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Truncated 7-demicube Cantic 7-cube
D₇ Coxeter plane projection
Type	uniform 7-polytope
Schläfli symbol	t{3,3^4,1} h₂{4,3,3,3,3,3}
Coxeter diagram
6-faces	142
5-faces	1428
4-faces	5656
Cells	11760
Faces	13440
Edges	7392
Vertices	1344
Vertex figure	( )v{ }x{3,3,3}
Coxeter groups	D₇, [3^4,1,1]
Properties	convex

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes representing active mirrors. A demihypercube is an alternation of a hypercube.

Its 3-dimensional analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube.

Alternate names

Truncated demihepteract
Truncated hemihepteract (thesa) (Jonathan Bowers)^[1]

Cartesian coordinates

The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±3,±3,±3,±3)

with an odd number of plus signs.

Images

It can be visualized as a 2-dimensional orthogonal projections, for example the a D₇ Coxeter plane, containing 12-gonal symmetry. Most visualizations in symmetric projections will contain overlapping vertices, so the colors of the vertices are changed based on how many vertices are at each projective position, here shown with red color for no overlaps.

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

Dimensional family of cantic n-cubes
n	3	4	5	6	7	8
Symmetry [1⁺,4,3^n-2]	[1⁺,4,3] = [3,3]	[1⁺,4,3²] = [3,3^1,1]	[1⁺,4,3³] = [3,3^2,1]	[1⁺,4,3⁴] = [3,3^3,1]	[1⁺,4,3⁵] = [3,3^4,1]	[1⁺,4,3⁶] = [3,3^5,1]
Cantic figure
Coxeter	=	=	=	=	=	=
Schläfli	h₂{4,3}	h₂{4,3²}	h₂{4,3³}	h₂{4,3⁴}	h₂{4,3⁵}	h₂{4,3⁶}

There are 95 uniform polytopes with D₆ symmetry, 63 are shared by the B₆ symmetry, and 32 are unique:

Notes

^ Klitzing, (x3x3o *b3o3o3o3o - thesa)

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. "7D uniform polytopes (polyexa) x3x3o *b3o3o3o3o – thesa".

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	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

[1] Klitzing, (x3x3o *b3o3o3o3o - thesa)

[1]