Runcinated 6-demicube

6-cube	Runcinated 6-demicube	Runcitruncated 6-demicube
	Runcicantellated 6-demicube	Runcicantitruncated 6-demicube
Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a runcinated 6-demicube is a convex uniform 6-polytope with 3rd order truncations (Runcination) of the uniform 6-demicube.

There are unique 4 runcinations of the 6-demicube, including permutations of truncations, and cantellations.

Contents 1 Runcinated 6-demicube 1.1 Alternate names 1.2 Cartesian coordinates 1.3 Images 2 Runcitruncated 6-demicube 2.1 Alternate names 2.2 Cartesian coordinates 2.3 Images 3 Runcicantellated 6-demicube 3.1 Alternate names 3.2 Cartesian coordinates 3.3 Images 4 Runcicantitruncated 6-demicube 4.1 Alternate names 4.2 Cartesian coordinates 4.3 Images 5 Related polytopes 6 Notes 7 References 8 External links

Runcinated 6-demicube

Runcinated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,3{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	3360
Vertices	480
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• runcinated demihexeract
• Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)^[1]

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcinated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B6
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Runcitruncated 6-demicube

Runcitruncated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,1,3{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	12960
Vertices	2880
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)^[2]

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a runcicantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B6
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Runcicantellated 6-demicube

Runcicantellated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,2,3{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	7680
Vertices	1920
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)^[3]

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantellated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B6
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Runcicantitruncated 6-demicube

Runcicantitruncated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,1,2,3{3,32,1}
Coxeter symbol	t0,1,2,3(131)
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	17280
Vertices	5760
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)^[4]

Cartesian coordinates

The Cartesian coordinates for the 960 vertices of a runcicantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections

Coxeter plane	B6
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ symmetry, and 16 are unique:

t0(131)	t0,1(131)	t0,2(131)	t0,3(131)	t0,4(131)	t0,1,2(131)	t0,1,3(131)	t0,1,4(131)
t0,2,3(131)	t0,2,4(131)	t0,3,4(131)	t0,1,2,3(131)	t0,1,2,4(131)	t0,1,3,4(131)	t0,2,3,4(131)	t0,1,2,3,4(131)

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, editied 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.
• Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax

External links

• Weisstein, Eric W., "Hypercube" from MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes