7-demicube

Demihepteract (7-demicube)
Petrie polygon projection
Type	Uniform 7-polytope
Family	demihypercube
Coxeter symbol	141
Schläfli symbol	{3,34,1} = h{4,35} s{21,1,1,1,1,1}
Coxeter diagrams	=
6-faces	78	14 {31,3,1} 64 {35}
5-faces	532	84 {31,2,1} 448 {34}
4-faces	1624	280 {31,1,1} 1344 {33}
Cells	2800	560 {31,0,1} 2240 {3,3}
Faces	2240	{3}
Edges	672
Vertices	64
Vertex figure	Rectified 6-simplex
Symmetry group	D7, [34,1,1] = [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 ${\displaystyle \left\{3{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}}$ or {3,3^4,1}.

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.

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]

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	fk	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		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		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		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		7	21	35	0	35	0	21	0	7	*	64		D7/A6 = 64*7!/7! = 64

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

D7 polytopes
t0(141)	t0,1(141)	t0,2(141)	t0,3(141)	t0,4(141)	t0,5(141)	t0,1,2(141)	t0,1,3(141)
t0,1,4(141)	t0,1,5(141)	t0,2,3(141)	t0,2,4(141)	t0,2,5(141)	t0,3,4(141)	t0,3,5(141)	t0,4,5(141)
t0,1,2,3(141)	t0,1,2,4(141)	t0,1,2,5(141)	t0,1,3,4(141)	t0,1,3,5(141)	t0,1,4,5(141)	t0,2,3,4(141)	t0,2,3,5(141)
t0,2,4,5(141)	t0,3,4,5(141)	t0,1,2,3,4(141)	t0,1,2,3,5(141)	t0,1,2,4,5(141)	t0,1,3,4,5(141)	t0,2,3,4,5(141)	t0,1,2,3,4,5(141)

• 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".

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