List of H4 polytopes

120-cell

600-cell

In 4-dimensional geometry, there are 15 uniform polytopes with H₄ symmetry. Two of these, the 120-cell and 600-cell, are regular.

Visualizations

Each can be visualized as symmetric orthographic projections in Coxeter planes of the H₄ Coxeter group, and other subgroups.

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

#	Name	Coxeter plane projections						Schlegel diagrams		Net
		F4 [12]	[20]	H4 [30]	H3 [10]	A3 [4]	A2 [3]	Dodecahedron centered	Tetrahedron centered
32	120-cell {5,3,3}
33	rectified 120-cell r{5,3,3}
34	rectified 600-cell r{3,3,5}
35	600-cell {3,3,5}
36	truncated 120-cell t{5,3,3}
37	cantellated 120-cell rr{5,3,3}
38	runcinated 120-cell (also runcinated 600-cell) t0,3{5,3,3}
39	bitruncated 120-cell (also bitruncated 600-cell) t1,2{5,3,3}
40	cantellated 600-cell t0,2{3,3,5}
41	truncated 600-cell t{3,3,5}
42	cantitruncated 120-cell tr{5,3,3}
43	runcitruncated 120-cell t0,1,3{5,3,3}
44	runcitruncated 600-cell t0,1,3{3,3,4}
45	cantitruncated 600-cell tr{3,3,5}
46	omnitruncated 120-cell (also omnitruncated 600-cell) t0,1,2,3{5,3,3}

Diminished forms

#	Name	Coxeter plane projections						Schlegel diagrams		Net
		F4 [12]	[20]	H4 [30]	H3 [10]	A3 [4]	A2 [3]	Dodecahedron centered	Tetrahedron centered
47	20-diminished 600-cell (grand antiprism)
31	24-diminished 600-cell (snub 24-cell)
Nonuniform	Bi-24-diminished 600-cell
Nonuniform	120-diminished rectified 600-cell

Coordinates

The coordinates of uniform poyltopes from the H₄ family are complicated. The regular ones can be expressed in terms of the golden ratio φ = (1 + √5)/2 and σ = (3√5 + 1)/2. Coxeter expressed them as 5-dimensional coordinates.^[1]

n	120-cell	600-cell
4D	The 600 vertices of the 120-cell include all permutations of:[2] (0, 0, ±2, ±2) (±1, ±1, ±1, ±√5) (±φ^−2, ±φ, ±φ, ±φ) (±φ^−1, ±φ^−1, ±φ^−1, ±φ^2) and all even permutations of (0, ±φ^−2, ±1, ±φ^2) (0, ±φ^−1, ±φ, ±√5) (±φ^−1, ±1, ±φ, ±2)	The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ (where φ = (1+√5) /2 is the golden ratio), can be given as follows: 16 vertices of the form:[3] (±½, ±½, ±½, ±½), and 8 vertices obtained from (0, 0, 0, ±1) by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of ½ (±φ, ±1, ±1/φ, 0).
5D	Zero-sum permutation: (30): (√5,√5,0,-√5,-√5) (10): ±(4,-1,-1,-1,-1) (40): ±(φ^−1,φ^−1,φ^−1,2,-σ) (40): ±(φ,φ,φ,-2,-(σ-1)) (120): ±(φ√5,0,0,φ^−1√5,-√5) (120): ±(2,2,φ^−1√5,-φ,-3) (240): ±(φ^2,2φ^−1,φ^−2,-1,-2φ)	Zero-sum permutation: (20): (√5,0,0,0,-√5) (40): ±(φ^2,φ^−2,-1,-1,-1) (60): ±(2,φ^−1,φ^−1,-φ,-φ)

References

• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (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, ISBN 978-0-471-01003-6 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
  • (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

1. Coxeter, Regular and Semi-Regular Polytopes II, Four-dimensional polytopes', p.296-298
2. Weisstein, Eric W., "120-cell", MathWorld.
3. Weisstein, Eric W., "600-cell", MathWorld.

External links

• Richard Klitzing, 4D, uniform 4-polytopes
• Uniform, convex polytopes in four dimensions:, Marco Möller (German)
  • 2004 Dissertation Four-dimensional Archimedean polytopes (German)
• Uniform Polytopes in Four Dimensions, George Olshevsky.
  • Convex uniform polychora based on the 120-cell/600-cell, George Olshevsky.
• H4 uniform polytopes with coordinates: {5,3,3}, {3,3,5}, r{5,3,3},r{3,3,5}, t{3,3,5}, t{5,3,3}, rr{3,3,5}, rr{5,3,3}, tr{3,3,5}, tr{5,3,3}, 2t{5,3,3}, t03{5,3,3}, t013{3,3,5}, t013{5,3,3}, t0123{5,3,3}, grand antiprism