h    		where:
         		h = x3o3o *b3o (hex)
H       H		H = o3o3x *b3o (gyro hex)

This segmentopeta is derived from hax as its monostratic segment, in fact its hex-first rotunda. Even though it thus clearly omits all the 8 vertices of its diametrally opposite hex, there is a different and more symmetrical octadiminished hemihexeract as well, which therefore better deserves the according name: odhax.

Incidence matrix according to Dynkin symbol

ox xo3oo3oo *c3ox&#x   → height = 1/2
(hex || gyro hexip)

o. o.3o.3o. *c3o.    | 8  * ♦  6  8 0  0 | 12  4 24 12  0  0 | 4 4 12  6 24 12  8  0  0  0 | 1 12  6  4  8  8  2 0 0 0 | 4 4 1 2 0
.o .o3.o3.o *c3.o    | * 16 |  0  4 1  6 |  0  4  6 12  6 12 | 0 0  6 12  4  6 12 12  4  4 | 0  4  6 12  1  4  4 4 4 1 | 1 4 4 1 1
---------------------+------+------------+-------------------+-----------------------------+---------------------------+----------
.. x. .. ..    ..    | 2  0 | 24  * *  * ♦  4  0  4  0  0  0 | 2 2  2  0  8  2  0  0  0  0 | 1  4  1  0  4  4  0 0 0 0 | 2 2 0 2 0
oo oo3oo3oo *c3oo&#x | 1  1 |  * 64 *  * |  0  1  3  3  0  0 | 0 0  3  3  3  3  3  0  0  0 | 0  3  3  3  1  3  1 0 0 0 | 1 3 1 1 0
.x .. .. ..    ..    | 0  2 |  *  * 8  * |  0  4  0  0  6  0 | 0 0  6 12  0  0  0 12  0  0 | 0  4  6 12  0  0  0 4 4 0 | 1 4 4 0 1
.. .. .. ..    .x    | 0  2 |  *  * * 48 |  0  0  0  2  1  4 | 0 0  0  2  0  1  4  4  2  2 | 0  0  1  4  0  2  2 2 2 1 | 0 2 2 1 1
---------------------+------+------------+-------------------+-----------------------------+---------------------------+----------
.. x.3o. ..    ..    | 3  0 |  3  0 0  0 | 32  *  *  *  *  * | 1 1  0  0  2  0  0  0  0  0 | 1  1  0  0  2  2  0 0 0 0 | 1 1 0 2 0
ox .. .. ..    ..&#x | 1  2 |  0  2 1  0 |  * 32  *  *  *  * | 0 0  3  3  0  0  0  0  0  0 | 0  3  3  3  0  0  0 0 0 0 | 1 3 1 0 0
.. xo .. ..    ..&#x | 2  1 |  1  2 0  0 |  *  * 96  *  *  * | 0 0  1  0  2  1  0  0  0  0 | 0  2  1  0  1  2  0 0 0 0 | 1 2 0 1 0
.. .. .. ..    ox&#x | 1  2 |  0  2 0  1 |  *  *  * 96  *  * | 0 0  0  1  0  1  2  0  0  0 | 0  0  1  2  0  2  1 0 0 0 | 0 2 1 1 0
.x .. .. ..    .x    | 0  4 |  0  0 2  2 |  *  *  *  * 24  * | 0 0  0  2  0  0  0  4  0  0 | 0  0  1  4  0  0  0 2 2 0 | 0 2 2 0 1
.. .. .o .. *c3.x    | 0  3 |  0  0 0  3 |  *  *  *  *  * 64 | 0 0  0  0  0  0  1  1  1  1 | 0  0  0  1  0  1  1 1 1 1 | 0 1 1 1 1
---------------------+------+------------+-------------------+-----------------------------+---------------------------+----------
.. x.3o.3o.    ..    ♦ 4  0 |  6  0 0  0 |  4  0  0  0  0  0 | 8 *  *  *  *  *  *  *  *  * | 1  0  0  0  2  0  0 0 0 0 | 1 0 0 2 0
.. x.3o. .. *c3o.    ♦ 4  0 |  6  0 0  0 |  4  0  0  0  0  0 | * 8  *  *  *  *  *  *  *  * | 1  0  0  0  0  2  0 0 0 0 | 0 1 0 2 0
ox xo .. ..    ..&#x ♦ 2  2 |  1  4 1  0 |  0  2  2  0  0  0 | * * 48  *  *  *  *  *  *  * | 0  2  1  0  0  0  0 0 0 0 | 1 2 0 0 0
ox .. .. ..    ox&#x ♦ 1  4 |  0  4 2  2 |  0  2  0  2  1  0 | * *  * 48  *  *  *  *  *  * | 0  0  1  2  0  0  0 0 0 0 | 0 2 1 0 0
.. xo3oo ..    ..&#x ♦ 3  1 |  3  3 0  0 |  1  0  3  0  0  0 | * *  *  * 64  *  *  *  *  * | 0  1  0  0  1  1  0 0 0 0 | 1 1 0 1 0
.. xo .. ..    ox&#x ♦ 2  2 |  1  4 0  1 |  0  0  2  2  0  0 | * *  *  *  * 48  *  *  *  * | 0  0  1  0  0  2  0 0 0 0 | 0 2 0 1 0
.. .. oo .. *c3ox&#x ♦ 1  3 |  0  3 0  3 |  0  0  0  3  0  1 | * *  *  *  *  * 64  *  *  * | 0  0  0  1  0  1  1 0 0 0 | 0 1 1 1 0
.x .. .o .. *c3.x    ♦ 0  6 |  0  0 3  6 |  0  0  0  0  3  2 | * *  *  *  *  *  * 32  *  * | 0  0  0  1  0  0  0 1 1 0 | 0 1 1 0 1
.. .o3.o .. *c3.x    ♦ 0  4 |  0  0 0  6 |  0  0  0  0  0  4 | * *  *  *  *  *  *  * 16  * | 0  0  0  0  0  1  0 1 0 1 | 0 1 0 1 1
.. .. .o3.o *c3.x    ♦ 0  4 |  0  0 0  6 |  0  0  0  0  0  4 | * *  *  *  *  *  *  *  * 16 | 0  0  0  0  0  0  1 0 1 1 | 0 0 1 1 1
---------------------+------+------------+-------------------+-----------------------------+---------------------------+----------
.. x.3o.3o. *c3o.    ♦ 8  0 | 24  0 0  0 | 32  0  0  0  0  0 | 8 8  0  0  0  0  0  0  0  0 | 1  *  *  *  *  *  * * * * | 0 0 0 2 0
ox xo3oo ..    ..&#x ♦ 3  2 |  3  6 1  0 |  1  3  6  0  0  0 | 0 0  3  0  2  0  0  0  0  0 | * 32  *  *  *  *  * * * * | 1 1 0 0 0
ox xo .. ..    ox&#x ♦ 2  4 |  1  8 2  2 |  0  4  4  4  1  0 | 0 0  2  2  0  2  0  0  0  0 | *  * 24  *  *  *  * * * * | 0 2 0 0 0
ox .. oo .. *c3ox&#x ♦ 1  6 |  0  6 3  6 |  0  3  0  6  3  2 | 0 0  0  3  0  0  2  1  0  0 | *  *  * 32  *  *  * * * * | 0 1 1 0 0
.. xo3oo3oo    ..&#x ♦ 4  1 |  6  4 0  0 |  4  0  6  0  0  0 | 1 0  0  0  4  0  0  0  0  0 | *  *  *  * 16  *  * * * * | 1 0 0 1 0
.. xo3oo .. *c3ox&#x ♦ 4  4 |  6 12 0  6 |  4  0 12 12  0  4 | 0 1  0  0  4  6  4  0  1  0 | *  *  *  *  * 16  * * * * | 0 1 0 1 0
.. .. oo3oo *c3ox&#x ♦ 1  4 |  0  4 0  6 |  0  0  0  6  0  4 | 0 0  0  0  0  0  4  0  0  1 | *  *  *  *  *  * 16 * * * | 0 0 1 1 0
.x .o3.o .. *c3.x    ♦ 0  8 |  0  0 4 12 |  0  0  0  0  6  8 | 0 0  0  0  0  0  0  4  2  0 | *  *  *  *  *  *  * 8 * * | 0 1 0 0 1
.x .. .o3.o *c3.x    ♦ 0  8 |  0  0 4 12 |  0  0  0  0  6  8 | 0 0  0  0  0  0  0  4  0  2 | *  *  *  *  *  *  * * 8 * | 0 0 1 0 1
.. .o3.o3.o *c3.x    ♦ 0  8 |  0  0 0 24 |  0  0  0  0  0 32 | 0 0  0  0  0  0  0  0  8  8 | *  *  *  *  *  *  * * * 2 | 0 0 0 1 1
---------------------+------+------------+-------------------+-----------------------------+---------------------------+----------
ox xo3oo3oo    ..&#x ♦ 4  2 |  6  8 1  0 |  4  4 12  0  0  0 | 1 0  6  0  8  0  0  0  0  0 | 0  4  0  0  2  0  0 0 0 0 | 8 * * * *
ox xo3oo .. *c3ox&#x ♦ 4  8 |  6 24 4 12 |  4 12 24 24  6  8 | 0 1 12 12  8 12  8  4  2  0 | 0  4  6  4  0  2  0 1 0 0 | * 8 * * *
ox .. oo3oo *c3ox&#x ♦ 1  8 |  0  8 4 12 |  0  4  0 12  6  8 | 0 0  0  6  0  0  8  4  0  2 | 0  0  0  4  0  0  2 0 1 0 | * * 8 * *
.. xo3oo3oo *c3ox&#x ♦ 8  8 | 24 32 0 24 | 32  0 48 48  0 32 | 8 8  0  0 32 24 32  0  8  8 | 1  0  0  0  8  8  8 0 0 1 | * * * 2 *
.x .o3.o3.o *c3.x    ♦ 0 16 |  0  0 8 48 |  0  0  0  0 24 64 | 0 0  0  0  0  0  0 32 16 16 | 0  0  0  0  0  0  0 8 8 2 | * * * * 1