For P>4 this is a hypercompact hyperbolic tesselation, which uses order P square tilings in the sense of an infinite bollohedron for some of its cells.

Incidence matrix according to Dynkin symbol

x4oPo4x   (P parametrisable, N,M,K → ∞)

. . . . | 8NMK ♦     P     P |     P    2P     P |   1    P    P   1
--------+------+-------------+-----------------+------------------
x . . . |    2 | 4PNMK     * |     2     2     0 |   1    2    1   0
. . . x |    2 |     * 4PNMK |     0     2     2 |   0    1    2   1
--------+------+-------------+-----------------+------------------
x4o . . |    4 |     4     0 | 2PNMK     *     * |   1    1    0   0
x . . x |    4 |     2     2 |     * 4PNMK     * |   0    1    1   0
. . o4x |    4 |     0     4 |     *     * 2PNMK |   0    0    1   1
--------+------+-------------+-----------------+------------------
x4oPo . ♦   4M |   2PM     0 |    PM     0     0 | 2NK    *    *   *
x4o . x ♦    8 |     8     4 |     2     4     0 |   * PNMK    *   *
x . o4x ♦    8 |     4     8 |     0     4     2 |   *    * PNMK   *
. oPo4x ♦   4K |     0   2PK |     0     0    PK |   *    *    * 2NM

or
. . . .    | 4NM ♦   2P |   2P   2P |  2  2P
-----------+-----+------+-----------+-------
x . . .  & |   2 | 4PNM |    2    2 |  1   3
-----------+-----+------+-----------+-------
x4o . .  & |   4 |    4 | 2PNM    * |  1   1
x . . x    |   4 |    4 |    * 2PNM |  0   2
-----------+-----+------+-----------+-------
x4oPo .  & ♦  4M |  2PM |   PM    0 | 2N   *
x4o . x  & ♦   8 |   12 |    2    4 |  * PNM

snubbed forms: s4oPo4s