Acronym quishexah Name hyperbolic quarter order 4 hexagonal-tiling honeycomb Circumradius sqrt(-2/3) = 0.816497 i

This non-compact hyperbolic tesselation uses both the euclidean tilings that and trat in the sense of infinite horohedra as some of its cells.

Note, this figure can be considered as being s4o3o6s' too. This is quite surprising here, as in general sequential applications of alternated facetings is not commutative! But here we have

• on the one hand: x4o3o6xs4o3o6x (= x3o3o *b6x) → x3o3o *b6s' (cf. below)
• on the other hand: x4o3o6xx4o3o6s' (= x4o3x3o3*b) → s4o3x3o3*b (cf. below)

(where both end forms are described below, in turn being equivalent with x3x3o3o3*a3*c). So this provides a rare commutative example, which does not base on an additional symmetry of the starting symbol!

Incidence matrix according to Dynkin symbol

```x3x3o3o3*a3*c   (N,M,K → ∞)

. . . .       | 4NMK |     6    3 |    6    3    3    3 |   3   3   1   1
--------------+------+------------+---------------------+----------------
x . . .       |    2 | 12NMK    * |    1    1    1    0 |   1   1   1   0
. x . .       |    2 |     * 6NMK |    2    0    0    2 |   2   1   0   1
--------------+------+------------+---------------------+----------------
x3x . .       |    6 |     3    3 | 4NMK    *    *    * |   1   1   0   0
x . o . *a3*c |    3 |     3    0 |    * 4NMK    *    * |   1   0   1   0
x . . o3*a    |    3 |     3    0 |    *    * 4NMK    * |   0   1   1   0
. x3o .       |    3 |     0    3 |    *    *    * 4NMK |   1   0   0   1
--------------+------+------------+---------------------+----------------
x3x3o . *a3*c ♦   3M |    3M   3M |    M    M    0    M | 4NK   *   *   *
x3x . o3*a    ♦   12 |    12    6 |    4    0    4    0 |   * NMK   *   *
x . o3o3*a3*c ♦    K |    3K    0 |    0    K    K    0 |   *   * 4NM   *
. x3o3o       ♦    4 |     0    6 |    0    0    0    4 |   *   *   * NMK
```

```x3o3o *b6s   (N,M,K → ∞)

demi( . . .    . ) | 4NMK |    3     6 |    3    3    6    3 |   1   3   1   3
-------------------+------+------------+---------------------+----------------
demi( x . .    . ) |    2 | 6NMK     * |    2    0    2    0 |   1   2   0   1
sefa( . o . *b6s ) |    2 |    * 12NMK |    0    1    1    1 |   0   1   1   1
-------------------+------+------------+---------------------+----------------
demi( x3o .    . ) |    3 |    3     0 | 4NMK    *    *    * |   1   1   0   0
. o . *b6s   |    3 |    0     3 |    * 4NMK    *    * |   0   1   1   0
sefa( x3o . *b6s ) |    6 |    3     3 |    *    * 4NMK    * |   0   1   0   1
sefa( . o3o *b6s ) |    3 |    0     3 |    *    *    * 4NMK |   0   0   1   1
-------------------+------+------------+---------------------+----------------
demi( x3o3o    . ) ♦    4 |    6     0 |    4    0    0    0 | NMK   *   *   *
x3o . *b6s   ♦   3M |   3M    3M |    M    M    M    0 |   * 4NK   *   *
. o3o *b6s   ♦    K |    0    3K |    0    K    0    K |   *   * 4NM   *
sefa( x3o3o *b6s ) ♦   12 |    6    12 |    0    0    4    4 |   *   *   * NMK

starting figure: x3o3o *b6x
```

```s4o3x3o3*b   (N,M,K → ∞)

demi( . . . .    ) | 4NMK |     6    3 |    3    3    6    3 |   1   3   1   3
-------------------+------+------------+---------------------+----------------
demi( . . x .    ) |    2 | 12NMK    * |    1    1    1    0 |   1   1   0   1
s4o . .      |    2 |     * 6NMK |    0    0    2    2 |   0   1   1   2
-------------------+------+------------+---------------------+----------------
demi( . o3x .    ) |    3 |     3    0 | 4NMK    *    *    * |   1   1   0   0
demi( . . x3o    ) |    3 |     3    0 |    * 4NMK    *    * |   1   0   0   1
sefa( s4o3x .    ) |    6 |     3    3 |    *    * 4NMK    * |   0   1   0   1
sefa( s4o . o3*b ) |    3 |     0    3 |    *    *    * 4NMK |   0   0   1   1
-------------------+------+------------+---------------------+----------------
demi( . o3x3o3*b ) ♦    M |    3M    0 |    M    M    0    0 | 4NK   *   *   *
s4o3x .      ♦   12 |    12    6 |    4    0    4    0 |   * NMK   *   *
s4o . o3*b   ♦    4 |     0    6 |    0    0    0    4 |   *   * NMK   *
sefa( s4o3x3o3*b ) ♦   3K |    3K   3K |    0    K    K    K |   *   *   * 4NM

starting figure: x4o3x3o3*b
```

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