated 5-orthoplex

Runcicantellated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3{3,3,3,4} t0,2,3{3,3,31,1}
Coxeter-Dynkin diagram
4-faces	162
Cells	1200
Faces	2960
Edges	2880
Vertices	960
Vertex figure
Coxeter group	B5 [4,3,3,3] D5 [32,1,1]
Properties	convex

Alternate names

• Runcicantellated pentacross
• Prismatorhombated triacontiditeron (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

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Runcicantitruncated 5-orthoplex

Runcicantitruncated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	t0,1,2,3{3,3,3,4}
Coxeter-Dynkin diagram
4-faces	162
Cells	1440
Faces	4160
Edges	4800
Vertices	1920
Vertex figure	Irregular 5-cell
Coxeter groups	B5 [4,3,3,3] D5 [32,1,1]
Properties	convex, isogonal

Alternate names

• Runcicantitruncated pentacross
• Great prismated triacontiditeron (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:

[math]\displaystyle{ \left(0, 1, 2, 3, 4\right) }[/math]

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

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 [3^2,1,1]⁺ 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

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

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, ISBN 978-0-471-01003-6 [1]
    • (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. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary