B5 polytope explained

In 5-dimensional geometry, there are 31 uniform polytopes with B₅ symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube.

They can be visualized as symmetric orthographic projections in Coxeter planes of the B₅ Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 32 polytopes can be made in the B₅, B₄, B₃, B₂, A₃, Coxeter planes. A_k has [k+1] symmetry, and B_k has [2k] symmetry.

These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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, ^[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]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

Notes and References

1. http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter