7-demicube explained

bgcolor=#e7dcc3 colspan=3	Demihepteract (7-demicube)
bgcolor=#ffffff align=center colspan=3	Petrie polygon projection
Type	Uniform 7-polytope
Family	demihypercube
Coxeter symbol	141
Schläfli symbol	= h s
Coxeter diagrams	=
6-faces	78	14 64
5-faces	532	84 448
4-faces	1624	280 1344
Cells	2800	560 2240
Faces	2240
Edges	672
Vertices	64
Vertex figure	Rectified 6-simplex
Symmetry group	D7, [3<sup>4,1,1</sup>] = [1<sup>+</sup>,4,3<sup>5</sup>] [2<sup>6</sup>]^+
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.

\left\{ 3 \begin{array}{l} 3 , 3 , 3 , 3 \\ 3 \end{array}\right\} or .

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.

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.

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		D7/A6 = 64*7!/7	= 64
A4A1A1			f1	2	672	10	5	20	10	20	10	10	5	2		D7/A4A1A1 = 64*7!/5	/2/2 = 672
A3A2		100	f2	3	3	2240	1	4	4	6	6	4	4	1		D7/A3A2 = 64*7!/4	/3! = 2240
A3A3		101	f3	4	6	4	560		4	0	6	0	4	0		D7/A3A3 = 64*7!/4	/4! = 560
A3A2		110		4	6	4		2240	1	3	3	3	3	1		D7/A3A2 = 64*7!/4	/3! = 2240
D4A2		111	f4	8	24	32	8	8	280		3	0	3	0		D7/D4A2 = 64*7!/8/4	/2 = 280
A4A1		120		5	10	10	0	5		1344	1	2	2	1		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

Related polytopes

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

References

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition,, 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (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, (Chapter 26. pp. 409: Hemicubes: 1_n1)

External links

• • Multi-dimensional Glossary

Notes and References

1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117