Acronym ...
Name family of (generally hypercompact) hyperbolic o4x4o2Qo honeycombs (Q≥2)
Confer
more general:
o4x4oPo (not necessarily even P)  
Especially o4x4o4o (Q=2, paracompact family member)  

This in general hypercompact hyperbolic tesselation uses squats (order 4 square tilings) and also order 2Q square tilings, in the sense of infinite horohedra (i.e. with euclidean curvature) resp. in the sense of infinite bollohedra (i.e. with hyperbolic curvature itself), as its cell types.

For sure, the case Q=2 would be contained here too, resulting in o4x4o4x*a, but this one would be just paracompact (the order 2Q square tilings then would become horohedra too). Moreover it would become regular then.

We might consider for using rational Q=n/d as well. Clearly, in order to not get a Grünbaumian vertex figure, d then has to be odd in general (in the o4x4o4x4*aQ*c setup; for the o4x4o2Qo one rather cf. to o4x4oPo instead). Again Q=n/d>2 here would be hypercompact. And 1<Q=n/d≤2 theoretically would be paracompact. But the subgroup o4o4o-n/d-*a for 1<n/d<2 belongs to spherical geometry and allows just for a single finite realization, cf. the list of Schwarz triangles, where n/d=3/2. But that one would then contradict to our restriction on d.


Incidence matrix according to Dynkin symbol

o4x4o2Qo   (Q parametrisable, N,M,K → ∞)

. . .  . | 2NMK |    4Q |   2Q    4Q |   2Q   2  (vertex figure: uniform 2Q-prism, scaled by sqrt2)
---------+------+-------+------------+---------
. x .  . |    2 | 4QNMK |    1     2 |    2   1
---------+------+-------+------------+---------
o4x .  . |    4 |     4 | QNMK     * |    2   0
. x4o  . |    4 |     4 |    * 2QNMK |    1   1
---------+------+-------+------------+---------
o4x4o  . |   2M |    4M |    M     M | 2QNK   *  (horohedron with vert. config. 4^4, i.e. squat)
. x4o2Qo |   2K |   2QK |    0    QK |    * 2NM  (bollohedron with vert. config. 4^(2Q))

snubbed forms: o4s4o2Qo

o4x4o4x4*aQ*c   (Q parametrisable; N,M,K,L,P → ∞)

. . . .       | 4NMKLP |      2Q      2Q |      Q      Q      Q      2Q      Q |    1     Q    1     Q  (vertex figure: uniform 2Q-prism, scaled by sqrt2)
--------------+--------+-----------------+-------------------------------------+----------------------
. x . .       |      2 | 4QNMKLP       * |      1      0      1       1      0 |    1     1    0     1
. . . x       |      2 |       * 4QNMKLP |      0      1      0       1      1 |    0     1    1     1
--------------+--------+-----------------+-------------------------------------+----------------------
o4x . .       |      4 |       4       0 | QNMKLP      *      *       *      * |    1     1    0     0
o . . x4*a    |      4 |       0       4 |      * QNMKLP      *       *      * |    0     1    1     0
. x4o .       |      4 |       4       0 |      *      * QNMKLP       *      * |    1     0    0     1
. x . x       |      4 |       2       2 |      *      *      * 2QNMKLP      * |    0     1    0     1
. . o4x       |      4 |       0       4 |      *      *      *       * QNMKLP |    0     0    1     1
--------------+--------+-----------------+-------------------------------------+----------------------
o4x4o . *aQ*c |     4M |     4QM       0 |     QM      0     QM       0      0 | NKLP     *    *     *  (bollohedron with vert. config. 4^(2Q))
o4x . x4*a    |     4K |      4K      4K |      K      K      0      2K      0 |    * QNMLP    *     *  (horohedron with vert. config. 4^4, i.e. squat)
o . o4x4*aQ*c |     4L |       0     4QL |      0     QL      0       0     QL |    *     * NMKP     *  (bollohedron with vert. config. 4^(2Q))
. x4o4x       |     4P |      4P      4P |      0      0      P      2P      P |    *     *    * QNMKL  (horohedron with vert. config. 4^4, i.e. squat)

snubbed forms: o4s4o4s4*aQ*c

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