6-demicube

Cartesian coordinates

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: : (±1,±1,±1,±1,±1,±1) with an odd number of plus signs.

As a configuration

This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-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.

Images

Related polytopes

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.

Skew icosahedron

Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 12, Section 1.8 Configurations,
  • 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, p. 12, Section 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
3. {{KlitzingPolytopes. ../incmats/hax.htm. x3o3o *b3o3o3o - hax
4. "The beauty of geometry : twelve essays". Dover Publications.
5. (2000). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices". Advanced Studies in Pure Mathematics.