Acronym batch Name bitruncated cubic honeycomb,truncated-octahedral honeycomb,Voronoi complex of body-centered cubic (bcc) lattice ` © ©   ©` Vertex figure ` ©` Dual bichon Confer related tesselations: Delone complex of primitive cubic lattice   Voronoi complex of primitive cubic lattice   Delone complex of fcc lattice   general polytopal classes: partial Stott expansions   noble polytopes Externallinks
` ©   `

As can be read from the matrices below, at every edge there are 2 hexagons. Thus we get as pseudo cells something with hexagons only. From the vertex incidence we further read off that this pseudo tiling happens to use 4 hexagons per vertex. From the here truely being used cells (toe) it is deduced, that any straight edge sequence of that seeming x6o4o needs to be mod-wrapped to square holes. Therefore those pseudo cells rather are the skew polyhedron x6o4o|4 instead.

The facial subset, obtained when neglecting one tetrahedral subset of hexagons within each toe (e.g. the hexagons x3x . . and . . x3x in the representation x3x3x3x3*a), by itself describes the Wythoffian infinite skew polyhedron x6o6x|6, a modwrap of the hyperbolical tiling x6o6x.

By virtue of an outer symmetry this is a non-quasiregular monotoxal honeycomb, that is all edges belong to the same equivalence class.

Incidence matrix according to Dynkin symbol

```o4x3x4o   (N → ∞)

. . . . | 12N |   2   2 |  1  4  1 | 2 2
--------+-----+---------+----------+----
. x . . |   2 | 12N   * |  1  2  0 | 2 1
. . x . |   2 |   * 12N |  0  2  1 | 1 2
--------+-----+---------+----------+----
o4x . . |   4 |   4   0 | 3N  *  * | 2 0
. x3x . |   6 |   3   3 |  * 8N  * | 1 1
. . x4o |   4 |   0   4 |  *  * 3N | 0 2
--------+-----+---------+----------+----
o4x3x . ♦  24 |  24  12 |  6  8  0 | N *
. x3x4o ♦  24 |  12  24 |  0  8  6 | * N
```
```or
. . . .    | 6N |   4 |  2  4 | 4
-----------+----+-----+-------+--
. x . .  & |  2 | 12N |  1  2 | 3
-----------+----+-----+-------+--
o4x . .  & |  4 |   4 | 3N  * | 2
. x3x .    |  6 |   6 |  * 4N | 2
-----------+----+-----+-------+--
o4x3x .  & ♦ 24 |  36 |  6  8 | N

snubbed forms: o4s3s4o
```

```o4x3x4/3o   (N → ∞)

. . .   . | 12N |   2   2 |  1  4  1 | 2 2
----------+-----+---------+----------+----
. x .   . |   2 | 12N   * |  1  2  0 | 2 1
. . x   . |   2 |   * 12N |  0  2  1 | 1 2
----------+-----+---------+----------+----
o4x .   . |   4 |   4   0 | 3N  *  * | 2 0
. x3x   . |   6 |   3   3 |  * 8N  * | 1 1
. . x4/3o |   4 |   0   4 |  *  * 3N | 0 2
----------+-----+---------+----------+----
o4x3x   . ♦  24 |  24  12 |  6  8  0 | N *
. x3x4/3o ♦  24 |  12  24 |  0  8  6 | * N
```

```o4/3x3x4/3o   (N → ∞)

.   . .   . | 12N |   2   2 |  1  4  1 | 2 2
------------+-----+---------+----------+----
.   x .   . |   2 | 12N   * |  1  2  0 | 2 1
.   . x   . |   2 |   * 12N |  0  2  1 | 1 2
------------+-----+---------+----------+----
o4/3x .   . |   4 |   4   0 | 3N  *  * | 2 0
.   x3x   . |   6 |   3   3 |  * 8N  * | 1 1
.   . x4/3o |   4 |   0   4 |  *  * 3N | 0 2
------------+-----+---------+----------+----
o4/3x3x   . ♦  24 |  24  12 |  6  8  0 | N *
.   x3x4/3o ♦  24 |  12  24 |  0  8  6 | * N
```
```or
.   . .   .    | 6N |   4 |  2  4 | 4
---------------+----+-----+-------+--
.   x .   .  & |  2 | 12N |  1  2 | 3
---------------+----+-----+-------+--
o4/3x .   .  & |  4 |   4 | 3N  * | 2
.   x3x   .    |  6 |   6 |  * 4N | 2
---------------+----+-----+-------+--
o4/3x3x   .  & ♦ 24 |  36 |  6  8 | N
```

```x3x3x *b4o   (N → ∞)

. . .    . | 24N |   1   2   1 |  2  1  2  1 |  2 1 1
-----------+-----+-------------+-------------+-------
x . .    . |   2 | 12N   *   * |  2  1  0  0 |  2 1 0
. x .    . |   2 |   * 24N   * |  1  0  1  1 |  1 1 1
. . x    . |   2 |   *   * 12N |  0  1  2  0 |  2 0 1
-----------+-----+-------------+-------------+-------
x3x .    . |   6 |   3   3   0 | 8N  *  *  * |  1 1 0
x . x    . |   4 |   2   0   2 |  * 6N  *  * |  2 0 0
. x3x    . |   6 |   0   3   3 |  *  * 8N  * |  1 0 1
. x . *b4o |   4 |   0   4   0 |  *  *  * 6N |  0 1 1
-----------+-----+-------------+-------------+-------
x3x3x    . ♦  24 |  12  12  12 |  4  6  4  0 | 2N * *
x3x . *b4o ♦  24 |  12  24   0 |  8  0  0  6 |  * N *
. x3x *b4o ♦  24 |   0  24  12 |  0  0  8  6 |  * * N

snubbed forms: s3s3s *b4o
```

```x3x3x *b4/3o   (N → ∞)

. . .      . | 24N |   1   2   1 |  2  1  2  1 |  2 1 1
-------------+-----+-------------+-------------+-------
x . .      . |   2 | 12N   *   * |  2  1  0  0 |  2 1 0
. x .      . |   2 |   * 24N   * |  1  0  1  1 |  1 1 1
. . x      . |   2 |   *   * 12N |  0  1  2  0 |  2 0 1
-------------+-----+-------------+-------------+-------
x3x .      . |   6 |   3   3   0 | 8N  *  *  * |  1 1 0
x . x      . |   4 |   2   0   2 |  * 6N  *  * |  2 0 0
. x3x      . |   6 |   0   3   3 |  *  * 8N  * |  1 0 1
. x . *b4/3o |   4 |   0   4   0 |  *  *  * 6N |  0 1 1
-------------+-----+-------------+-------------+-------
x3x3x      . ♦  24 |  12  12  12 |  4  6  4  0 | 2N * *
x3x . *b4/3o ♦  24 |  12  24   0 |  8  0  0  6 |  * N *
. x3x *b4/3o ♦  24 |   0  24  12 |  0  0  8  6 |  * * N
```

```x3x3x3x3*a   (N → ∞)

. . . .    | 24N |   1   1   1   1 |  1  1  1  1  1  1 | 1 1 1 1
-----------+-----+-----------------+-------------------+--------
x . . .    |   2 | 12N   *   *   * |  1  1  1  0  0  0 | 1 1 1 0
. x . .    |   2 |   * 12N   *   * |  1  0  0  1  1  0 | 1 1 0 1
. . x .    |   2 |   *   * 12N   * |  0  1  0  1  0  1 | 1 0 1 1
. . . x    |   2 |   *   *   * 12N |  0  0  1  0  1  1 | 0 1 1 1
-----------+-----+-----------------+-------------------+--------
x3x . .    |   6 |   3   3   0   0 | 4N  *  *  *  *  * | 1 1 0 0
x . x .    |   4 |   2   0   2   0 |  * 6N  *  *  *  * | 1 0 1 0
x . . x3*a |   6 |   3   0   0   3 |  *  * 4N  *  *  * | 0 1 1 0
. x3x .    |   6 |   0   3   3   0 |  *  *  * 4N  *  * | 1 0 0 1
