Acronym hexathex Name (degenerate) hex atop thex Circumradius ∞   i.e. flat in euclidean space Confer general polytopal classes: lace simplices   decomposition

It either can be thought of as a degenerate 5D segmentotope with zero height, or as a 4D euclidean decomposition of the larger base into smaller bits.

Incidence matrix according to Dynkin symbol

```xx3ox3oo4oo&#x   → height = 0
(hex || thex)

o.3o.3o.4o.    | 8  * |  6  6  0  0 | 12  6 12  0  0 |  8 12  8  0 0 | 1  8 1 0
.o3.o3.o4.o    | * 48 |  0  1  1  4 |  0  1  4  4  4 |  0  4  4  4 1 | 0  4 1 1
---------------+------+-------------+----------------+---------------+---------
x. .. .. ..    | 2  0 | 24  *  *  * |  4  1  0  0  0 |  4  4  0  0 0 | 1  4 0 0
oo3oo3oo4oo&#x | 1  1 |  * 48  *  * |  0  1  4  0  0 |  0  4  4  0 0 | 0  4 1 0
.x .. .. ..    | 0  2 |  *  * 24  * |  0  1  0  4  0 |  0  4  0  4 0 | 0  4 0 1
.. .x .. ..    | 0  2 |  *  *  * 96 |  0  0  1  1  2 |  0  1  2  2 1 | 0  2 1 1
---------------+------+-------------+----------------+---------------+---------
x.3o. .. ..    | 3  0 |  3  0  0  0 | 32  *  *  *  * |  2  1  0  0 0 | 1  2 0 0
xx .. .. ..&#x | 2  2 |  1  2  1  0 |  * 24  *  *  * |  0  4  0  0 0 | 0  4 0 0
.. ox .. ..&#x | 1  2 |  0  2  0  1 |  *  * 96  *  * |  0  1  2  0 0 | 0  2 1 0
.x3.x .. ..    | 0  6 |  0  0  3  3 |  *  *  * 32  * |  0  1  0  2 0 | 0  2 0 1
.. .x3.o ..    | 0  3 |  0  0  0  3 |  *  *  *  * 64 |  0  0  1  1 1 | 0  1 1 1
---------------+------+-------------+----------------+---------------+---------
x.3o.3o. ..    ♦ 4  0 |  6  0  0  0 |  4  0  0  0  0 | 16  *  *  * * | 1  1 0 0
xx3ox .. ..&#x ♦ 3  6 |  3  6  3  3 |  1  3  3  1  0 |  * 32  *  * * | 0  2 0 0
.. ox3oo ..&#x ♦ 1  3 |  0  3  0  3 |  0  0  3  0  1 |  *  * 64  * * | 0  1 1 0
.x3.x3.o ..    ♦ 0 12 |  0  0  6 12 |  0  0  0  4  4 |  *  *  * 16 * | 0  1 0 1
.. .x3.o4.o    ♦ 0  6 |  0  0  0 12 |  0  0  0  0  8 |  *  *  *  * 8 | 0  0 1 1
---------------+------+-------------+----------------+---------------+---------
x.3o.3o.4o.    ♦ 8  0 | 24  0  0  0 | 32  0  0  0  0 | 16  0  0  0 0 | 1  * * *
xx3ox3oo ..&#x ♦ 4 12 |  6 12  6 12 |  4  6 12  4  4 |  1  4  4  1 0 | * 16 * *
.. ox3oo4oo&#x ♦ 1  6 |  0  6  0 12 |  0  0 12  0  8 |  0  0  8  0 1 | *  * 8 *
.x3.x3.o4.o    ♦ 0 48 |  0  0 24 96 |  0  0  0 32 64 |  0  0  0 16 8 | *  * * 1
```

```oo3ox3oo *b3xx&#x   → height = 0
(hex || thex)

o.3o.3o. *b3o.    | 8  * |  6  6  0  0 | 12 12  6  0  0  0 | 4 4  4  4 12 0 0 0 | 1 1 4 4 0
.o3.o3.o *b3.o    | * 48 |  0  1  4  1 |  0  4  1  2  2  4 | 0 0  2  2  4 1 2 2 | 0 1 2 2 1
