Acronym ...
Name family of hyperbolic o4s4oPo honeycombs (P≠2)
Especially o4s4o3o (P=3, paracompact family member)   o4s4o4o (P=4, paracompact family member)   o4s4o2Qo (P even)  

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

In the case of P=3 the order P {P} tiling clearly becomes a tet. And for P=4 that one would become a further squat. Both cases then result in paracompact hyperbolic honeycombs only.

The vertex figure here generalizes a cuboctahedron by replacing one of its 4-fold axes by a P-fold one. Its lateral faces still are squares, resting on their tips. It also could be considered as the rectification of the P-prism.

Even so being formally derived as mere alternated faceting (of vertices) from o4x4oPo, it clearly is a uniform snub (in general, i.e. for any P), because all of its edges were derived from square diagonals and therefore all have the same size.

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: o4s4o2Qo = o4s4o4s4*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 one cell type, while the latter then would ask for a Grünbaumian 6/2-prism.


Incidence matrix according to Dynkin symbol

o4s4oPo   (P parametrisable, N,M,K → ∞ in general;   resp. K=2 if P=3)

demi( . . . . ) | 2NMK |    P    2P |    4P   2P |    P   2   2P
----------------+------+------------+------------+--------------
      o4s . .   |    2 | PNMK     * |     4    0 |    2   0    2
      . s4o .   |    2 |    * 2PNMK |     2    2 |    1   1    2
----------------+------+------------+------------+--------------
sefa( o4s4o . ) |    4 |    2     2 | 2PNMK    * |    1   0    1
sefa( . s4oPo ) |    P |    0     P |     * 4NMK |    0   1    1
----------------+------+------------+------------+--------------
      o4s4o .   |    M |    M     M |     M    0 | 2PNK   *    *  (horohedron squat)
      . s4oPo   |   2K |    0    PK |     0   2K |    * 2NM    *  (polyhedron xPoPo)
sefa( o4s4oPo ) |   2P |    P    2P |     P    2 |    *   * 2NMK  (P-prism)

starting figure: o4x4oPo

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