o x3x    
             
             
x o3x   x x3o

This polyteron can be obtained from a wedge of rat. In fact use the opeaco and the diminish the co base on either side down to the equatorial hexagon. These diminishing planes will scratch the other base just tangential.

Incidence matrix according to Dynkin symbol

oxx xox3xxo&#x   → height(1,2) = height(1,3) = sqrt(5/12) = 0.645497
                   height(2,3) = sqrt(2/3) = 0.816497
                   height(1,23) = 1/2 (ope || {6})

o.. o..3o..    | 6 * * | 1 1  2  2 0 0  0 0 0 | 1 1 2 2  2 1 2 2 0 0 0 0 0 0 0 | 1 1 2 1 2 2 1 1 2 0 0 0 0 0 | 1 1 1 2 1 0
.o. .o.3.o.    | * 6 * | 0 0  2  0 1 2  2 0 0 | 0 2 1 2  2 0 0 0 2 1 2 1 2 0 0 | 1 2 1 2 1 2 0 0 0 1 1 2 1 0 | 1 1 2 1 0 1
..o ..o3..o    | * * 6 | 0 0  0  2 0 0  2 1 2 | 0 0 0 0  2 2 2 1 0 0 2 2 1 2 1 | 0 0 0 2 2 1 2 1 1 0 2 1 1 1 | 0 2 1 1 1 1
---------------+-------+----------------------+--------------------------------+-----------------------------+------------
... x.. ...    | 2 0 0 | 3 *  *  * * *  * * * | 1 0 2 0  0 0 2 0 0 0 0 0 0 0 0 | 1 0 2 0 2 0 1 0 2 0 0 0 0 0 | 1 1 0 2 1 0
... ... x..    | 2 0 0 | * 3  *  * * *  * * * | 1 0 0 2  0 0 0 2 0 0 0 0 0 0 0 | 0 1 2 0 0 2 0 1 2 0 0 0 0 0 | 1 0 1 2 1 0
oo. oo.3oo.&#x | 1 1 0 | * * 12  * * *  * * * | 0 1 1 1  1 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | 1 1 1 1 0 0
o.o o.o3o.o&#x | 1 0 1 | * *  * 12 * *  * * * | 0 0 0 0  1 1 1 1 0 0 0 0 0 0 0 | 0 0 0 1 1 1 1 1 1 0 0 0 0 0 | 0 1 1 1 1 0
.x. ... ...    | 0 2 0 | * *  *  * 3 *  * * * | 0 2 0 0  0 0 0 0 2 0 2 0 0 0 0 | 1 2 0 2 0 0 0 0 0 1 1 2 0 0 | 1 1 2 0 0 1
... ... .x.    | 0 2 0 | * *  *  * * 6  * * * | 0 0 0 1  0 0 0 0 1 1 0 0 1 0 0 | 0 1 1 0 0 1 0 0 0 1 0 1 1 0 | 1 0 1 1 0 1
.oo .oo3.oo&#x | 0 1 1 | * *  *  * * * 12 * * | 0 0 0 0  1 0 0 0 0 0 1 1 1 0 0 | 0 0 0 1 1 1 0 0 0 0 1 1 1 0 | 0 1 1 1 0 1
..x ... ...    | 0 0 2 | * *  *  * * *  * 3 * | 0 0 0 0  0 2 0 0 0 0 2 0 0 2 0 | 0 0 0 2 0 0 2 2 0 0 2 1 0 1 | 0 2 1 0 1 1
... ..x ...    | 0 0 2 | * *  *  * * *  * * 6 | 0 0 0 0  0 0 1 0 0 0 0 1 0 1 1 | 0 0 0 0 1 0 1 0 1 0 1 0 1 1 | 0 1 0 1 1 1
---------------+-------+----------------------+--------------------------------+-----------------------------+------------
