H4_polytope

H<sub>4</sub> polytope

H₄ polytope

Four-dimensional geometric objects

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

120-cell	600-cell

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[1] [1]
Coxeter, Regular and Semi-Regular Polytopes II, Four-dimensional polytopes', p. 296–298

[2] [2]
Weisstein, Eric W. "120-cell". MathWorld.

[3] [3]
Weisstein, Eric W. "600-cell". MathWorld.

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[2]

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#	Name	Coxeter plane projections						Schlegel diagrams		Net
#	Name	F4 [12]	[20]	H4 [30]	H3 [10]	A3 [4]	A2 [3]	Dodecahedron centered	Tetrahedron centered	Net
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) t_0,3{5,3,3}
8	bitruncated 120-cell (also bitruncated 600-cell) t_1,2{5,3,3}
9	cantellated 600-cell t_0,2{3,3,5}
10	truncated 600-cell t{3,3,5}
11	cantitruncated 120-cell tr{5,3,3}
12	runcitruncated 120-cell t_0,1,3{5,3,3}
13	runcitruncated 600-cell t_0,1,3{3,3,5}
14	cantitruncated 600-cell tr{3,3,5}
15	omnitruncated 120-cell (also omnitruncated 600-cell) t_0,1,2,3{5,3,3}

#	Name	Coxeter plane projections						Schlegel diagrams		Net
#	Name	F4 [12]	[20]	H4 [30]	H3 [10]	A3 [4]	A2 [3]	Dodecahedron centered	Tetrahedron centered	Net
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

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] 1/2 (±1, ±1, ±1, ±1), and 8 vertices obtained from (0, 0, 0, ±1) by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of 1/2 (±φ, ±1, ±1/φ, 0).
5D	Zero-sum permutation: (30): √5 (1, 1, 0, −1, −1) (10): ±(4, −1, −1, −1, −1) (40): ±(φ⁻¹, φ⁻¹, φ⁻¹, 2, −σ) (40): ±(φ, φ, φ, −2, −(σ−1)) (120): ±√5 (φ, 0, 0, φ⁻¹, −1) (120): ±(2, 2, φ⁻¹√5, −φ, −3) (240): ±(φ², 2φ⁻¹, φ⁻², −1, −2φ)	Zero-sum permutation: (20): √5 (1, 0, 0, 0, −1) (40): ±(φ², φ⁻², −1, −1, −1) (60): ±(2, φ⁻¹, φ⁻¹, −φ, −φ)

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

H4_polytope

H₄ polytope

Visualizations

Coordinates

References

Notes

External links

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