8-demicube

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, 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 HM₈ for an 8-dimensional half measure polytope.

Coxeter named this polytope as 1₅₁ from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or {3,3^5,1}.

Acronym: hocto (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube:

with an odd number of plus signs.

Related polytopes and honeycombs

This polytope is the vertex figure for the uniform tessellation, 2₅₁ with Coxeter-Dynkin diagram:

Images

Notes

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: 1_n1,
• x3o3o *b3o3o3o3o3o - hocto

External links

• Multi-dimensional Glossary