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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}
2 rectified 120-cell

r{5,3,3}
3 rectified 600-cell

r{3,3,5}
4 600-cell

{3,3,5}
5 truncated 120-cell

t{5,3,3}
6 cantellated 120-cell

rr{5,3,3}
7 runcinated 120-cell
(also runcinated 600-cell)

t0,3{5,3,3}
8 bitruncated 120-cell
(also bitruncated 600-cell)

t1,2{5,3,3}
9 cantellated 600-cell

t0,2{3,3,5}
10 truncated 600-cell

t{3,3,5}
11 cantitruncated 120-cell

tr{5,3,3}
12 runcitruncated 120-cell

t0,1,3{5,3,3}
13 runcitruncated 600-cell

t0,1,3{3,3,5}
14 cantitruncated 600-cell

tr{3,3,5}
15 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
16 20-diminished 600-cell
(grand antiprism)
17 24-diminished 600-cell
(snub 24-cell)
18
Nonuniform 		Bi-24-diminished 600-cell
19
Nonuniform 		120-diminished rectified 600-cell

Coordinates

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.[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, ±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[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): ±(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

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

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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