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Demihepteract (7-demicube)
Petrie polygon projection
Type	Uniform 7-polytope
Family	demihypercube
Coxeter symbol	1₄₁
Schläfli symbol	{3,3^4,1} = h{4,3⁵} s{2^1,1,1,1,1,1}
Coxeter diagrams	=
6-faces	78	14 {3^1,3,1} 64 {3⁵}
5-faces	532	84 {3^1,2,1} 448 {3⁴}
4-faces	1624	280 {3^1,1,1} 1344 {3³}
Cells	2800	560 {3^1,0,1} 2240 {3,3}
Faces	2240	{3}
Edges	672
Vertices	64
Vertex figure	Rectified 6-simplex
Symmetry group	D₇, [3^4,1,1] = [1⁺,4,3⁵] [2⁶]⁺
Dual	?
Properties	convex

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₇ for a 7-dimensional half measure polytope.

Coxeter named this polytope as 1₄₁ from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol $\left\{3{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$ or {3,3^4,1}.

Cartesian coordinates

Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:

(±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images

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]

As a configuration

This configuration matrix represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[1]^[2]

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^[3]

D₇	k-face	f_k	f₀	f₁	f₂	f₃		f₄		f₅		f₆		k-figures	notes
A₆	( )	f₀	64	21	105	35	140	35	105	21	42	7	7	0₄₁	D₇/A₆ = 64*7!/7! = 64
A₄A₁A₁	{ }	f₁	2	672	10	5	20	10	20	10	10	5	2	{ }×{3,3,3}	D₇/A₄A₁A₁ = 64*7!/5!/2/2 = 672
A₃A₂	1₀₀	f₂	3	3	2240	1	4	4	6	6	4	4	1	{3,3}v( )	D₇/A₃A₂ = 64*7!/4!/3! = 2240
A₃A₃	1₀₁	f₃	4	6	4	560	*	4	0	6	0	4	0	{3,3}	D₇/A₃A₃ = 64*7!/4!/4! = 560
A₃A₂	1₁₀	f₃	4	6	4	*	2240	1	3	3	3	3	1	{3}v( )	D₇/A₃A₂ = 64*7!/4!/3! = 2240
D₄A₂	1₁₁	f₄	8	24	32	8	8	280	*	3	0	3	0	{3}	D₇/D₄A₂ = 64*7!/8/4!/2 = 280
A₄A₁	1₂₀	f₄	5	10	10	0	5	*	1344	1	2	2	1	{ }v( )	D₇/A₄A₁ = 64*7!/5!/2 = 1344
D₅A₁	1₂₁	f₅	16	80	160	40	80	10	16	84	*	2	0	{ }	D₇/D₅A₁ = 64*7!/16/5!/2 = 84
A₅	1₃₀	f₅	6	15	20	0	15	0	6	*	448	1	1	{ }	D₇/A₅ = 64*7!/6! = 448
D₆	1₃₁	f₆	32	240	640	160	480	60	192	12	32	14	*	( )	D₇/D₆ = 64*7!/32/6! = 14
A₆	1₄₀	f₆	7	21	35	0	35	0	21	0	7	*	64	( )	D₇/A₆ = 64*7!/7! = 64

Related polytopes

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

References

^ Coxeter, Regular Polytopes, sec 1.8 Configurations
^ Coxeter, Complex Regular Polytopes, p.117
^ Klitzing, Richard. "x3o3o *b3o3o3o - hax".

H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- 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]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".

External links

Olshevsky, George. "Demihepteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Multi-dimensional Glossary

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