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