------------------+------+-------------+-------------------+--------------------+----------
.. .. ..    x.    | 2  0 | 24  *  *  * |  4  0  1  0  0  0 | 2 2  0  0  4 0 0 0 | 1 0 2 2 0
oo3oo3oo *b3oo&#x | 1  1 |  * 48  *  * |  0  4  1  0  0  0 | 0 0  2  2  4 0 0 0 | 0 1 2 2 0
.. .x ..    ..    | 0  2 |  *  * 96  * |  0  1  0  1  1  1 | 0 0  1  1  1 1 1 1 | 0 1 1 1 1
.. .. ..    .x    | 0  2 |  *  *  * 24 |  0  0  1  0  0  4 | 0 0  0  0  4 0 2 2 | 0 0 2 2 1
------------------+------+-------------+-------------------+--------------------+----------
.. o. .. *b3x.    | 3  0 |  3  0  0  0 | 32  *  *  *  *  * | 1 1  0  0  1 0 0 0 | 1 0 1 1 0
.. ox ..    ..&#x | 1  2 |  0  2  1  0 |  * 96  *  *  *  * | 0 0  1  1  1 0 0 0 | 0 1 1 1 0
.. .. ..    xx&#x | 2  2 |  1  2  0  1 |  *  * 24  *  *  * | 0 0  0  0  4 0 0 0 | 0 0 2 2 0
.o3.x ..    ..    | 0  3 |  0  0  3  0 |  *  *  * 32  *  * | 0 0  1  0  0 1 1 0 | 0 1 1 0 1
.. .x3.o    ..    | 0  3 |  0  0  3  0 |  *  *  *  * 32  * | 0 0  0  1  0 1 0 1 | 0 1 0 1 1
.. .x .. *b3.x    | 0  6 |  0  0  3  3 |  *  *  *  *  * 32 | 0 0  0  0  1 0 1 1 | 0 0 1 1 1
------------------+------+-------------+-------------------+--------------------+----------
o.3o. .. *b3x.    ♦ 4  0 |  6  0  0  0 |  4  0  0  0  0  0 | 8 *  *  *  * * * * | 1 0 1 0 0
.. o.3o. *b3x.    ♦ 4  0 |  6  0  0  0 |  4  0  0  0  0  0 | * 8  *  *  * * * * | 1 0 0 1 0
oo3ox ..    ..&#x ♦ 1  3 |  0  3  3  0 |  0  3  0  1  0  0 | * * 32  *  * * * * | 0 1 1 0 0
.. ox3oo    ..&#x ♦ 1  3 |  0  3  3  0 |  0  3  0  0  1  0 | * *  * 32  * * * * | 0 1 0 1 0
.. ox .. *b3xx&#x ♦ 3  6 |  3  6  3  3 |  1  3  3  0  0  1 | * *  *  * 32 * * * | 0 0 1 1 0
.o3.x3.o    ..    ♦ 0  6 |  0  0 12  0 |  0  0  0  4  4  0 | * *  *  *  * 8 * * | 0 1 0 0 1
.o3.x .. *b3.x    ♦ 0 12 |  0  0 12  6 |  0  0  0  4  0  4 | * *  *  *  * * 8 * | 0 0 1 0 1
.. .x3.o *b3.x    ♦ 0 12 |  0  0 12  6 |  0  0  0  0  4  4 | * *  *  *  * * * 8 | 0 0 0 1 1
------------------+------+-------------+-------------------+--------------------+----------
o.3o.3o. *b3x.    ♦ 8  0 | 24  0  0  0 | 32  0  0  0  0  0 | 8 8  0  0  0 0 0 0 | 1 * * * *
oo3ox3oo    ..&#x ♦ 1  6 |  0  6 12  0 |  0 12  0  4  4  0 | 0 0  4  4  0 1 0 0 | * 8 * * *
oo3ox .. *b3xx&#x ♦ 4 12 |  6 12 12  6 |  4 12  6  4  0  4 | 1 0  4  0  4 0 1 0 | * * 8 * *
.. ox3oo *b3xx&#x ♦ 4 12 |  6 12 12  6 |  4 12  6  0  4  4 | 0 1  0  4  4 0 0 1 | * * * 8 *
.o3.x3.o *b3.x    ♦ 0 48 |  0  0 96 24 |  0  0  0 32 32 32 | 0 0  0  0  0 8 8 8 | * * * * 1
```