Runcinated 5-simplexes

5-simplex	Runcinated 5-simplex	Runcitruncated 5-simplex
Birectified 5-simplex	Runcicantellated 5-simplex	Runcicantitruncated 5-simplex
Coxeter plane

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

There are 4 unique runcinations of the 5-simplex with

.

Runcinated 5-simplex

Runcinated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t0,3{3,3,3} 20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3}
Cells	255	45 {3,3} 180 { }×{3} 30 r{3,3}
Faces	420	240 {4}
Edges	270
Vertices	60
Vertex figure
Coxeter group	A5 [3,3,3,3], order 720
Properties	convex

Alternate names

• Runcinated hexateron
• Small prismated hexateron (Acronym: spix) (Jonathan Bowers)^[1]

Coordinates

The vertices of the runcinated 5-simplex can be most simply constructed on a

respectively.

Images

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Runcitruncated 5-simplex

Runcitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,1,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 {3}×{6} 15 { }×r{3,3} 6 rr{3,3,3}
Cells	315
Faces	720
Edges	630
Vertices	180
Vertex figure
Coxeter group	A5 [3,3,3,3], order 720
Properties	convex, isogonal

Alternate names

• Runcitruncated hexateron
• Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)^[2]

Coordinates

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,1,2,3)

This construction exists as one of 64

.

Images

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Runcicantellated 5-simplex

Runcicantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47
Cells	255
Faces	570
Edges	540
Vertices	180
Vertex figure
Coxeter group	A5 [3,3,3,3], order 720
Properties	convex, isogonal

Alternate names

• Runcicantellated hexateron
• Biruncitruncated 5-simplex/hexateron
• Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)^[3]

Coordinates

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,2,2,3)

This construction exists as one of 64

.

Images

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Runcicantitruncated 5-simplex

Runcicantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,1,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 {}×t{3,3} 6 tr{3,3,3}
Cells	315	45 t0,1,2{3,3} 120 { }×{3} 120 { }×{6} 30 t{3,3}
Faces	810	120 {3} 450 {4} 240 {6}
Edges	900
Vertices	360
Vertex figure	Irregular 5-cell
Coxeter group	A5 [3,3,3,3], order 720
Properties	convex, isogonal

Alternate names

• Runcicantitruncated hexateron
• Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)^[4]

Coordinates

The coordinates can be made in 6-space, as 360 permutations of:

(0,0,1,2,3,4)

This construction exists as one of 64

.

Images

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Related uniform 5-polytopes

These polytopes are in a set of 19

. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

A5 polytopes
t0	t1	t2	t0,1	t0,2	t1,2	t0,3
t1,3	t0,4	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4
t0,2,4	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,1,2,3,4

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,
    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)". x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix

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	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: List of regular polytopes and compounds