Acronym squah Name hyperbolic order 3 square-tiling honeycomb,hyperbolic honeycomb with seed point in center of right-angled octahedron domain ` ©` Circumradius 1/sqrt(-2) = 0.707107 i Dual x3o4o4o Confer more general: o4x4oPo   o4x4o2Po   general polytopal classes: regular   noble polytopes Externallinks

This regular, non-compact hyperbolic tesselation uses squats in the sense of an infinite horohedron as its single cell type.

This regular hyperbolic honeycomb is the single paracompact member of the general family o4x4o2Qo, which would be generally hypercompact for Q>2 (and no longer regular then).

Incidence matrix according to Dynkin symbol

```o3o4o4x   (N,M → ∞)

. . . . | NM ♦   8 |  12 |  6
--------+----+-----+-----+---
. . . x |  2 | 4NM |   3 |  3
--------+----+-----+-----+---
. . o4x |  4 |   4 | 3NM |  2
--------+----+-----+-----+---
. o4o4x ♦  M |  2M |   M | 6N

snubbed forms: o3o4o4s
```

```o4x4o4o   (N,M,K → ∞)

. . . . | NMK ♦    8 |   4    8 |   4   2
--------+-----+------+----------+--------
. x . . |   2 | 4NMK |   1    2 |   2   1
--------+-----+------+----------+--------
o4x . . |   4 |    4 | NMK    * |   2   0
. x4o . |   4 |    4 |   * 2NMK |   1   1
--------+-----+------+----------+--------
o4x4o . ♦  2M |   4M |   M    M | 2NK   *
. x4o4o ♦   K |   2K |   0    K |   * 2NM

snubbed forms: o4s4o4o
```

```o4x4o *b4o   (N,M,K,L → ∞)

. . .    . | NMKL ♦     8 |    4    4    4 |   2   2   2
-----------+------+-------+----------------+------------
. x .    . |    2 | 4NMKL |    1    1    1 |   1   1   1
-----------+------+-------+----------------+------------
o4x .    . |    4 |     4 | NMKL    *    * |   1   1   0
. x4o    . |    4 |     4 |    * NMKL    * |   1   0   1
. x . *b4o |    4 |     4 |    *    * NMKL |   0   1   1
-----------+------+-------+----------------+------------
o4x4o    . ♦   2M |    4M |    M    M    0 | NKL   *   *
o4x . *b4o ♦   2K |    4K |    K    0    K |   * NML   *
. x4o *b4o ♦   2L |    4L |    0    L    L |   *   * NMK

snubbed forms: o4s4o *b4o
```

```x4o4x *b4o   (N,M,K,L → ∞)

. . .    . | NMKL ♦     4     4 |    4    4    4 |   4   1   1
-----------+------+-------------+----------------+------------
x . .    . |    2 | 2NMKL     * |    2    1    0 |   2   1   0
. . x    . |    2 |     * 2NMKL |    0    1    2 |   2   0   1
-----------+------+-------------+----------------+------------
x4o .    . |    4 |     4     0 | NMKL    *    * |   1   1   0
x . x    . |    4 |     2     2 |    * NMKL    * |   2   0   0
. o4x    . |    4 |     0     4 |    *    * NMKL |   1   0   1
-----------+------+-------------+----------------+------------
x4o4x    . ♦   4M |    4M    4M |    M   2M    M | NKL   *   *
x4o . *b4o ♦    K |    2K     0 |    K    0    0 |   * NML   *
. o4x *b4o ♦    L |     0    2L |    0    0    L |   *   * NMK

snubbed forms: s4o4s *b4o
```

```x4o4x4o4*a   (N,M,K,L,P → ∞)

. . . .    | 2NMKLP ♦      4      4 |     2      4     2     2     2 |    2    1    2    1
-----------+--------+---------------+--------------------------------+--------------------
x . . .    |      2 | 4NMKLP      * |     1      1     1     0     0 |    1    1    1    0
. . x .    |      2 |      * 4NMKLP |     0      1     0     1     1 |    1    0    1    1
-----------+--------+---------------+--------------------------------+--------------------
x4o . .    |      4 |      4      0 | NMKLP      *     *     *     * |    1    1    0    0
x . x .    |      4 |      2      2 |     * 2NMKLP     *     *     * |    1    0    1    0
x . . o4*a |      4 |      4      0 |     *      * NMKLP     *     * |    0    1    1    0
. o4x .    |      4 |      0      4 |     *      *     * NMKLP     * |    1    0    0    1
. . x4o    |      4 |      0      4 |     *      *     *     * NMKLP |    0    0    1    1
-----------+--------+---------------+--------------------------------+--------------------
x4o4x .    ♦     4M |     4M     4M |     M     2M     0     M     0 | NKLP    *    *    *
x4o . o4*a ♦     2K |     4K      0 |     K      0     K     0     0 |    * NMLP    *    *
x . x4o4*a ♦     4L |     4L     4L |     0     2L     L     0     L |    *    * NMKP    *
. o4x4o    ♦     2P |      0     4P |     0      0     0     P     P |    *    *    * NMKL

snubbed forms: s4o4s4o4*a
```

```o4xØx4*a4xØx4*a

. . .    . .    | 4NMKLPQ ♦       2       2       2       2 |      1      1      1      1       2       2       2       2 |     1     1     1     1      2
----------------+---------+---------------------------------+-------------------------------------------------------------+-------------------------------
. x .    . .    |       2 | 4NMKLPQ       *       *       * |      1      0      0      0       1       1       0       0 |     1     1     0     0      1
. . x    . .    |       2 |       * 4NMKLPQ       *       * |      0      1      0      0       0       0       1       1 |     0     0     1     1      1
. . .    x .    |       2 |       *       * 4NMKLPQ       * |      0      0      1      0       1       0       1       0 |     1     0     1     0      1
. . .    . x    |       2 |       *       *       * 4NMKLPQ |      0      0      0      1       0       1       0       1 |     0     1     0     1      1
