Acronym	bipach
Name	biprismatocubic honeycomb
Confer	general polytopal classes: isogonal
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This isogonal honeycomb cannot be made uniform, i.e. having all 3 edge types at the same size.

The incidence matrix below shows that the vertex figure is a tri-apiculated triangular pyramid.

Note that the below provided value of c (obtained from the zero height requirement of the tegum sum), when inserted into the formula of the dihedral angle of the lacing edge of the recta results in arccos(1/[4c²/(b-a)²-1]) = arccos(1/2) = 60° = 360°/6, i.e. the below provided seemingly huge number of 6 trapezoprisms around the c-edges indeed is exact here. Note moreover that the calculated value does no longer depend on the chosen ratio of a:b !

Sure, asking that the lacing edges of the pyramid are outside to the burried pseudo edges of the medial layer triangle, might give some further restriction to the a:b ratio, which so far has not been evaluated.

Squares and rectangles here are coplanar. In fact, the cubes are situated in the vertex positions of a body centered cubic lattice, and the square prisms are situated in the edge positions of the 2 interwoven inscribed cubic honeycombs. The recta then simply interconnect those.

Incidence matrix according to Dynkin symbol

ab4oo3oo4ba&#zc   (N → ∞)   → height = 0
                              a < b
                              c = |a-b| sqrt(3)/2

o.4o.3o.3o.     & | 8N |   3   3  1 |  3   6   6 | 1  3  6
------------------+----+------------+------------+--------
a. .. .. ..     & |  2 | 12N   *  * |  2   2   1 | 1  2  2
.. .. .. b.     & |  2 |   * 12N  * |  0   2   1 | 0  1  2
oo4oo3oo4oo&#c    |  2 |   *   * 4N |  0   0   6 | 0  0  6
------------------+----+------------+------------+--------
a.4o. .. ..     & |  4 |   4   0  0 | 6N   *   * | 1  1  0  square
a. .. .. b.     & |  4 |   2   2  0 |  * 12N   * | 0  1  1  rectangle
ab .. .. ..&#c  & |  4 |   1   1  2 |  *   * 12N | 0  0  2  trapezium
------------------+----+------------+------------+--------
a.4o.3o. ..     & |  8 |  12   0  0 |  6   0   0 | N  *  *  cube
a.4o. .. b.     & |  8 |   8   4  0 |  2   4   0 | * 3N  *  long square prism
ab .. .. ba&#c    |  8 |   4   4  4 |  0   2   4 | *  * 6N  recta