Acronym	tekah
Name	tetrakis hexahedron, apiculated cube
	©   ©
Inradius	2/sqrt(5) = 0.894427
Vertex figure	[t6], [T4]
Coordinates	(2/3, 2/3, 2/3)      & all permutations, all changes of sign (vertex inscribed 4/3-cube) (1, 0, 0)               & all permutations, all changes of sign (vertex inscribed sqrt(2)-oct)
Dihedral angles	at long edge:   arccos(-4/5) = 143.130102° at short edge:   arccos(-4/5) = 143.130102°
Dual	toe
Face vector	14, 36, 24
Confer	general polytopal classes: Catalan polyhedra
External links

©

The triangles {(t,t,T)} have vertex angles t = arccos(2/3) = 48.189685° resp. T = arccos(1/9) = 83.620630°.

Edge sizes used here are tT = x = 1 (short) resp. tt = b = 4/3 = 1.333333 (long).

A nice construction is shown at the right: Start with a small co of edge size 1/q. Apiculating its squares by according squippies results in an oct of edge size 2/q = q, thus already providing exactly the above provided second vertex type coordinates. Also apiculating its triangles by according tets will provide a tip-to-tip diagonal of 1/q ⋅ q/h (scaling × tet-height) + 1/q ⋅ 2q/h (scaling × co-height between parallel triangles) + 1/q ⋅ q/h (opposite tet) = 4/3 h, i.e. exactly the vertex-to-vertex diagonal of the here being used b-scaled cube, hence giving rise to the above given first set of coordinates. Thus m3m4o then is just the hull of that all-apiculated co, even within its correct metricallity.

Incidence matrix according to Dynkin symbol

m3m4o =
ao3oo4ob&#zx   → height = 0
                 a = sqrt(2) = 1.414214
                 b = 4/3 = 1.333333

o.3o.4o.    | 6 * |  4  0 |  4  [T4]
.o3.o4.o    | * 8 |  3  3 |  6  [t6]
------------+-----+-------+---
oo3oo4oo&#x | 1 1 | 24  * |  2  x
.. .. .b    | 0 2 |  * 12 |  2  b
------------+-----+-------+---
.. .. ob&#x | 1 2 |  2  1 | 24  {(t,t,T)}

m3m3m =
coo3oao3ooc&#z(x,x,b)   → height = 0
                          a = sqrt(2) = 1.414214
                          b = lacing(1,3) = 4/3 = 1.333333
                          c = 4 sqrt(2)/3 = 1.885618

o..3o..3o..           | 4 * * |  3  3  0 |  6  [t6]
.o.3.o.3.o.           | * 6 * |  2  0  2 |  4  [T4]
..o3..o3..o           | * * 4 |  0  3  3 |  6  [t6]
----------------------+-------+----------+---
oo.3oo.3oo.&#x        | 1 1 0 | 12  *  * |  2  x
o.o3o.o3o.o&#b        | 1 0 1 |  * 12  * |  2  b
.oo3.oo3.oo&#x        | 0 1 1 |  *  * 12 |  2  x
----------------------+-------+----------+---
ooo3ooo3ooo&#r(x,x,b) | 1 1 1 |  1  1  1 | 24  {(t,t,T)}