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Four-dimensional geometric objects From Wikipedia, the free encyclopedia
In 4-dimensional geometry, there are 15 uniform polytopes with H4 symmetry. Two of these, the 120-cell and 600-cell, are regular.
The article's lead section may need to be rewritten. (October 2023) |
120-cell |
600-cell |
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} |
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2 | rectified 120-cell r{5,3,3} |
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3 | rectified 600-cell r{3,3,5} |
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4 | 600-cell {3,3,5} |
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5 | truncated 120-cell t{5,3,3} |
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6 | cantellated 120-cell rr{5,3,3} |
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7 | runcinated 120-cell (also runcinated 600-cell) t0,3{5,3,3} |
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8 | bitruncated 120-cell (also bitruncated 600-cell) t1,2{5,3,3} |
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9 | cantellated 600-cell t0,2{3,3,5} |
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10 | truncated 600-cell t{3,3,5} |
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11 | cantitruncated 120-cell tr{5,3,3} |
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12 | runcitruncated 120-cell t0,1,3{5,3,3} |
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13 | runcitruncated 600-cell t0,1,3{3,3,5} |
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14 | cantitruncated 600-cell tr{3,3,5} |
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15 | omnitruncated 120-cell (also omnitruncated 600-cell) t0,1,2,3{5,3,3} |
# | 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) |
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17 | 24-diminished 600-cell (snub 24-cell) |
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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.[1]
n | 120-cell | 600-cell |
---|---|---|
4D |
The 600 vertices of the 120-cell include all permutations of[2]
and all even permutations of
|
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]
and 8 vertices obtained from
The remaining 96 vertices are obtained by taking even permutations of
|
5D |
Zero-sum permutation:
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Zero-sum permutation:
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