Runcinated 5-orthoplexes

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	t_0,3{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 [3^2,1,1]
Properties	convex

Alternate names

Runcinated pentacross
Small prismated triacontaditeron (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	t_0,1,3{3,3,3,4} t_0,1,3{3,3^1,1}
Coxeter-Dynkin diagrams
4-faces	162
Cells	1440
Faces	3680
Edges	3360
Vertices	960
Vertex figure
Coxeter groups	B₅, [3,3,3,4] D₅, [3^2,1,1]
Properties	convex

Alternate names

Runcitruncated pentacross
Prismatotruncated triacontaditeron (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	t_0,2,3{3,3,3,4} t_0,2,3{3,3,3^1,1}
Coxeter-Dynkin diagram
4-faces	162
Cells	1200
Faces	2960
Edges	2880
Vertices	960
Vertex figure
Coxeter group	B₅ [4,3,3,3] D₅ [3^2,1,1]
Properties	convex

Alternate names

Runcicantellated pentacross
Prismatorhombated triacontaditeron (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	t_0,1,2,3{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 [3^2,1,1]
Properties	convex, isogonal

Alternate names

Runcicantitruncated pentacross
Great prismated triacontaditeron (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:

\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 [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

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. 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) with acronyms". 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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsspathtm_(x3o3o3x4o_-_spat)]-1] Klitzing, (x3o3o3x4o - spat).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatspattithtm_(x3x3o3x4o_-_pattit)]-2] Klitzing, (x3x3o3x4o - pattit).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatspirthtm_(x3o3x3x4o_-_pirt)]-3] Klitzing, (x3o3x3x4o - pirt).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsgippithtm_(x3x3x3x4o_-_gippit)]-4] Klitzing, (x3x3x3x4o - gippit).

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B5 polytopes
β₅	t₁β₅	t₂γ₅	t₁γ₅	γ₅	t_0,1β₅	t_0,2β₅	t_1,2β₅
t_0,3β₅	t_1,3γ₅	t_1,2γ₅	t_0,4γ₅	t_0,3γ₅	t_0,2γ₅	t_0,1γ₅	t_0,1,2β₅
t_0,1,3β₅	t_0,2,3β₅	t_1,2,3γ₅	t_0,1,4β₅	t_0,2,4γ₅	t_0,2,3γ₅	t_0,1,4γ₅	t_0,1,3γ₅
t_0,1,2γ₅	t_0,1,2,3β₅	t_0,1,2,4β₅	t_0,1,3,4γ₅	t_0,1,2,4γ₅	t_0,1,2,3γ₅	t_0,1,2,3,4γ₅