... x..3x..    | 6 0 0 | 3 3  0  0 0 0  0 0 0 | 1 * * *  * * * * * * * * * * * | 0 0 2 0 0 0 0 0 2 0 0 0 0 0 | 1 0 0 2 1 0
ox. ... ...&#x | 1 2 0 | 0 0  2  0 1 0  0 0 0 | * 6 * *  * * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
... xo. ...&#x | 2 1 0 | 1 0  2  0 0 0  0 0 0 | * * 6 *  * * * * * * * * * * * | 1 0 1 0 1 0 0 0 0 0 0 0 0 0 | 1 1 0 1 0 0
... ... xx.&#x | 2 2 0 | 0 1  2  0 0 1  0 0 0 | * * * 6  * * * * * * * * * * * | 0 1 1 0 0 1 0 0 0 0 0 0 0 0 | 1 0 1 1 0 0
ooo ooo3ooo&#x | 1 1 1 | 0 0  1  1 0 0  1 0 0 | * * * * 12 * * * * * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0
o.x ... ...&#x | 1 0 2 | 0 0  0  2 0 0  0 1 0 | * * * *  * 6 * * * * * * * * * | 0 0 0 1 0 0 1 1 0 0 0 0 0 0 | 0 1 1 0 1 0
... x.x ...&#x | 2 0 2 | 1 0  0  2 0 0  0 0 1 | * * * *  * * 6 * * * * * * * * | 0 0 0 0 1 0 1 0 1 0 0 0 0 0 | 0 1 0 1 1 0
... ... x.o&#x | 2 0 1 | 0 1  0  2 0 0  0 0 0 | * * * *  * * * 6 * * * * * * * | 0 0 0 0 0 1 0 1 1 0 0 0 0 0 | 0 0 1 1 1 0
.x. ... .x.    | 0 4 0 | 0 0  0  0 2 2  0 0 0 | * * * *  * * * * 3 * * * * * * | 0 1 0 0 0 0 0 0 0 1 0 1 0 0 | 1 0 1 0 0 1
... .o.3.x.    | 0 3 0 | 0 0  0  0 0 3  0 0 0 | * * * *  * * * * * 2 * * * * * | 0 0 1 0 0 0 0 0 0 1 0 0 1 0 | 1 0 0 1 0 1
.xx ... ...&#x | 0 2 2 | 0 0  0  0 1 0  2 1 0 | * * * *  * * * * * * 6 * * * * | 0 0 0 1 0 0 0 0 0 0 1 1 0 0 | 0 1 1 0 0 1
... .ox ...&#x | 0 1 2 | 0 0  0  0 0 0  2 0 1 | * * * *  * * * * * * * 6 * * * | 0 0 0 0 1 0 0 0 0 0 1 0 1 0 | 0 1 0 1 0 1
... ... .xo&#x | 0 2 1 | 0 0  0  0 0 1  2 0 0 | * * * *  * * * * * * * * 6 * * | 0 0 0 0 0 1 0 0 0 0 0 1 1 0 | 0 0 1 1 0 1
..x ..x ...    | 0 0 4 | 0 0  0  0 0 0  0 2 2 | * * * *  * * * * * * * * * 3 * | 0 0 0 0 0 0 1 0 0 0 1 0 0 1 | 0 1 0 0 1 1
... ..x3..o    | 0 0 3 | 0 0  0  0 0 0  0 0 3 | * * * *  * * * * * * * * * * 2 | 0 0 0 0 0 0 0 0 1 0 0 0 1 1 | 0 0 0 1 1 1
---------------+-------+----------------------+--------------------------------+-----------------------------+------------
ox. xo. ...&#x ♦ 2 2 0 | 1 0  4  0 1 0  0 0 0 | 0 2 2 0  0 0 0 0 0 0 0 0 0 0 0 | 3 * * * * * * * * * * * * * | 1 1 0 0 0 0
ox. ... xx.&#x ♦ 2 4 0 | 0 1  4  0 2 2  0 0 0 | 0 2 0 2  0 0 0 0 1 0 0 0 0 0 0 | * 3 * * * * * * * * * * * * | 1 0 1 0 0 0
... xo.3xx.&#x ♦ 6 3 0 | 3 3  6  0 0 3  0 0 0 | 1 0 3 3  0 0 0 0 0 1 0 0 0 0 0 | * * 2 * * * * * * * * * * * | 1 0 0 1 0 0
