©

A---B---A---B---
| / | \ | / | \ 
C---+---C---+---
| \ | / | \ | / 
A---B---A---B---
| / | \ | / | \ 
C---+---C---+---
| \ | / | \ | / 

(vertical B-B lines of each column resp. 
 horizontal C-C lines of each row are to be identified)

It is known that ico can be given within 2 different, dual positionings and that tes can be vertex inscribed into it. This gives rise to a 4-coloring of the tes vertices similar to the above mentioned coordinates, in fact in here simply one color is being omitted, yielding their new cells underneath, q-tets. Each cube as a cell remains in here, but it will become a diamond wedge: 2 diametral vertices of one of the square prism bases will be cut off.

Incidence matrix

4 * * | 2 2 0  0 0 | 1 1  4 0 0 | 2 2 0  A
* 4 * | 2 0 1  4 0 | 1 0  4 4 2 | 2 2 2  B
* * 4 | 0 2 0  4 1 | 0 1  4 2 4 | 2 2 2  C
------+------------+------------+------
1 1 0 | 8 * *  * * | 1 0  2 0 0 | 2 1 0  x
1 0 1 | * 8 *  * * | 0 1  2 0 0 | 1 2 0  x
0 2 0 | * * 2  * * | 0 0  0 4 0 | 0 2 2  q
0 1 1 | * * * 16 * | 0 0  1 1 1 | 1 1 1  q
0 0 2 | * * *  * 2 | 0 0  0 0 4 | 2 0 2  q
------+------------+------------+------
2 2 0 | 4 0 0  0 0 | 2 *  * * * | 2 0 0  (ABAB) x4o
2 0 2 | 0 4 0  0 0 | * 2  * * * | 0 2 0  (ACAC) x4o
1 1 1 | 1 1 0  1 0 | * * 16 * * | 1 1 0  (ABC) oq&#x
0 2 1 | 0 0 1  2 0 | * *  * 8 * | 0 1 1  (BBC) q3o
0 1 2 | 0 0 0  2 1 | * *  * * 8 | 1 0 1  (BCC) q3o
------+------------+------------+------
2 2 2 | 4 2 0  4 1 | 1 0  4 0 2 | 4 * *  bidiminished cube, diamond wedge
2 2 2 | 2 4 1  4 0 | 0 1  4 2 0 | * 4 *  bidiminished cube, diamond wedge
0 2 2 | 0 0 1  4 1 | 0 0  0 2 2 | * * 4  q-tet

or
4 * |  4 0  0 | 2  4  0 | 4 0  A
* 8 |  2 1  4 | 1  4  6 | 4 2  B+C
----+---------+---------+----
1 1 | 16 *  * | 1  2  0 | 3 0  x
0 2 |  * 4  * | 0  0  4 | 2 2  q
0 2 |  * * 16 | 0  1  2 | 2 1  q
----+---------+---------+----
2 2 |  4 0  0 | 4  *  * | 2 0  x4o
1 2 |  2 0  1 | * 16  * | 2 0  oq&#x
0 3 |  0 1  2 | *  * 16 | 1 1  q3o
----+---------+---------+----
2 4 |  6 1  4 | 1  4  2 | 8 *  bidiminished cube, diamond wedge
0 4 |  0 2  4 | 0  0  4 | * 4  q-tet