o o3x  
              
              
              
x o3x    x x3o

This wedge of dot can be obtained from its tet-first half, tetaope, as a diminishing. In fact the cutting hyperplane rasps the top tet from one of its tips down to its opposite triangle, whereas at the opposite base it just scratches tangentially, that is, the ope there remains unchanged.

None the same it well can be extrapolated towards any general N-gonal symmetry other than just 3, then resulting in (N-gon, N-p, gyro N-p)-trigonic.

Incidence matrix according to Dynkin symbol

oxx oox3xxo&#x   → height(1,2) = sqrt(3)/2 = 0.866025
                   height(1,3) = sqrt(5/12) = 0.645497
                   height(2,3) = sqrt(2/3) = 0.816497

o.. o..3o..    | 3 * * | 2 2  4 0 0  0 0 0 | 1 1 4  4 2 2 4 0 0 0 0 0 0 0 | 2 2 2 2 4 1 2 2 0 0 0 0 0 | 1 1 2 2 1 0
.o. .o.3.o.    | * 6 * | 0 1  0 1 2  2 0 0 | 0 1 2  2 0 0 0 2 1 2 1 2 0 0 | 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  2 0 0  2 1 2 | 0 0 0  2 2 2 1 0 0 2 2 1 2 1 | 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 | 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 | * 6  * * *  * * * | 0 1 2  2 0 0 0 0 0 0 0 0 0 0 | 2 1 2 1 2 0 0 0 0 0 0 0 0 | 1 1 2 1 0 0
o.o o.o3o.o&#x | 1 0 1 | * * 12 * *  * * * | 0 0 0  1 1 1 1 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 1 0  0 0 0 0 2 0 2 0 0 0 0 | 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 1  0 0 0 0 1 1 0 0 1 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  1 0 0 0 0 0 1 1 1 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 2 0 0 0 0 2 0 0 2 0 | 0 0 2 0 0 2 1 0 0 2 1 0 1 | 0 2 1 0 1 1
... ..x ...    | 0 0 2 | * *  * * *  * * 6 | 0 0 0  0 0 1 0 0 0 0 1 0 1 1 | 0 0 0 1 0 1 0 1 0 1 0 1 1 | 0 1 0 1 1 1
---------------+-------+-------------------+------------------------------+---------------------------+------------
... o..3x..    | 3 0 0 | 3 0  0 0 0  0 0 0 | 1 * *  * * * * * * * * * * * | 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 2  0 1 0  0 0 0 | * 3 *  * * * * * * * * * * * | 2 0 2 0 0 0 0 0 0 0 0 0 0 | 1 1 2 0 0 0
... ... xx.&#x | 2 2 0 | 1 2  0 0 1  0 0 0 | * * 6  * * * * * * * * * * * | 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 1  1 0 0  1 0 0 | * * * 12 * * * * * * * * * * | 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  2 0 0  0 1 0 | * * *  * 6 * * * * * * * * * | 0 0 1 0 0 1 1 0 0 0 0 0 0 | 0 1 1 0 1 0
... o.x ...&#x | 1 0 2 | 0 0  2 0 0  0 0 1 | * * *  * * 6 * * * * * * * * | 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 | 1 0  2 0 0  0 0 0 | * * *  * * * 6 * * * * * * * | 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 2 2  0 0 0 | * * *  * * * * 3 * * * * * * | 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 3  0 0 0 | * * *  * * * * * 2 * * * * * | 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 1 0  2 1 0 | * * *  * * * * * * 6 * * * * | 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  2 0 1 | * * *  * * * * * * * 6 * * * | 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 1  2 0 0 | * * *  * * * * * * * * 6 * * | 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 2 2 | * * *  * * * * * * * * * 3 * | 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 3 | * * *  * * * * * * * * * * 2 | 0 0 0 0 0 0 0 1 0 0 0 1 1 | 0 0 0 1 1 1
---------------+-------+-------------------+------------------------------+---------------------------+------------
ox. ... xx.&#x ♦ 2 4 0 | 1 4  0 2 2  0 0 0 | 0 2 2  0 0 0 0 1 0 0 0 0 0 0 | 3 * * * * * * * * * * * * | 1 0 1 0 0 0
... oo.3xx.&#x ♦ 3 3 0 | 3 3  0 0 3  0 0 0 | 1 0 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 2  2 1 0  2 1 0 | 0 1 0  2 1 0 0 0 0 1 0 0 0 0 | * * 6 * * * * * * * * * * | 0 1 1 0 0 0
... oox ...&#x ♦ 1 1 2 | 0 1  2 0 0  2 0 1 | 0 0 0  2 0 1 0 0 0 0 1 0 0 0 | * * * 6 * * * * * * * * * | 0 1 0 1 0 0
... ... xxo&#x ♦ 2 2 1 | 1 2  2 0 1  2 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 o.x ...&#x ♦ 1 0 4 | 0 0  4 0 0  0 2 2 | 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 | 1 0  4 0 0  0 1 0 | 0 0 0  0 2 0 2 0 0 0 0 0 0 0 | * * * * * * 3 * * * * * * | 0 0 1 0 1 0
... o.x3x.o&#x ♦ 3 0 3 | 3 0  6 0 0  0 0 3 | 1 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 3 6  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 1 0  4 2 2 | 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 2 2  4 1 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 3  6 0 3 | 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 3 6 | 0 0 0  0 0 0 0 0 0 0 0 0 3 2 | * * * * * * * * * * * * 1 | 0 0 0 0 1 1
---------------+-------+-------------------+------------------------------+---------------------------+------------
ox. oo.3xx.&#x ♦ 3 6 0 | 3 6  0 3 6  0 0 0 | 1 3 6  0 0 0 0 3 2 0 0 0 0 0 | 3 2 0 0 0 0 0 0 1 0 0 0 0 | 1 * * * * *
oxx oox ...&#x ♦ 1 2 4 | 0 2  4 1 0  4 2 2 | 0 1 0  4 2 2 0 0 0 2 2 0 1 0 | 0 0 2 2 0 1 0 0 0 1 0 0 0 | * 3 * * * *
oxx ... xxo&#x ♦ 2 4 2 | 1 4  4 2 2  4 1 0 | 0 2 2  4 2 0 2 1 0 2 0 2 0 0 | 1 0 2 0 2 0 1 0 0 0 1 0 0 | * * 3 * * *
... oox3xxo&#x ♦ 3 3 3 | 3 3  6 0 3  6 0 3 | 1 0 3  6 0 3 3 0 1 0 3 3 0 1 | 0 1 0 3 3 0 0 1 0 0 0 1 0 | * * * 2 * *
o.x o.x3x.o&#x ♦ 3 0 6 | 3 0 12 0 0  0 3 6 | 1 0 0  0 6 6 6 0 0 0 0 0 3 2 | 0 0 0 0 0 3 3 2 0 0 0 0 1 | * * * * 1 *
.xx .ox3.xo&#x ♦ 0 6 6 | 0 0  0 3 6 12 3 6 | 0 0 0  0 0 0 0 3 2 6 6 6 3 2 | 0 0 0 0 0 0 0 0 1 3 3 2 1 | * * * * * 1