Acronym ...
Name hyperbolic x4o4o4o tesselation,
hyperbolic honeycomb with seed point at vertex of right-angled octahedron domain
 
 ©
Circumradius 0 i
Dual (selfdual)
Vertex figure
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Confer
general polytopal classes:
regular   noble polytopes  
External
links
wikipedia

This regular, non-compact hyperbolic tesselation uses the (q scaled) euclidean tiling squat in the sense of an infinite horohedron as vertex figure. An (unit) squat in the sense of an infinite horohedron is its single cell type too.


Incidence matrix according to Dynkin symbol

x4o4o4o   (N,M,K → ∞)

. . . . | 2NK    M |  2M |   M
--------+-----+-----+-----+----
x . . . |   2 | NMK |   4 |   4
--------+-----+-----+-----+----
x4o . . |   4 |   4 | NMK |   2
--------+-----+-----+-----+----
x4o4o .    K |  2K |   K | 2NM

snubbed forms: x4o4o4o

x4o4o *b4o   (N,M,K,L → ∞)

. . .    . | NKL    2M |   4M |   M   M
-----------+-----+------+------+--------
x . .    . |   2 | NMKL |    4 |   2   2
-----------+-----+------+------+--------
x4o .    . |   4 |    4 | NMKL |   1   1
-----------+-----+------+------+--------
x4o4o    .    K |   2K |    K | NML   *
x4o . *b4o    L |   2L |    L |   * NMK

snubbed forms: s4o4o *b4o

x4o4o4o4*a   (N,M,K,L,P → ∞)

. . . .    | NKLP      4M |    4M    4M |    M   2M    M
-----------+------+--------+-------------+---------------
x . . .    |    2 | 2NMKLP |     2     2 |    1    2    1
-----------+------+--------+-------------+---------------
x4o . .    |    4 |      4 | NMKLP     * |    1    1    0
x . . o4*a |    4 |      4 |     * NMKLP |    0    1    1
-----------+------+--------+-------------+---------------
x4o4o .        K |     2K |     K     0 | NMLP    *    *
x4o . o4*a    2L |     4L |     L     L |    * NMKP    *
x . o4o4*a     P |     2P |     0     P |    *    * NMKL

snubbed forms: s4o4o4o4*a

s4o4o4o   (N,M,K,L → ∞)

demi( . . . . ) | NKL    2M |   4M |   M   M
----------------+-----+------+------+--------
      s4o . .   |   2 | NMKL |    4 |   2   2
----------------+-----+------+------+--------
sefa( s4o4o . ) |   4 |    4 | NMKL |   1   1
----------------+-----+------+------+--------
      s4o4o .      K |   2K |    K | NML   *
sefa( s4o4o4o )    L |   2L |    L |   * NMK

starting figure: x4o4o4o

s4o4o *b4o   (N,M,K,L,P → ∞)

demi( . . .    . ) | NMKLP      4M |    4M    4M |    M    M   2M
-------------------+-------+--------+-------------+---------------
      s4o .    .   |     2 | 2NMKLP |     2     2 |    1    1    2
-------------------+-------+--------+-------------+---------------
sefa( s4o4o    . ) |     4 |      4 | NMKLP     * |    1    0    1
sefa( s4o . *b4o ) |     4 |      4 |     * NMKLP |    0    1    1
-------------------+-------+--------+-------------+---------------
      s4o4o    .        K |     2K |     K     0 | NMLP    *    *
      s4o . *b4o        L |     2L |     0     L |    * NMKP    *
sefa( s4o4o *b4o )     2P |     4P |     P     P |    *    * NMKL

starting figure: x4o4o *b4o

s4o4o4o4*a   (N,M,K,L,P,Q → ∞)

demi( . . . .    ) | NMKLPQ       4M      4M |     4M      8M     4M |     M     2M     M    4M
-------------------+--------+-----------------+-----------------------+-------------------------
      s4o . .      |      2 | 2NMKLPQ       * |      2       2      0 |     1      1     0     2
      s . . o4*a   |      2 |       * 2NMKLPQ |      0       2      2 |     0      1     1     2
-------------------+--------+-----------------+-----------------------+-------------------------
sefa( s4o4o .    ) |      4 |       4       0 | NMKLPQ       *      * |     1      0     0     1
sefa( s4o . o4*a ) |      4 |       2       2 |      * 2NMKLPQ      * |     0      1     0     1
sefa( s . o4o4*a ) |      4 |       0       4 |      *       * NMKLPQ |     0      0     1     1
-------------------+--------+-----------------+-----------------------+-------------------------
      s4o4o .            K |      2K       0 |      K       0      0 | NMLPQ      *     *     *
      s4o . o4*a         L |       L       L |      0       L      0 |     * 2NMKPQ     *     *
      s . o4o4*a         P |       0      2P |      0       0      P |     *      * NMKLQ     *
sefa( s4o4o4o4*a )      4Q |      4Q      4Q |      Q      2Q      Q |     *      *     * NMKLP

starting figure: x4o4o4o4*a

x4oØo4*a4oØo4*a   (N,M,K,L,P,Q → ∞)

. . .    . .    | 2NKLPQ       4M |     2M     2M     2M     2M |     M     M     M     M
----------------+--------+---------+-----------------------------+------------------------
x . .    . .    |      2 | 4NMKLPQ |      1      1      1      1 |     1     1     1     1
----------------+--------+---------+-----------------------------+------------------------
x4o .    . .    |      4 |       4 | NMKLPQ      *      *      * |     1     1     0     0
x . o4*a . .    |      4 |       4 |      * NMKLPQ      *      * |     0     0     1     1
x . . *a4o .    |      4 |       4 |      *      * NMKLPQ      * |     1     0     1     0
x . .    . o4*a |      4 |       4 |      *      *      * NMKLPQ |     0     1     0     1
----------------+--------+---------+-----------------------------+------------------------
x4o . *a4o .         2K |      4K |      K      0      K      0 | NMLPQ     *     *     *
x4o .    . o4*a      2L |      4L |      L      0      0      L |     * NMKPQ     *     *
x . o4*a4o .         2P |      4P |      0      P      P      0 |     *     * NMKLQ     *
x . o4*a . o4*a      2Q |      4Q |      0      Q      0      Q |     *     *     * NMKLP

o4xØo4*a4xØo4*a   (N,M,K,L,P,Q → ∞)

. . .    . .    | NKLPQ       4M      4M |     4M     4M      8M |    4M     M     M     2M
----------------+-------+-----------------+-----------------------+-------------------------
. x .    . .    |     2 | 2NMKLPQ       * |      2      0       2 |     2     1     0      1
. . .    x .    |     2 |       * 2NMKLPQ |      0      2       2 |     2     0     1      1
----------------+-------+-----------------+-----------------------+-------------------------
o4x .    . .    |     4 |       4       0 | NMKLPQ      *       * |     1     1     0      0
o . . *a4x .    |     4 |       0       4 |      * NMKLPQ       * |     1     0     1      0
. x .    x .    |     4 |       2       2 |      *      * 2NMKLPQ |     1     0     0      1
----------------+-------+-----------------+-----------------------+-------------------------
o4x . *a4x .        4K |      4K      4K |      K      K      2K | NMLPQ     *     *      *
o4x .    . o4*a      L |      2L       0 |      L      0       0 |     * NMKPQ     *      *
o . o4*a4x .         P |       0      2P |      0      P       0 |     *     * NMKLQ      *
. xØo    xØo         Q |       Q       Q |      0      0       Q |     *     *     * 2NMKLP

octahedral Coxeter domain with boundary pattern:

  b    e
g    a  
        
  h    c
d    f  

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

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


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