# Runcinated 5-orthoplexes

(Redirected from Runcinated 5-orthoplex)
 Orthogonal projections in B5 Coxeter plane 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

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)[1]

### Coordinates

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)[2]

### 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)[3]

### 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-Dynkindiagram 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)[4]

### 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:

${\displaystyle \left(0,1,2,3,4\right)}$

### 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 [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

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

## Notes

1. ^ Klitzing, (x3o3o3x4o - spat)
2. ^ Klitzing, (x3x3o3x4o - pattit)
3. ^ Klitzing, (x3o3x3x4o - pirt)
4. ^ Klitzing, (x3x3x3x4o - gippit)

## 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)". x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit