Acronym	bithon
Name	bitetrahedral honeycomb
Confer	general polytopal classes: isogonal
External links

Note that the here being used edge ratio y:x = sqrt(3/8) is just the ratio of of circumradius versus edge length of the regular tetrahedron.

This isogonal honeycomb thus can be derived from an octet when its full symmetry gets broken, the octahedra become concentrical half symmetrical aligned further tetrahedra of same edge size, while every second of the former tetrahedra gets replaced by its center, which thereby are nothing but the vertices of those new tetrahedra. In between the dually arranged parallel regular triangle faces of those 2 types of remaining tetrahedra then shallow triangular antiprisms come in. At their obtuse isoceles lacing triangles those antiprisms will attach to one another.

The vertex figure here can be derived from the dual of the truncated tetrahedron, the triakis tetrahedron, when just its 3-fold vertices become truncated.

Incidence matrix according to Dynkin symbol

xo3oo3oo3ox3*a&#zy   (N → ∞)   → height = 0, where y = sqrt(3/8) = 0.612372

o.3o.3o.3o.3*a     | N * | 12  4  0 | 12 12  6  0 | 4 12 0
.o3.o3.o3.o3*a     | * N |  0  4 12 |  0  6 12 12 | 0 12 4
-------------------+-----+----------+-------------+-------
x. .. .. ..        | 2 0 | 6N  *  * |  2  1  0  0 | 1  2 0  x
oo3oo3oo3oo3*a&#y  | 1 1 |  * 4N  * |  0  3  3  0 | 0  6 0  y
.. .. .. .x        | 0 2 |  *  * 6N |  0  0  1  2 | 0  2 1  x
-------------------+-----+----------+-------------+-------
x. .. .. o.3*a     | 3 0 |  3  0  0 | 4N  *  *  * | 1  1 0
xo .. .. ..   &#y  | 2 1 |  1  2  0 |  * 6N  *  * | 0  2 0
.. .. .. ox   &#y  | 1 2 |  0  2  1 |  *  * 6N  * | 0  2 0
.o .. .. .x3*a     | 0 3 |  0  0  3 |  *  *  * 4N | 0  1 1
-------------------+-----+----------+-------------+-------
x. .. o.3o.3*a     | 4 0 |  6  0  0 |  4  0  0  0 | N  * *  tet
xo .. .. ox3*a&#y  | 3 3 |  3  6  3 |  1  3  3  1 | * 4N *  shallow 3ap
.o3.o .. .x3*a     | 0 4 |  0  0  6 |  0  0  0  4 | *  * N  tet

or
o.3o.3o.3o.3*a     & | N | 12  4 | 12 18 | 4 12
---------------------+---+-------+-------+-----
x. .. .. ..        & | 2 | 6N  * |  2  1 | 1  2  x
oo3oo3oo3oo3*a&#y    | 2 |  * 2N |  0  6 | 0  6  y
---------------------+---+-------+-------+-----
x. .. .. o.3*a     & | 3 |  3  0 | 4N  * | 1  1
xo .. .. ..   &#y  & | 3 |  1  2 |  * 6N | 0  2
---------------------+---+-------+-------+-----
x. .. o.3o.3*a     & | 4 |  6  0 |  4  0 | N  *  tet
xo .. .. ox3*a&#y    | 6 |  6  6 |  2  6 | * 2N  shallow 3ap