. x . x    |   4 |   0   2   0   2 |  *  *  *  * 6N  * | 0 1 0 1
. . x3x    |   6 |   0   0   3   3 |  *  *  *  *  * 4N | 0 0 1 1
-----------+-----+-----------------+-------------------+--------
x3x3x .    ♦  24 |  12  12  12   0 |  4  6  0  4  0  0 | N * * *
x3x . x3*a ♦  24 |  12  12   0  12 |  4  0  4  0  6  0 | * N * *
x . x3x3*a ♦  24 |  12   0  12  12 |  0  6  4  0  0  4 | * * N *
. x3x3x    ♦  24 |   0  12  12  12 |  0  0  0  4  6  4 | * * * N

snubbed forms: s3s3s3s3*a
```

```s4x3x4o   (N → ∞)

demi( . . . . ) | 24N |   1   2   1 |  2  1  1  2 | 1  2 1
----------------+-----+-------------+-------------+-------
demi( . x . . ) |   2 | 12N   *   * |  2  0  1  0 | 1  2 0
demi( . . x . ) |   2 |   * 24N   * |  1  1  0  1 | 1  1 1
sefa( s4x . . ) |   2 |   *   * 12N |  0  0  1  2 | 0  2 1
----------------+-----+-------------+-------------+-------
demi( . x3x . ) |   6 |   3   3   0 | 8N  *  *  * | 1  1 0
demi( . . x4o ) |   4 |   0   4   0 |  * 6N  *  * | 1  0 1
s4x . .   |   4 |   2   0   2 |  *  * 6N  * | 0  2 0
sefa( s4x3x . ) |   6 |   0   3   3 |  *  *  * 8N | 0  1 1
----------------+-----+-------------+-------------+-------
demi( . x3x4o ) ♦  24 |  12  24   0 |  8  6  0  0 | N  * *
s4x3x .   ♦  24 |  12  12  12 |  4  0  6  4 | * 2N *
sefa( s4x3x4o ) ♦  24 |   0  24  12 |  0  6  0  8 | *  * N

starting figure: x4x3x4o
```

```:qooo:4:xuxu:4:ooqo:&##x   (N → ∞)   → all heights = 1/sqrt(2) = 0.707107

o... 4 o... 4 o...      | 4N  *  *  * |  2  1  0  0  0  1 | 1  2  1  2  0 0 | 2 2
.o.. 4 .o.. 4 .o..      |  * 2N  *  * |  0  2  2  0  0  0 | 0  4  1  0  1 0 | 2 2
..o. 4 ..o. 4 ..o.      |  *  * 4N  * |  0  0  1  2  1  0 | 0  2  0  2  1 1 | 2 2
...o 4 ...o 4 ...o      |  *  *  * 2N |  0  0  0  0  2  2 | 0  0  1  4  1 0 | 2 2
-------------------------+-------------+-------------------+-----------------+----
....   x...   ....      |  2  0  0  0 | 4N  *  *  *  *  * | 1  1  0  1  0 0 | 1 2
oo.. 4 oo.. 4 oo.. &#x  |  1  1  0  0 |  * 4N  *  *  *  * | 0  2  1  0  0 0 | 2 1
.oo. 4 .oo. 4 .oo. &#x  |  0  1  1  0 |  *  * 4N  *  *  * | 0  2  0  0  1 0 | 1 2
....   ..x.   ....      |  0  0  2  0 |  *  *  * 4N  *  * | 0  1  0  1  0 1 | 2 1
..oo 4 ..oo 4 ..oo &#x  |  0  0  1  1 |  *  *  *  * 4N  * | 0  0  0  2  1 0 | 1 2
:o..o:4:o..o:4:o..o:&#x  |  1  0  0  1 |  *  *  *  *  * 4N | 0  0  1  2  0 0 | 2 1
-------------------------+-------------+-------------------+-----------------+----
....   x... 4 o...      |  4  0  0  0 |  4  0  0  0  0  0 | N  *  *  *  * * | 0 2
....   xux.   .... &#xt |  2  2  2  0 |  1  2  2  1  0  0 | * 4N  *  *  * * | 1 1
:qo.o:  ....   .... &#xt |  2  1  0  1 |  0  2  0  0  0  2 | *  * 2N  *  * * | 2 0
....  :x.xu:  .... &#xt |  2  0  2  2 |  1  0  0  1  2  2 | *  *  * 4N  * * | 1 1
....   ....   .oqo &#xt |  0  1  2  1 |  0  0  2  0  2  0 | *  *  *  * 2N * | 0 2
..o. 4 ..x.   ....      |  0  0  4  0 |  0  0  0  4  0  0 | *  *  *  *  * N | 2 0
-------------------------+-------------+-------------------+-----------------+----
