This is the exact geometrical shape, a mere alternating faceting of which reproduces an h = sqrt(3) scaled regular octahedron.

Incidence matrix according to Dynkin symbol

q x6o

. . . | 12 | 1  2 | 2 1
------+----+------+----
q . . |  2 | 6  * | 2 0
. x . |  2 | * 12 | 1 1
------+----+------+----
q x . |  4 | 2  2 | 6 *
. x6o |  6 | 0  6 | * 2

q x3x

. . . | 12 | 1 1 1 | 1 1 1
------+----+-------+------
q . . |  2 | 6 * * | 1 1 0
. x . |  2 | * 6 * | 1 0 1
. . x |  2 | * * 6 | 0 1 1
------+----+-------+------
q x . |  4 | 2 2 0 | 3 * *
q . x |  4 | 2 0 2 | * 3 *
. x3x |  6 | 0 3 3 | * * 2

xx6oo&#q   → height = sqrt(2) = 1.414214
({6} || {6})

o.6o.    | 6 * | 2 1 0 | 1 2 0
.o6.o    | * 6 | 0 1 2 | 0 2 1
---------+-----+-------+------
x. ..    | 2 0 | 6 * * | 1 1 0
oo6oo&#q | 1 1 | * 6 * | 0 2 0
.x ..    | 0 2 | * * 6 | 0 1 1
---------+-----+-------+------
x.6o.    | 6 0 | 6 0 0 | 1 * *
xx ..&#q | 2 2 | 1 2 1 | * 6 *
.x6.o    | 0 6 | 0 0 6 | * * 1

xx3xx&#u   → height = sqrt(2) = 1.414214
({6} || {6})

o.3o.    | 6 * | 1 1 1 0 0 | 1 1 1 0
.o3.o    | * 6 | 0 0 1 1 1 | 0 1 1 1
---------+-----+-----------+--------
x. ..    | 2 0 | 3 * * * * | 1 1 0 0
.. x.    | 2 0 | * 3 * * * | 1 0 1 0
oo3oo&#q | 1 1 | * * 6 * * | 0 1 1 0
.x ..    | 0 2 | * * * 3 * | 0 1 0 1
.. .x    | 0 2 | * * * * 3 | 0 0 1 1
---------+-----+-----------+--------
x.3x.    | 6 0 | 3 3 0 0 0 | 1 * * *
xx ..&#q | 2 2 | 1 0 2 1 0 | * 3 * *
.. xx&#q | 2 2 | 0 1 2 0 1 | * * 3 *
.x3.x    | 0 6 | 0 0 0 3 3 | * * * 1

xux qqq&#xt   → both heights = sqrt(3)/2 = 0.866025
((x,q)-{4} || pseudo (u,q)-{4} || (x,q)-{4})

o.. o..     | 4 * * | 1 1 1 0 0 0 0 | 1 1 1 0 0
.o. .o.     | * 4 * | 0 0 1 1 1 0 0 | 0 1 1 1 0
..o ..o     | * * 4 | 0 0 0 0 1 1 1 | 0 0 1 1 1
------------+-------+---------------+----------
x.. ...     | 2 0 0 | 2 * * * * * * | 1 0 1 0 0
... q..     | 2 0 0 | * 2 * * * * * | 1 1 0 0 0
oo. oo.&#x  | 1 1 0 | * * 4 * * * * | 0 1 1 0 0
... .q.     | 0 2 0 | * * * 2 * * * | 0 1 0 1 0
.oo .oo&#x  | 0 1 1 | * * * * 4 * * | 0 0 1 1 0
..x ...     | 0 0 2 | * * * * * 2 * | 0 0 1 0 1
... ..q     | 0 0 2 | * * * * * * 2 | 0 0 0 1 1
------------+-------+---------------+----------
x.. q..     | 4 0 0 | 2 2 0 0 0 0 0 | 1 * * * *
... qq.&#x  | 2 2 0 | 0 1 2 1 0 0 0 | * 2 * * *
xux ...&#xt | 2 2 2 | 1 0 2 0 2 1 0 | * * 2 * *
... .qq&#x  | 0 2 2 | 0 0 0 1 2 0 1 | * * * 2 *
..x ..q     | 0 0 4 | 0 0 0 0 0 2 2 | * * * * 1

or
o.. o..      & | 8 * | 1 1 1 0 | 1 1 1
.o. .o.        | * 4 | 0 0 2 1 | 0 2 1
---------------+-----+---------+------
x.. ...      & | 2 0 | 4 * * * | 1 0 1
... q..      & | 2 0 | * 4 * * | 1 1 0
oo. oo.&#x   & | 1 1 | * * 8 * | 0 1 1
... .q.        | 0 2 | * * * 2 | 0 2 0
---------------+-----+---------+------
x.. q..      & | 4 0 | 2 2 0 0 | 2 * *
... qq.&#x   & | 2 2 | 0 1 2 1 | * 4 *
xux ...&#xt    | 4 2 | 2 0 4 0 | * * 2