H4 polytope

Short description: Four-dimensional geometric objects

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
1	120-cell {5,3,3}		60px	60px	60px	60px
2	rectified 120-cell r{5,3,3}		60px	60px	60px	60px
3	rectified 600-cell r{3,3,5}		60px	60px	60px	60px
4	600-cell {3,3,5}		60px	60px	60px	60px
5	truncated 120-cell t{5,3,3}		60px	60px	60px	60px
6	cantellated 120-cell rr{5,3,3}					60px
7	runcinated 120-cell (also runcinated 600-cell) t0,3{5,3,3}					60px
8	bitruncated 120-cell (also bitruncated 600-cell) t1,2{5,3,3}					60px
9	cantellated 600-cell t0,2{3,3,5}		60px	60px	60px	60px
10	truncated 600-cell t{3,3,5}		60px	60px	60px	60px
11	cantitruncated 120-cell tr{5,3,3}					60px
12	runcitruncated 120-cell t0,1,3{5,3,3}					60px
13	runcitruncated 600-cell t0,1,3{3,3,5}					60px
14	cantitruncated 600-cell tr{3,3,5}					60px
15	omnitruncated 120-cell (also omnitruncated 600-cell) t0,1,2,3{5,3,3}					60px

Diminished forms

#	Name	Coxeter plane projections						Schlegel diagrams		Net
		F4 [12]	[20]	H4 [30]	H3 [10]	A3 [4]	A2 [3]	Dodecahedron centered	Tetrahedron centered
16	20-diminished 600-cell (grand antiprism)
17	24-diminished 600-cell (snub 24-cell)					60px
18 Nonuniform	Bi-24-diminished 600-cell
19 Nonuniform	120-diminished rectified 600-cell

The coordinates of uniform polytopes 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.

n	120-cell	600-cell
4D	The 600 vertices of the 120-cell include all permutations of (0, 0, ±2, ±2) (±1, ±1, ±1, ±√5) (±φ−2, ±φ, ±φ, ±φ) (±φ−1, ±φ−1, ±φ−1, ±φ2) and all even permutations of (0, ±φ−2, ±1, ±φ2) (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 (±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): ±(φ2, 2φ−1, φ−2, −1, −2φ)	Zero-sum permutation: (20): (√5, 0, 0, 0, −√5) (40): ±(φ2, 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
• Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry 20 (3): 433–444. doi:10.1515/advgeom-2020-0005. 
• Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras (Springer Science and Business Media) 31 (3). doi:10.1007/s00006-021-01139-2. 

External links

• Klitzing, Richard. "4D uniform 4-polytopes". https://bendwavy.org/klitzing/dimensions/polychora.htm. 
• Uniform, convex polytopes in four dimensions:, Marco Möller (in German)
  • Möller, Marco (2004). Vierdimensionale Archimedische Polytope (PDF) (Doctoral dissertation) (in Deutsch). University of Hamburg.
• 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