complex polychoron xp-4-o2-3-o2-3-o2,
γp4
© p=3 p=4 p=5
- real space embedding:
- (p,p,p,p)-quadprism
- general polytopal classes:
- complex polytopes
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| Acronym | ... |
| Name |
Shephard's p-generalised tesseract, complex polychoron xp-4-o2-3-o2-3-o2, γp4 |
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| Vertex figure | tet |
| Coordinates | (εpn, εpm, εpk, εpl) for any 1≤n,m,k,l≤p, where εp=exp(2πi/p) |
| Dual | x2-3-o2--3-o2-4-op |
| Face vector | p4, 4p3, 6p2, 4p |
| Especially | x3-4-o2-3-o2-3-o2 (p=3) x4-4-o2-3-o2-3-o2 (p=4) x5-4-o2-3-o2-3-o2 (p=5) |
| Confer |
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External links |
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The (complex) faces are xp-4-o2, the (complex) cells are xp-4-o2-3-o2, and the vertex figure here throughout is just x2-3-o2-3-o2, i.e. nothing but the real space tet.
Those polytopes happen to be the (complex) 4-dimensional versions of Shephard's generalised hypercubes. For sure, p=2 not only returns into the real subspace only, but moreover becomes nothing but the well-known tes.
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 p-gons, (p,p)-dips, and (p,p,p)-tips from this real quadprism only.
Incidence matrix according to Dynkin symbol
xp-4-o2-3-o2-3-o2 . . . . | p4 ♦ 4 | 6 | 4 -----------------+----+-----+-----+--- xp . . . | p | 4p3 | 3 | 3 -----------------+----+-----+-----+--- xp-4-o2 . . ♦ p2 | 2p | 6p2 | 2 -----------------+----+-----+-----+--- xp-4-o2-3-o2 . ♦ p3 | 3p2 | 3p | 4p snubbed forms: sp-4-o2-3-o2-3-o2
xp xp-4-o2-3-o2 . . . . | p4 ♦ 1 3 | 3 3 | 3 1 -----------------+----+--------+---------+----- xp . . . | p | p3 * | 3 0 | 3 0 . xp . . | p | * 3p3 | 1 2 | 2 1 -----------------+----+--------+---------+----- xp xp . . ♦ p2 | p p | 3p2 * | 2 0 . xp-4-o2 . ♦ p2 | 0 2p | * 3p2 | 1 1 -----------------+----+--------+---------+----- xp xp-4-o2 . ♦ p3 | p2 2p2 | 2p p | 3p * . xp-4-o2-3-o2 ♦ p3 | 0 3p2 | 0 3p | * p
xp-4-o2 xp-4-o2 . . . . | p4 ♦ 2 2 | 1 4 1 | 2 2 -----------------+----+---------+-----------+------ xp . . . | p | 2p3 * | 1 2 0 | 2 1 . . xp . | p | * 2p3 | 0 2 1 | 1 2 -----------------+----+---------+-----------+------ xp-4-o2 . . ♦ p2 | 2p 0 | p2 * * | 2 0 xp . xp . ♦ p2 | p p | * 4p2 * | 1 1 . . xp-4-o2 ♦ p2 | 0 2p | * * p2 | 0 2 -----------------+----+---------+-----------+------ xp-4-o2 xp . ♦ p3 | 2p2 p2 | p 2p 0 | 2p * xp . xp-4-o2 ♦ p3 | p2 2p2 | 0 2p p | * 2p
xp xp xp-4-o2 . . . . | p4 ♦ 1 1 2 | 1 2 2 1 | 2 1 1 -----------------+----+-----------+--------------+------- xp . . . | p | p3 * * | 1 2 0 0 | 2 1 0 . xp . . | p | * p3 * | 1 0 2 0 | 2 0 1 . . xp . | p | * * 2p3 | 0 1 1 1 | 1 1 1 -----------------+----+-----------+--------------+------- xp xp . . ♦ p2 | p p 0 | p2 * * * | 2 0 0 xp . xp . ♦ p2 | p 0 p | * 2p2 * * | 1 1 0 . xp xp . ♦ p2 | 0 p p | * * 2p2 * | 1 0 1 . . xp-4-o2 ♦ p2 | 0 0 2p | * * * p2 | 0 1 1 -----------------+----+-----------+--------------+------- xp xp xp . ♦ p3 | p2 p2 p2 | p p p 0 | 2p * * xp . xp-4-o2 ♦ p3 | p2 0 2p2 | 0 2p 0 p | * p * . xp xp-4-o2 ♦ p3 | 0 p2 2p2 | 0 0 2p p | * * p
xp xp xp xp . . . . | p4 ♦ 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 -----------------+----+-------------+------------------+-------- xp . . . | p | p3 * * * | 1 1 1 0 0 0 | 1 1 1 0 . xp . . | p | * p3 * * | 1 0 0 1 1 0 | 1 1 0 1 . . xp . | p | * * p3 * | 0 1 0 1 0 1 | 1 0 1 1 . . . xp | p | * * * p3 | 0 0 1 0 1 1 | 0 1 1 1 -----------------+----+-------------+------------------+-------- xp xp . . ♦ p2 | p p 0 0 | p2 * * * * * | 1 1 0 0 xp . xp . ♦ p2 | p 0 p 0 | * p2 * * * * | 1 0 1 0 xp . . xp ♦ p2 | p 0 0 p | * * p2 * * * | 0 1 1 0 . xp xp . ♦ p2 | 0 p p 0 | * * * p2 * * | 1 0 0 1 . xp . xp ♦ p2 | 0 p 0 p | * * * * p2 * | 0 1 0 1 . . xp xp ♦ p2 | 0 0 p p | * * * * * p2 | 0 0 1 1 -----------------+----+-------------+------------------+-------- xp xp xp . ♦ p3 | p2 p2 p2 0 | p p 0 p 0 0 | p * * * xp xp . xp ♦ p3 | p2 p2 0 p2 | p 0 p 0 p 0 | * p * * xp . xp xp ♦ p3 | p2 0 p2 p2 | 0 p p 0 0 p | * * p * . xp xp xp ♦ p3 | 0 p2 p2 p2 | 0 0 0 p p p | * * * p
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