Acronym bamich Name biambocubic honeycomb Confer more general: sobath   general polytopal classes: isogonal Externallinks

Rectification, when applicable, results in using a specific point on each edge. Ambification simplifies that by using the midpoint always. Birectification, when applicable, results in using a specific point on each face. Biambification accordingly simplifies that by using the respective centerpoint. Clearly this applies here to either cubic honeycomb, to the original as well as to its dual.

Thus the vertex set here represents the union of the centers of the squares (face centers of the original) and the edge centers (face centers of the dual) of a single cubic honeycomb. In consequence the regular oct cells are centered in the positions of the body-centered cubic (bcc) lattice A3*. The shallow traps just buffer in between, joining then the mutually inverted triangles note that these shallow traps adjoin pairwise at their lacing isots.

This isogonal honeycomb can well be obtained from a q-rich = o4q3o4o: it keeps the . q3o4o octs, but dissects the q-coes o4q3o . into 1 inscribed further q-oct (then touching the centers of the squares) plus 8 shallow traps qo3oq&#x, one underneath each triangle of that former co.

The isosceles triangles happen to be right angled. They come in coplanar quadruples.

Incidence matrix according to Dynkin symbol

```oo4qo3oq3oo&#zx   (N → ∞)   → height = 0
(tegum sum of 2 inverted richs)

o.4o.3o.4o.     & | 3N |   8  4 |  8  12 | 2  8
------------------+----+--------+--------+-----
.. q. .. ..     & |  2 | 12N  * |  2   1 | 1  2
oo4oo3oo4oo&#x    |  2 |   * 6N |  0   4 | 0  4
------------------+----+--------+--------+-----
.. q.3o. ..     & |  3 |   3  0 | 8N   * | 1  1
.. qo .. ..&#x  & |  3 |   1  2 |  * 12N | 0  2
------------------+----+--------+--------+-----
.. q.3o.4o.     & |  6 |  12  0 |  8   0 | N  *  q-oct
.. qo3oq ..&#x    |  6 |   6  6 |  2   6 | * 4N  (q,x)-3ap
```

```qo3oq3qo3oq3*a&#zx   (N → ∞)   → height = 0

o.3o.3o.3o.3*a     & | 6N |   4   4   4 |  2  2  2  2   6   6 | 1 1  4  2  2
---------------------+----+-------------+---------------------+-------------
q. .. .. ..        & |  2 | 12N   *   * |  1  1  0  0   1   0 | 1 0  1  1  0 q
.. .. q. ..        & |  2 |   * 12N   * |  0  0  1  1   0   1 | 0 1  1  0  1 q
oo3oo3oo3oo3*a&#x    |  2 |   *   * 12N |  0  0  0  0   2   2 | 0 0  2  1  1 x
---------------------+----+-------------+---------------------+-------------
q.3o. .. ..        & |  3 |   3   0   0 | 4N  *  *  *   *   * | 1 0  1  0  0
q. .. .. o.3*a     & |  3 |   3   0   0 |  * 4N  *  *   *   * | 1 0  0  1  0
.. o.3q. ..        & |  3 |   0   3   0 |  *  * 4N  *   *   * | 0 1  0  0  1
.. .. q.3o.        & |  3 |   0   3   0 |  *  *  * 4N   *   * | 0 1  1  0  0
qo .. .. ..   &#x  & |  3 |   1   0   2 |  *  *  *  * 12N   * | 0 0  1  1  0
.. oq .. ..   &#x  & |  3 |   0   1   2 |  *  *  *  *   * 12N | 0 0  1  0  1
---------------------+----+-------------+---------------------+-------------
q.3o. .. o.3*a     & |  6 |  12   0   0 |  4  4  0  0   0   0 | N *  *  *  * q-oct
.. o.3q.3o.        & |  6 |   0  12   0 |  0  0  4  4   0   0 | * N  *  *  * q-oct
qo3oq .. ..   &#x  & |  6 |   3   3   6 |  1  0  0  1   3   3 | * * 4N  *  * (q,x)-3ap
qo .. .. oq3*a&#x    |  6 |   6   0   6 |  0  2  0  0   6   0 | * *  * 2N  * (q,x)-3ap
.. oq3qo ..   &#x    |  6 |   0   6   6 |  0  0  2  0   0   6 | * *  *  * 2N (q,x)-3ap
```
```or
o.3o.3o.3o.3*a     & | 3N |   8  4 |  8  12 | 2  8
---------------------+----+--------+--------+-----
q. .. .. ..        & |  2 | 12N  * |  2   1 | 1  2
oo3oo3oo3oo3*a&#x    |  2 |   * 6N |  0   4 | 0  4
---------------------+----+--------+--------+-----
q.3o. .. ..        & |  3 |   3  0 | 8N   * | 1  1
qo .. .. ..   &#x  & |  3 |   1  2 |  * 12N | 0  2
---------------------+----+--------+--------+-----
q.3o. .. o.3*a     & |  6 |  12  0 |  8   0 | N  * q-oct
qo3oq .. ..   &#x  & |  6 |   6  6 |  2   6 | * 4N (q,x)-3ap
```