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 {3,34,1}.

## 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 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

## Related polytopes

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

## References

1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
3. ^ 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: 1n1)
• Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds