Acronym
| ... |

Name
| (infinite series of) hyperbolic uniform honeycombs with pyritohedral vertex figure |

Vertex figure | |

Especially
| x4o3o5o (p=4) |

Columns of p-gonal prisms are aligned at each vertex within 3 mutually perpendicular directions, giving rise for 8 additional these connecting cubes.
Accordingly the vertex figure will be snit, i.e. `cao2aoc2oca&#zd` with `a = x(p)` and `d = x(4) = q`.
As even for `p = 3` this amounts for `c = (1+sqrt(13))/2 = 2.302776` and thus a circumradius of that vertex figure of `sqrt[(9+sqrt(13))/2] = 2.510533`,
which clealy is larger than 1, all members of this series would belong to hyperbolic geometry only.

For the case of `p=4` all cells would all become identical cubes and the honeycomb becomes regular, in fact it will be x4o3o5o.

This series of pyritohedral hyperbolic honeycombs was found in 1997 by W. Krieger.

(N → ∞) pN ♦ 12 | 24 6 | 8 12 ---+-----+--------+------ 2 | 6pN | 4 1 | 2 3 ---+-----+--------+------ 4 | 4 | 6pN * | 1 1 p | p | * 6N | 0 2 ---+-----+--------+------ 8 | 12 | 6 0 | pN * cube 2p | 3p | p 2 | * 6N p-gonal prism

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