Acronym ... Name family of (generally hypercompact) hyperbolic o4s4o2Qo honeycombs (Q≥2) Confer more general: o4s4oPo (not necessarily even P) Especially o4s4o4o (Q=2, paracompact family member)

This in general hypercompact hyperbolic tesselation uses squats (order 4 square tilings) and also order 2Q {2Q} 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 some of its cell types.

The vertex figure here generalizes a cuboctahedron by replacing one of its 4-fold axes by a 2Q-fold one. Its lateral faces still are squares, resting on their tips. It also could be considered as the rectification of the 2Q-prism.

Although being formally derived as mere alternated faceting (of vertices) from o4x4o4x4*aQ*c, it clearly is a uniform snub (in general, i.e. for any Q), because all of its edges were derived from square diagonals and therefore all have the same size.

For sure, the case Q=2 would be contained here too, resulting in o4s4o4s*a, but this one would be just paracompact (the order 2Q {2Q} tilings then would become horohedra too).

We might consider for using rational Q=n/d as well. (Note that the general ambiguity for edge rescalings when using rational link marks in the search for uniform snubs here does not apply, as described above.) Clearly, in order to not get a Grünbaumian snub cell (cf. sefa), d then has to be odd in general (in the o4s4o4s4*aQ*c setup; for the o4s4o2Qo one rather cf. to o4s4oPo 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 contradicts to our restriction on d.

Incidence matrix according to Dynkin symbol

o4s4o2Qo   (Q parametrisable, N,M,K → ∞)

demi( . . .  . ) | NMK |   2Q    4Q |    8Q   4Q |   2Q   2  4Q
-----------------+-----+------------+------------+-------------
o4s .  .   |   2 | QNMK     * |     4    0 |    2   0   2
. s4o  .   |   2 |    * 2QNMK |     2    2 |    1   1   2
-----------------+-----+------------+------------+-------------
sefa( o4s4o  . ) |   4 |    2     2 | 2QNMK    * |    1   0   1
sefa( . s4o2Qo ) |  2Q |    0    2Q |     * 2NMK |    0   1   1
-----------------+-----+------------+------------+-------------
o4s4o  .   |   M |    M     M |     M    0 | 2QNK   *   *  (horohedron with vert. config. 4^4, i.e. squat)
. s4o2Qo   |   K |    0    QK |     0    K |    * 2NM   *  (bollohedron with vert. config. (2Q)^(2Q), i.e. x2Qo2Qo)
sefa( o4s4o2Qo ) |  4Q |   2Q    4Q |    2Q    2 |    *   * NMK  (2Q-prism)

starting figure: o4x4o2Qo

o4s4o4s4*aQ*c   (Q parametrisable; N,M,K,L,P → ∞)

demi( . . . .       ) | 2NMKLP |      Q      Q      Q      2Q      Q |     2Q      4Q     2Q      4Q |    1     Q    1     Q     4Q
----------------------+--------+-------------------------------------+-------------------------------+-----------------------------
o4s . .         |      2 | QNMKLP      *      *       *      * |      2       2      0       0 |    1     1    0     0      2
o . . s4*a      |      2 |      * QNMKLP      *       *      * |      0       2      2       0 |    0     1    1     0      2
. s4o .         |      2 |      *      * QNMKLP       *      * |      2       0      0       2 |    1     0    0     1      2
. s 2 s         |      2 |      *      *      * 2QNMKLP      * |      0       2      0       2 |    0     1    0     1      2
. . o4s         |      2 |      *      *      *       * QNMKLP |      0       0      2       2 |    0     0    1     1      2
----------------------+--------+-------------------------------------+-------------------------------+-----------------------------
sefa( o4s4o . *aQ*c ) |     2Q |      Q      0      Q       0      0 | 2NMKLP       *      *       * |    1     0    0     0      1  {2Q}
sefa( o4s . s4*a    ) |      4 |      1      1      0       2      0 |      * 2QNMKLP      *       * |    0     1    0     0      1  {4}
sefa( o . o4s4*aQ*c ) |     2Q |      0      Q      0       0      Q |      *       * 2NMKLP       * |    0     0    1     0      1  {2Q}
sefa( . s4o4s       ) |      4 |      0      0      1       2      1 |      *       *      * 2QNMKLP |    0     0    0     1      1  {4}
----------------------+--------+-------------------------------------+-------------------------------+-----------------------------
o4s4o . *aQ*c   |     2M |     QM      0     QM       0      0 |     2M       0      0       0 | NKLP     *    *     *      *  (bollohedron with vert. config. (2Q)^(2Q), i.e. x2Qo2Qo)
o4s . s4*a      |     2K |      K      K      0      2K      0 |      0      2K      0       0 |    * QNMLP    *     *      *  (horohedron with vert. config. 4^4, i.e. squat)
o . o4s4*aQ*c   |     2L |      0     QL      0       0     QL |      0       0     2L       0 |    *     * NMKP     *      *  (bollohedron with vert. config. (2Q)^(2Q), i.e. x2Qo2Qo)
. s4o4s         |     2P |      0      0      P      2P      P |      0       0      0      2P |    *     *    * QNMKL      *  (horohedron with vert. config. 4^4, i.e. squat)
sefa( o4s4o4s4*aQ*c ) |     4Q |      Q      Q      Q      2Q      Q |      1       Q      1       Q |    *     *    *     * 2NMKLP  (2Q-prism)

starting figure: o4x4o4x4*aQ*c