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.

Incidence matrix

(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