Acronym ... Name hyperbolic x4s4o *b3s tesselation Circumradius ... i

This non-compact hyperbolic tesselation uses squat in the sense of an infinite horohedron as one of its cell types. No uniform representation is possible, as can be seen from the vertex figure.

Note, this figure can be considered as being s3s4o4s' 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: x3x4o4xx3x4o4s' (= x4x4o *b3x) → x4s4o *b3s (cf. below)
• on the other hand: x3x4o4xs3s4o4x → s3s4o4s' (s before s', cf. below)

(where both end forms are described below). So this provides a rare commutative example, which does not base on an additional symmetry of the starting symbol.

The second sequence here is esp. interesting, as it runs from a uniform figure, via an only scaliform one. Accordingly it is a bit harder to compute (cf. below: no alternate, merely Wythoffian Dynkin symbol does exist, which would allow the second step to be displayed independently). In fact, there the so far derived ike ( s3s4o . ) get alternated. Moreover, restricting its effect onto the sectioning figures of the privious step, i.e. the squacues ( sefa( s3s4o4x ) ), then the triangles of those would get alternated, thus resulting in trips.

Incidence matrix according to Dynkin symbol

x4s4o *b3s   (N,M → ∞)

demi( . . .    . ) | 12NM |   1   1    2    4 |   2   2   2    4    3 |  1  2  1   3
-------------------+------+-------------------+-----------------------+-------------
demi( x . .    . ) |    2 | 6NM   *    *    * |   2   0   0    2    0 |  1  2  0   1
. s4o    .   |    2 |   * 6NM    *    * |   0   0   2    0    2 |  1  0  1   2
sefa( x4s .    . ) |    2 |   *   * 12NM    * |   1   0   1    1    0 |  1  1  0   1
sefa( . s . *b3s ) |    2 |   *   *    * 24NM |   0   1   0    1    1 |  0  1  1   1
-------------------+------+-------------------+-----------------------+-------------
x4s .    .   |    4 |   2   0    2    0 | 6NM   *   *    *    * |  1  1  0   0
. s . *b3s   |    3 |   0   0    0    3 |   * 8NM   *    *    * |  0  1  1   0
sefa( x4s4o    . ) |    4 |   0   2    2    0 |   *   * 6NM    *    * |  1  0  0   1
sefa( x4s . *b3s ) |    4 |   1   0    1    2 |   *   *   * 12NM    * |  0  1  0   1
sefa( . s4o *b3s ) |    3 |   0   1    0    2 |   *   *   *    * 12NM |  0  0  1   1
-------------------+------+-------------------+-----------------------+-------------
x4s4o    .      2M |   M   M   2M    0 |   M   0   M    0    0 | 6N  *  *   *
x4s . *b3s      24 |  12   0   12   24 |   6   8   0   12    0 |  * NM  *   *
. s4o *b3s      12 |   0   6    0   24 |   0   8   0    0   12 |  *  * NM   *
sefa( x4s4o *b3s )     6 |   1   2    2    4 |   0   0   1    2    2 |  *  *  * 6NM

starting figure: x4x4o *b3x

s3s4o4s' (here: first s, next s';   N,M → ∞)

demi'( demi( . . . .  ) ) | 12NM |   1    4   1    2 |   2    3   2   2    4 |  1  1   3  2
--------------------------+------+-------------------+-----------------------+-------------
demi'(       . s4o .    ) |    2 | 6NM    *   *    * |   0    2   0   2    0 |  1  1   2  0
demi'( sefa( s3s . .  ) ) |    2 |   * 24NM   *    * |   1    1   0   0    1 |  1  0   1  1
demi( . . o4s' )   |    2 |   *    * 6NM    * |   0    0   2   0    2 |  0  1   1  2
sefa'( sefa( . s4o4s' ) ) |    2 |   *    *   * 12NM |   0    0   1   1    1 |  0  1   1  1
--------------------------+------+-------------------+-----------------------+-------------
demi'(       s3s . .    ) |    3 |   0    3   0    0 | 8NM    *   *   *    * |  1  0   0  1
demi'( sefa( s3s4o .  ) ) |    3 |   1    2   0    0 |   * 12NM   *   *    * |  1  0   1  0
sefa( . s4o4s' )   |    4 |   0    0   2    2 |   *    * 6NM   *    * |  0  1   1  0
sefa'(       . s4o4s'   ) |    4 |   2    0   0    2 |   *    *   * 6NM    * |  0  1   0  1
sefa'( sefa( s3s4o4s' ) ) |    4 |   0    2   1    1 |   *    *   *   * 12NM |  0  0   1  1
--------------------------+------+-------------------+-----------------------+-------------
demi'(       s3s4o .    )    12 |   6   24   0    0 |   8   12   0   0    0 | NM  *   *  *
. s4o4s'        2M |   M    0   M   2M |   0    0   M   M    0 |  * 6N   *  *
sefa( s3s4o4s' )       6 |   2    4   1    2 |   0    2   1   0    2 |  *  * 6NM  *
sefa'(       s3s4o4s'   )    24 |   0   24  12   12 |   8    0   0   6   12 |  *  *   * NM

starting figure: s3s4o4x