7-demicube

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.

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.

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

• 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