7-demicube

Demihepteract (7-demicube)
Petrie polygon projection
Type	Uniform 7-polytope
Family	demihypercube
Coxeter symbol	141
Schläfli symbol	{3,3^4,1} = h{4,3^5} s{2^1,1,1,1,1,1}
Coxeter diagrams	=
6-faces	78	14 {3^1,3,1} 64 {3^5}
5-faces	532	84 {3^1,2,1} 448 {3^4}
4-faces	1624	280 {3^1,1,1} 1344 {3^3}
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, [3^6,1,1] = [1^+,4,3^5] [2^6]^+
Dual	?
Properties	convex

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices truncated. 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]

Related polytopes

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)

References

• 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
    • (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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o *b3o3o3o3o - hesa

External links

• Olshevsky, George, Demihepteract at Glossary for Hyperspace.
• Multi-dimensional Glossary