----------------+---------+---------------------------------+-------------------------------------------------------------+-------------------------------
o4x .    . .    |       4 |       4       0       0       0 | NMKLPQ      *      *      *       *       *       *       * |     1     1     0     0      0
o . x4*a . .    |       4 |       0       4       0       0 |      * NMKLPQ      *      *       *       *       *       * |     0     0     1     1      0
o . . *a4x .    |       4 |       0       0       4       0 |      *      * NMKLPQ      *       *       *       *       * |     1     0     1     0      0
o . .    . x4*a |       4 |       0       0       0       4 |      *      *      * NMKLPQ       *       *       *       * |     0     1     0     1      0
. x .    x .    |       4 |       2       0       2       0 |      *      *      *      * 2NMKLPQ       *       *       * |     1     0     0     0      1
. x .    . x    |       4 |       2       0       0       2 |      *      *      *      *       * 2NMKLPQ       *       * |     0     1     0     0      1
. . x    x .    |       4 |       0       2       2       0 |      *      *      *      *       *       * 2NMKLPQ       * |     0     0     1     0      1
. . x    . x    |       4 |       0       2       0       2 |      *      *      *      *       *       *       * 2NMKLPQ |     0     0     0     1      1
----------------+---------+---------------------------------+-------------------------------------------------------------+-------------------------------
o4x . *a4x .    ♦      4M |      4M       0      4M       0 |      M      0      M      0      2M       0       0       0 | NKLPQ     *     *     *      *
o4x .    . x4*a ♦      4K |      4K       0       0      4K |      K      0      0      K       0      2K       0       0 |     * NMLPQ     *     *      *
o . x4*a4x .    ♦      4L |       0      4L      4L       0 |      0      L      L      0       0       0      2L       0 |     *     * NMKPQ     *      *
o . x4*a . x4*a ♦      4P |       0      4P       0      4P |      0      P      0      P       0       0       0      2P |     *     *     * NMKLQ      *
. xØx    xØx    ♦      4Q |      2Q      2Q      2Q      2Q |      0      0      0      0       Q       Q       Q       Q |     *     *     *     * 2NMKLP
```

```octahedral Coxeter domain with boundary pattern:

b    e
g    a

h    c
d    f

xØxØxØxØxØxØxØxØ*aØ*cØ*gØ*eØ*hØ*fØ*bØ*dØ*a   (N,M,K,L,P,Q,R → ∞)
a b c d e f g h

. . . . . . . .                            | 4NMKLPQR ♦        1        1        1        1        1        1        1        1 |       1       1       1       1       1       1       1       1       1       1       1       1 |      1      1      1      1      1      1
-------------------------------------------+----------+-------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+------------------------------------------
x . . . . . . .                            |        2 | 2NMKLPQR        *        *        *        *        *        *        * |       1       1       1       0       0       0       0       0       0       0       0       0 |      1      1      1      0      0      0
. x . . . . . .                            |        2 |        * 2NMKLPQR        *        *        *        *        *        * |       0       0       0       1       1       1       0       0       0       0       0       0 |      1      0      0      1      1      0
. . x . . . . .                            |        2 |        *        * 2NMKLPQR        *        *        *        *        * |       0       0       0       0       0       0       1       1       1       0       0       0 |      0      1      0      1      0      1
. . . x . . . .                            |        2 |        *        *        * 2NMKLPQR        *        *        *        * |       0       0       0       0       0       0       0       0       0       1       1       1 |      0      0      1      0      1      1
. . . . x . . .                            |        2 |        *        *        *        * 2NMKLPQR        *        *        * |       1       0       0       1       0       0       1       0       0       0       0       0 |      1      1      0      1      0      0
. . . . . x . .                            |        2 |        *        *        *        *        * 2NMKLPQR        *        * |       0       1       0       0       0       0       0       1       0       1       0       0 |      0      1      1      0      0      1
. . . . . . x .                            |        2 |        *        *        *        *        *        * 2NMKLPQR        * |       0       0       1       0       1       0       0       0       0       0       1       0 |      1      0      1      0      1      0
. . . . . . . x                            |        2 |        *        *        *        *        *        *        * 2NMKLPQR |       0       0       0       0       0       1       0       0       1       0       0       1 |      0      0      0      1      1      1
