o   
 q   o
o   q 

The name cubera is a portmanteau of the words 'cube' and 'chimera'. The cubera achieves its self-duality as the result of being, essentially, half cube and half octahedron (the dual polyhedron of a cube). The cubera has three square facets joined at a vertex (the cube 'half') and four triangles, with three of them adjoining a central triangle (the octahedron 'half'). And indeed the first matrix below has rotational symmetry, displaying a combinatorical duality.

It should be mentioned, that both below provided orientations allow for a dimensional analogy. Clearly there is the fully mono-truncated hypercube in any dimension. But also that hypercubical-corner-simplex atop full-hypercube is possible in general. For dimensions other than 3 the latter differs from the former. Moreover, as proved by Dan Moore here, the latter happens to be combinatorically dual in general! See also the 4D version of those (corner) hypercubera.

Incidence matrix according to Dynkin symbol

oqo3ooq&#tx   → both heights = 1/sqrt(3) = 0.577350
(pt || pseudo q-{3} || dual q-{3})

o..3o..     | 1 * * | 3 0 0 | 3 0 0
.o.3.o.     | * 3 * | 1 2 0 | 2 1 0
..o3..o     | * * 3 | 0 2 2 | 1 2 1
------------+-------+-------+------
oo.3oo.&#x  | 1 1 0 | 3 * * | 2 0 0
.oo3.oo&#x  | 0 1 1 | * 6 * | 1 1 0
... ..q     | 0 0 2 | * * 3 | 0 1 1
------------+-------+-------+------
oqo ...&#xt | 1 2 1 | 2 2 0 | 3 * *
... .oq&#x  | 0 1 2 | 0 2 1 | * 3 *  verf(squippy)
..o3..q     | 0 0 3 | 0 0 3 | * * 1  verf(cube)

oq&#x || oqo&#xt   → height = 1

o.      ...     | 1 * * * * | 2 1 0 0 0 0 0 | 1 2 0 0 0
.o      ...     | * 2 * * * | 1 0 1 1 1 0 0 | 1 1 1 1 0
..      o..     | * * 1 * * | 0 1 0 0 0 2 0 | 0 2 0 0 1
..      .o.     | * * * 2 * | 0 0 0 1 0 1 1 | 0 1 0 1 1
..      ..o     | * * * * 1 | 0 0 0 0 2 0 2 | 0 0 1 2 1
----------------+-----------+---------------+----------
oo&#x   ...     | 1 1 0 0 0 | 2 * * * * * * | 1 1 0 0 0  x
o.    || o..     | 1 0 1 0 0 | * 1 * * * * * | 0 2 0 0 0  x
.q      ...     | 0 2 0 0 0 | * * 1 * * * * | 1 0 1 0 0  q
.o    || .o.     | 0 1 0 1 0 | * * * 2 * * * | 0 1 0 1 0  x
.o    || ..o     | 0 1 0 0 1 | * * * * 2 * * | 0 0 1 1 0  q
..      oo.&#x  | 0 0 1 1 0 | * * * * * 2 * | 0 1 0 0 1  x
..      .oo&#x  | 0 0 0 1 1 | * * * * * * 2 | 0 0 0 1 1  x
----------------+-----------+---------------+----------
oq&#x   ...     | 1 2 0 0 0 | 2 0 1 0 0 0 0 | 1 * * * *  top half-square
oo&#x || oo.&#x  | 1 1 1 1 0 | 1 1 0 1 0 1 0 | * 2 * * *  2D-cubera = {4}
.q    || ..o     | 0 2 0 0 1 | 0 0 1 0 2 0 0 | * * 1 * *  straight q-line || pt = q-{3}
.o    || .oo&#x  | 0 1 0 1 1 | 0 0 0 1 1 0 1 | * * * 2 *  oblique pt || line = half-square
..      oqo&#x  | 0 0 1 2 1 | 0 0 0 0 0 2 2 | * * * * 1  bottom {4}