Acronym
| ... |

Name
| family of hyperbolic o4x4oPo honeycombs (P≠2) |

Especially
| o4x4o3o (P=3, paracompact family member) o4x4o4o (P=4, paracompact family member) o4x4o2Qo (P even) |

This in general hypercompact hyperbolic tesselation uses squats (order 4 square tilings) and also order P square tilings, in the sense of infinite horohedra (i.e. with euclidean curvature) resp. in the sense of infinite bollohedra (for P>4, i.e. with hyperbolic curvature itself), as its cell types.

In the case of P=3 the order P square tiling clearly becomes a cube. And for P=4 that one would become a further squat. Esp. it would become regular then. Both these cases then result in paracompact hyperbolic honeycombs only.

It should be noted that for P with even numerator (and thus odd denominator, i.e. P=2n/d) there is an alternative symmetry group description too:
`o4x4o2Qo = o4x4o4x4*aQ*c`, obviously using then Q=n/d (with still d being odd).
– But on the other hand e.g. P=3 would *not* be similarily equivalent to Q=3/2, as the former would provide the usual
trip as its vertex figure, while the latter then would ask for a Grünbaumian 6/2-prism.

Incidence matrix according to Dynkin symbol

o4x4oPo (P parametrisable, N,M,K → ∞ in general; resp. K=2 if P=3) . . . . | 4NMK | 2P | P 2P | P 2 (vertex figure: uniform P-prism, scaled by sqrt2) --------+------+-------+------------+--------- . x . . | 2 | 4PNMK | 1 2 | 2 1 --------+------+-------+------------+--------- o4x . . | 4 | 4 | PNMK * | 2 0 . x4o . | 4 | 4 | * 2PNMK | 1 1 --------+------+-------+------------+--------- o4x4o . | 2M | 4M | M M | 2PNK * (horohedron with vert. config. 4^4, i.e. squat) . x4oPo | 4K | 2PK | 0 PK | * 2NM (polyhedron with vert. config. 4^P) snubbed forms: o4s4oPo

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