D4 polytope

In 4-dimensional geometry, there are 7 uniform 4-polytopes with reflections of D₄ symmetry, all are shared with higher symmetry constructions in the B₄ or F₄ symmetry families. there is also one half symmetry alternation, the snub 24-cell.

Visualizations

Each can be visualized as symmetric orthographic projections in Coxeter planes of the D₄ Coxeter group, and other subgroups. The B₄ coxeter planes are also displayed, while D₄ polytopes only have half the symmetry. They can also be shown in perspective projections of Schlegel diagrams, centered on different cells.

Coordinates

The base point can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be . Some polytopes have two possible generator points. Points are prefixed by Even to imply only an even count of sign permutations should be included.

References

• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, pp. 38–39, 1965
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, 2008, , (Chapter 26)
• 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, 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]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

External links

• Uniform, convex polytopes in four dimensions, Marco Möller