:qooo:4:xuxu:  .... &#xt ♦  8  4  8  4 |  4  8  4  8  4  8 | 0  4  4  4  0 2 | N *
....  :xuxu:4:ooqo:&#xt ♦  8  4  8  4 |  8  4  8  4  8  4 | 2  4  0  4  4 0 | * N
```

```:xxu:3:xux:3:uxx:3*a&##x   (N → ∞)   → all heights = sqrt(2/3) = 0.816497

o.. 3 o.. 3 o.. 3*a     | 6N  *  * |  1  1  1  0  0  0  0  0  1 | 1  1  1 0  0 0  1  1  1 | 2 1 1
.o. 3 .o. 3 .o. 3*a     |  * 6N  * |  0  0  1  1  1  1  0  0  0 | 0  1  1 1  1 0  1  0  1 | 1 2 1
..o 3 ..o 3 ..o 3*a     |  *  * 6N |  0  0  0  0  0  1  1  1  1 | 0  0  1 0  1 1  1  1  1 | 1 1 2
-------------------------+----------+----------------------------+-------------------------+------
x..   ...   ...         |  2  0  0 | 3N  *  *  *  *  *  *  *  * | 1  1  0 0  0 0  1  0  0 | 2 1 0
...   x..   ...         |  2  0  0 |  * 3N  *  *  *  *  *  *  * | 1  0  1 0  0 0  0  1  0 | 2 0 1
oo. 3 oo. 3 oo. 3*a&#x  |  1  1  0 |  *  * 6N  *  *  *  *  *  * | 0  1  1 0  0 0  0  0  1 | 1 1 1
.x.   ...   ...         |  0  2  0 |  *  *  * 3N  *  *  *  *  * | 0  1  0 1  0 0  1  0  0 | 1 2 0
...   ...   .x.         |  0  2  0 |  *  *  *  * 3N  *  *  *  * | 0  0  0 1  1 0  0  0  1 | 0 2 1
.oo 3 .oo 3 .oo 3*a&#x  |  0  1  1 |  *  *  *  *  * 6N  *  *  * | 0  0  1 0  1 0  1  0  0 | 1 1 1
...   ..x   ...         |  0  0  2 |  *  *  *  *  *  * 3N  *  * | 0  0  1 0  0 1  0  1  0 | 1 0 2
...   ...   ..x         |  0  0  2 |  *  *  *  *  *  *  * 3N  * | 0  0  0 0  1 1  0  0  1 | 0 1 2
:o.o:3:o.o:3:o.o:3*a&#x  |  1  0  1 |  *  *  *  *  *  *  *  * 6N | 0  0  0 0  0 0  1  1  1 | 1 1 1
-------------------------+----------+----------------------------+-------------------------+------
x.. 3 x..   ...         |  6  0  0 |  3  3  0  0  0  0  0  0  0 | N  *  * *  * *  *  *  * | 2 0 0
xx.   ...   ...    &#x  |  2  2  0 |  1  0  2  1  0  0  0  0  0 | * 3N  * *  * *  *  *  * | 1 1 0
...   xux   ...    &#xt |  2  2  2 |  0  1  2  0  0  2  1  0  0 | *  * 3N *  * *  *  *  * | 1 0 1
.x.   ...   .x. 3*a     |  0  6  0 |  0  0  0  3  3  0  0  0  0 | *  *  * N  * *  *  *  * | 0 2 0
...   ...   .xx    &#x  |  0  2  2 |  0  0  0  0  1  2  0  1  0 | *  *  * * 3N *  *  *  * | 0 1 1
...   ..x 3 ..x         |  0  0  6 |  0  0  0  0  0  0  3  3  0 | *  *  * *  * N  *  *  * | 0 0 2
:xxu:  ...   ...    &#xt |  2  2  2 |  1  0  0  1  0  2  0  0  2 | *  *  * *  * * 3N  *  * | 1 1 0
...  :x.x:  ...    &#x  |  2  0  2 |  0  1  0  0  0  0  1  0  2 | *  *  * *  * *  * 3N  * | 1 0 1
...   ...  :uxx:   &#xt |  2  2  2 |  0  0  2  0  1  0  0  1  2 | *  *  * *  * *  *  * 3N | 0 1 1
-------------------------+----------+----------------------------+-------------------------+------
:xxu:3:xux:  ...    &#xt ♦ 12  6  6 |  6  6  6  3  0  6  3  0  6 | 2  3  3 0  0 0  3  3  0 | N * *
:xxu:  ...  :uxx:3*a&#xt ♦  6 12  6 |  3  0  6  6  6  6  0  3  6 | 0  3  0 2  3 0  3  0  3 | * N *
...  :xux:3:uxx:   &#xt ♦  6  6 12 |  0  3  6  0  3  6  6  6  6 | 0  0  3 0  3 2  0  3  3 | * * N
```