complex polychoron x5-4-o2-3-o2-3-o2,
γ54
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- more general:
- xp-4-o2-3-o2-3-o2
- real space embedding:
- piquid
- general polytopal classes:
- complex polytopes
links
| Acronym | ... |
| Name |
Shephard's 5-generalised tesseract, complex polychoron x5-4-o2-3-o2-3-o2, γ54 |
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| Vertex figure | tet |
| Coordinates | (ε5n, ε5m, ε5k, ε5l) for any 1≤n,m,k,l≤5, where ε5=exp(2πi/5) |
| Dual | x2-3-o2-3-o2-4-o5 |
| Face vector | 625, 500, 150, 20 |
| Confer |
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External links |
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The (complex) faces are x5-4-o2, the (complex) cells are x5-4-o2-3-o2, and the vertex figure here is just x2-3-o2-3-o2, i.e. nothing but the real space tet.
The below various incidence representations are direct, next dimensional consequences from what was explained already at xp-4-o2 = xp xp. I.e. the according cartesian or prism product applies for complex polytopes alike. In fact, these complex polychora simply re-use the even-dimensional elements of all the pentagons, pedips, and pettips from this real quadprism only.
Incidence matrix according to Dynkin symbol
x5-4-o2-3-o2-3-o2 . . . . | 625 ♦ 4 | 6 | 4 -----------------+-----+-----+-----+--- x5 . . . | 5 | 500 | 3 | 3 -----------------+-----+-----+-----+--- x5-4-o2 . . ♦ 25 | 10 | 150 | 2 -----------------+-----+-----+-----+--- x5-4-o2-3-o2 . ♦ 125 | 75 | 15 | 20
x5 x5-4-o2-3-o2 . . . . | 625 ♦ 1 3 | 3 3 | 3 1 -----------------+-----+---------+--------+----- x5 . . . | 5 | 125 * | 3 0 | 3 0 . x5 . . | 5 | * 375 | 1 2 | 2 1 -----------------+-----+---------+--------+----- x5 x5 . . ♦ 25 | 5 5 | 75 * | 2 0 . x5-4-o2 . ♦ 25 | 0 10 | * 75 | 1 1 -----------------+-----+---------+--------+----- x5 x5-4-o2 . ♦ 125 | 25 50 | 10 5 | 15 * . x5-4-o2-3-o2 ♦ 125 | 0 75 | 0 15 | * 5
x5-4-o2 x5-4-o2 . . . . | 625 ♦ 2 2 | 1 4 1 | 2 2 -----------------+-----+---------+-----------+------ x5 . . . | 5 | 250 * | 1 2 0 | 2 1 . . x5 . | 5 | * 250 | 0 2 1 | 1 2 -----------------+-----+---------+-----------+------ x5-4-o2 . . ♦ 25 | 10 0 | 25 * * | 2 0 x5 . x5 . ♦ 25 | 5 5 | * 100 * | 1 1 . . x5-4-o2 ♦ 25 | 0 10 | * * 25 | 0 2 -----------------+-----+---------+-----------+------ x5-4-o2 x5 . ♦ 125 | 50 25 | 5 10 0 | 10 * x5 . x5-4-o2 ♦ 125 | 25 50 | 0 10 5 | * 10
x5 x5 x5-4-o2 . . . . | 625 ♦ 1 1 2 | 1 2 2 1 | 2 1 1 -----------------+-----+-------------+-------------+------- x5 . . . | 5 | 125 * * | 1 2 0 0 | 2 1 0 . x5 . . | 5 | * 125 * | 1 0 2 0 | 2 0 1 . . x5 . | 5 | * * 250 | 0 1 1 1 | 1 1 1 -----------------+-----+-------------+-------------+------- x5 x5 . . ♦ 25 | 5 5 0 | 25 * * * | 2 0 0 x5 . x5 . ♦ 25 | 5 0 5 | * 50 * * | 1 1 0 . x5 x5 . ♦ 25 | 0 5 5 | * * 50 * | 1 0 1 . . x5-4-o2 ♦ 25 | 0 0 10 | * * * 25 | 0 1 1 -----------------+-----+-------------+-------------+------- x5 x5 x5 . ♦ 125 | 25 25 25 | 5 5 5 0 | 10 * * x5 . x5-4-o2 ♦ 125 | 25 0 50 | 0 10 0 5 | * 5 * . x5 x5-4-o2 ♦ 125 | 0 25 50 | 0 0 10 5 | * * 5
x5 x5 x5 xp . . . . | 625 ♦ 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 -----------------+-----+-----------------+-------------------+-------- x5 . . . | 5 | 125 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x5 . . | 5 | * 125 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x5 . | 5 | * * 125 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x5 | 5 | * * * 125 | 0 0 1 0 1 1 | 0 1 1 1 -----------------+-----+-----------------+-------------------+-------- x5 x5 . . ♦ 25 | 5 5 0 0 | 25 * * * * * | 1 1 0 0 x5 . x5 . ♦ 25 | 5 0 5 0 | * 25 * * * * | 1 0 1 0 x5 . . x5 ♦ 25 | 5 0 0 5 | * * 25 * * * | 0 1 1 0 . x5 x5 . ♦ 25 | 0 5 5 0 | * * * 25 * * | 1 0 0 1 . x5 . x5 ♦ 25 | 0 5 0 5 | * * * * 25 * | 0 1 0 1 . . x5 x5 ♦ 25 | 0 0 5 5 | * * * * * 25 | 0 0 1 1 -----------------+-----+-----------------+-------------------+-------- x5 x5 x5 . ♦ 125 | 25 25 25 0 | 5 5 0 5 0 0 | 5 * * * x5 x5 . x5 ♦ 125 | 25 25 0 25 | 5 0 5 0 5 0 | * 5 * * x5 . x5 x5 ♦ 125 | 25 0 25 25 | 0 5 5 0 0 5 | * * 5 * . x5 x5 x5 ♦ 125 | 0 25 25 25 | 0 0 0 5 5 5 | * * * 5
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