-------------------------------------------+----------+-------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+------------------------------------------
x . . . x . . .                            |        4 |        2        0        0        0        2        0        0        0 | NMKLPQR       *       *       *       *       *       *       *       *       *       *       * |      1      1      0      0      0      0
x . . . . x . .                            |        4 |        2        0        0        0        0        2        0        0 |       * NMKLPQR       *       *       *       *       *       *       *       *       *       * |      0      1      1      0      0      0
x . . . . . x .                            |        4 |        2        0        0        0        0        0        2        0 |       *       * NMKLPQR       *       *       *       *       *       *       *       *       * |      1      0      1      0      0      0
. x . . x . . .                            |        4 |        0        2        0        0        2        0        0        0 |       *       *       * NMKLPQR       *       *       *       *       *       *       *       * |      1      0      0      1      0      0
. x . . . . x .                            |        4 |        0        2        0        0        0        0        2        0 |       *       *       *       * NMKLPQR       *       *       *       *       *       *       * |      1      0      0      0      1      0
. x . . . . . x                            |        4 |        0        2        0        0        0        0        0        2 |       *       *       *       *       * NMKLPQR       *       *       *       *       *       * |      0      0      0      1      1      0
. . x . x . . .                            |        4 |        0        0        2        0        2        0        0        0 |       *       *       *       *       *       * NMKLPQR       *       *       *       *       * |      0      1      0      0      0      0
. . x . . x . .                            |        4 |        0        0        2        0        0        2        0        0 |       *       *       *       *       *       *       * NMKLPQR       *       *       *       * |      0      1      0      1      0      1
. . x . . . . x                            |        4 |        0        0        2        0        0        0        0        2 |       *       *       *       *       *       *       *       * NMKLPQR       *       *       * |      0      0      0      1      0      1
. . . x . x . .                            |        4 |        0        0        0        2        0        2        0        0 |       *       *       *       *       *       *       *       *       * NMKLPQR       *       * |      0      0      1      0      0      1
. . . x . . x .                            |        4 |        0        0        0        2        0        0        2        0 |       *       *       *       *       *       *       *       *       *       * NMKLPQR       * |      0      0      1      0      1      0
. . . x . . . x                            |        4 |        0        0        0        2        0        0        0        2 |       *       *       *       *       *       *       *       *       *       *       * NMKLPQR |      0      0      0      0      1      1
-------------------------------------------+----------+-------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+------------------------------------------
xØx . . x . x .       *gØ*e                ♦       4M |       2M       2M        0        0       2M        0       2M        0 |       M       0       M       M       M       0       0       0       0       0       0       0 | NKLPQR      *      *      *      *      *
x . x . xØx . . *aØ*c                      ♦       4K |       2K        0       2K        0       2K       2K        0        0 |       K       K       0       0       0       0       K       K       0       0       0       0 |      * NMLPQR      *      *      *      *
x . . x . xØx .                      *dØ*a ♦       4L |       2L        0        0       2L        0       2L       2L        0 |       0       L       L       0       0       0       0       0       0       L       L       0 |      *      * NMKPQR      *      *      *
. xØx . x . . x          *eØ*h             ♦       4P |        0       2P       2P        0       2P        0        0       2P |       0       0       0       P       0       P       0       P       P       0       0       0 |      *      *      * NMKLQR      *      *
. x . x . . xØx                   *bØ*d    ♦       4Q |        0       2Q        0       2Q        0        0       2Q       2Q |       0       0       0       0       Q       Q       0       0       0       0       Q       Q |      *      *      *      * NMKLPR      *
. . xØx . x . x             *hØ*f          ♦       4R |        0        0       2R       2R        0       2R        0       2R |       0       0       0       0       0       0       0       R       R       R       0       R |      *      *      *      *      * NMKLPQ
```