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7-demicube

Uniform 7-polytope

Uniform 7-polytope

Demihepteract(7-demicube)
[[File:Demihepteract ortho petrie.svg280px]]Petrie polygon projection
Type
Family
Coxeter symbol
Schläfli symbol
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Symmetry group
Dual
Properties

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 {3,34,1}.

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

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.

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*-facef*k*f0f1f2colspan=2f3colspan=2f4colspan=2f5colspan=2f6*k*-figuresNotesf0f1f2f3f4f5f6
A6( )**64**2110535140
A4A1A1{ }2**672**10520
A3A210033**2240**14
A3A3101464**560***
A3A2110464***2240**
D4A2[111](16-cell)8243288
A4A1[120](5-cell)5101005
D5A1[121](5-demicube)16801604080
A5[130](5-simplex)61520015
D6[131](6-demicube)32240640160480
A6[140](6-simplex)72135035

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, 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, wiley.com,
      • (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, p. 409, Hemicubes: 1n1,

References

  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117
  3. {{KlitzingPolytopes. ../incmats/hesa.htm. x3o3o *b3o3o3o3o - hesa
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