oxx ... ...&#x ♦ 1 2 2 | 0 0  2  2 1 0  2 1 0 | 0 1 0 0  2 1 0 0 0 0 1 0 0 0 0 | * * * 6 * * * * * * * * * * | 0 1 1 0 0 0
... xox ...&#x ♦ 2 1 2 | 1 0  2  2 0 0  2 0 1 | 0 0 1 0  2 0 1 0 0 0 0 1 0 0 0 | * * * * 6 * * * * * * * * * | 0 1 0 1 0 0
... ... xxo&#x ♦ 2 2 1 | 0 1  2  2 0 1  2 0 0 | 0 0 0 1  2 0 0 1 0 0 0 0 1 0 0 | * * * * * 6 * * * * * * * * | 0 0 1 1 0 0
o.x x.x ...&#x ♦ 2 0 4 | 1 0  0  4 0 0  0 2 2 | 0 0 0 0  0 2 2 0 0 0 0 0 0 1 0 | * * * * * * 3 * * * * * * * | 0 1 0 0 1 0
o.x ... x.o&#x ♦ 2 0 2 | 0 1  0  4 0 0  0 2 0 | 0 0 0 0  0 2 0 2 0 0 0 0 0 0 0 | * * * * * * * 3 * * * * * * | 0 0 1 0 1 0
... x.x3x.o&#x ♦ 6 0 3 | 3 3  0  6 0 0  0 0 3 | 1 0 0 0  0 0 3 3 0 0 0 0 0 0 1 | * * * * * * * * 2 * * * * * | 0 0 0 1 1 0
.x. .o.3.x.    ♦ 0 6 0 | 0 0  0  0 3 6  0 0 0 | 0 0 0 0  0 0 0 0 3 2 0 0 0 0 0 | * * * * * * * * * 1 * * * * | 1 0 0 0 0 1
.xx .ox ...&#x ♦ 0 2 4 | 0 0  0  0 1 0  4 2 2 | 0 0 0 0  0 0 0 0 0 0 2 2 0 1 0 | * * * * * * * * * * 3 * * * | 0 1 0 0 0 1
.xx ... .xo&#x ♦ 0 4 2 | 0 0  0  0 2 2  4 1 0 | 0 0 0 0  0 0 0 0 1 0 2 0 2 0 0 | * * * * * * * * * * * 3 * * | 0 0 1 0 0 1
... .ox3.xo&#x ♦ 0 3 3 | 0 0  0  0 0 3  6 0 3 | 0 0 0 0  0 0 0 0 0 1 0 3 3 0 1 | * * * * * * * * * * * * 2 * | 0 0 0 1 0 1
..x ..x3..o    ♦ 0 0 6 | 0 0  0  0 0 0  0 3 6 | 0 0 0 0  0 0 0 0 0 0 0 0 0 3 2 | * * * * * * * * * * * * * 1 | 0 0 0 0 1 1
---------------+-------+----------------------+--------------------------------+-----------------------------+------------
ox. xo.3xx.&#x ♦ 6 6 0 | 3 3 12  0 3 6  0 0 0 | 1 6 6 6  0 0 0 0 3 2 0 0 0 0 0 | 3 3 2 0 0 0 0 0 0 1 0 0 0 0 | 1 * * * * *
oxx xox ...&#x ♦ 2 2 4 | 1 0  4  4 1 0  4 2 2 | 0 2 2 0  4 2 2 0 0 0 2 2 0 1 0 | 1 0 0 2 2 0 1 0 0 0 1 0 0 0 | * 3 * * * *
oxx ... xxo&#x ♦ 2 4 2 | 0 1  4  4 2 2  4 1 0 | 0 2 0 2  4 2 0 2 1 0 2 0 2 0 0 | 0 1 0 2 0 2 0 1 0 0 0 1 0 0 | * * 3 * * *
... xox3xxo&#x ♦ 6 3 3 | 3 3  6  6 0 3  6 0 3 | 1 0 3 3  6 0 3 3 0 1 0 3 3 0 1 | 0 0 1 0 3 3 0 0 1 0 0 0 1 0 | * * * 2 * *
o.x x.x3x.o&#x ♦ 6 0 6 | 3 3  0 12 0 0  0 3 6 | 1 0 0 0  0 6 6 6 0 0 0 0 0 3 2 | 0 0 0 0 0 0 3 3 2 0 0 0 0 1 | * * * * 1 *
.xx .ox3.xo&#x ♦ 0 6 6 | 0 0  0  0 3 6 12 3 6 | 0 0 0 0  0 0 0 0 3 2 6 6 6 3 2 | 0 0 0 0 0 0 0 0 0 1 3 3 2 1 | * * * * * 1