Acronym hexasc Name hexadecachoric scalene,hexadecachoron-pyramid pyramid,line atop fully orthogonal hexadecachoron,1/4-lune of hexacontatetrapeton Circumradius 1/sqrt(2) = 0.707107 Lace cityin approx. ASCII-art ```o where: o = o3o3o4o (point) H o H = x3o3o4o (hex) ``` Coordinates (0, 0, 0, 0, 0, 1/sqrt(2)) top-tip (0, 0, 0, 0, 1/sqrt(2), 0) right-tip (1/sqrt(2), 0, 0, 0, 0, 0)   & permutations and changes of sign in all but last coordinate hexadecachoron base Dihedral angles at pen between hix and hix:   arccos(-2/3) = 131.810315° at hex between hexpy and hexpy:   90° at pen between hix and hexpy:   arccos[1/sqrt(6)] = 65.905157° Confer uniform relative: gee   related segmentopeta: tacpy   octete   squepe   general polytopal classes: segmentopeta   scalene

Incidence matrix according to Dynkin symbol

```xo ox3oo3oo3oo4oo&#x   → height = 1/2
(line || perp hex)

o. o.3o.3o.4o.    | 2 * ♦ 1  8  0 | 8 24  0 | 24 32  0 | 32 16 0 | 16 1
.o .o3.o3.o4.o    | * 8 ♦ 0  2  6 | 1 12 12 |  6 24  8 | 12 16 1 |  8 2
------------------+-----+---------+---------+----------+---------+-----
x. .. .. .. ..    | 2 0 | 1  *  * ♦ 8  0  0 | 24  0  0 | 32  0 0 | 16 0
oo oo3oo3oo4oo&#x | 1 1 | * 16  * ♦ 1  6  0 |  6 12  0 | 12  8 0 |  8 1
.. .x .. .. ..    | 0 2 | *  * 24 | 0  2  4 |  1  8  4 |  4  8 1 |  4 2
------------------+-----+---------+---------+----------+---------+-----
xo .. .. .. ..&#x | 2 1 | 1  2  0 | 8  *  * ♦  6  0  0 | 12  0 0 |  8 0
.. ox .. .. ..&#x | 1 2 | 0  2  1 | * 48  * ♦  1  4  0 |  4  4 0 |  4 1
.. .x3.o .. ..    | 0 3 | 0  0  3 | *  * 32 |  0  2  2 |  1  4 1 |  2 2
------------------+-----+---------+---------+----------+---------+-----
xo ox .. .. ..&#x ♦ 2 2 | 1  4  1 | 2  2  0 | 24  *  * |  4  0 0 |  4 0
.. ox3oo .. ..&#x ♦ 1 3 | 0  3  3 | 0  3  1 |  * 64  * |  1  2 0 |  2 1
.. .x3.o3.o ..    ♦ 0 4 | 0  0  6 | 0  0  4 |  *  * 16 |  0  2 1 |  1 2
------------------+-----+---------+---------+----------+---------+-----
xo ox3oo .. ..&#x ♦ 2 3 | 1  6  3 | 3  6  1 |  3  2  0 | 32  * * |  2 0
.. ox3oo3oo ..&#x ♦ 1 4 | 0  4  6 | 0  6  4 |  0  4  1 |  * 32 * |  1 1
.. .x3.o3.o4.o    ♦ 0 8 | 0  0 24 | 0  0 32 |  0  0 16 |  *  * 1 |  0 2
------------------+-----+---------+---------+----------+---------+-----
xo ox3oo3oo ..&#x ♦ 2 4 | 1  8  6 | 4 12  4 |  6  8  1 |  4  2 0 | 16 *
.. ox3oo3oo4oo&#x ♦ 1 8 | 0  8 24 | 0 24 32 |  0 32 16 |  0 16 1 |  * 2
```

```xo ox3oo3oo *c3oo&#x   → height = 1/2
(line || perp hex)

o. o.3o.3o. *c3o.    | 2 * ♦ 1  8  0 | 8 24  0 | 24 32 0 0 | 32  8  8 0 | 8 8 1
.o .o3.o3.o *c3.o    | * 8 ♦ 0  2  6 | 1 12 12 |  6 24 4 4 | 12  8  8 1 | 4 4 2
---------------------+-----+---------+---------+-----------+------------+------
x. .. .. ..    ..    | 2 0 | 1  *  * ♦ 8  0  0 | 24  0 0 0 | 32  0  0 0 | 8 8 0
oo oo3oo3oo *c3oo&#x | 1 1 | * 16  * ♦ 1  6  0 |  6 12 0 0 | 12  4  4 0 | 4 4 1
.. .x .. ..    ..    | 0 2 | *  * 24 | 0  2  4 |  1  8 2 2 |  4  4  4 1 | 2 2 2
---------------------+-----+---------+---------+-----------+------------+------
xo .. .. ..    ..&#x | 2 1 | 1  2  0 | 8  *  * ♦  6  0 0 0 | 12  0  0 0 | 4 4 0
.. ox .. ..    ..&#x | 1 2 | 0  2  1 | * 48  * ♦  1  4 0 0 |  4  2  2 0 | 2 2 1
.. .x3.o ..    ..    | 0 3 | 0  0  3 | *  * 32 |  0  2 1 1 |  1  2  2 1 | 1 1 2
---------------------+-----+---------+---------+-----------+------------+------
xo ox .. ..    ..&#x ♦ 2 2 | 1  4  1 | 2  2  0 | 24  * * * |  4  0  0 0 | 2 2 0
.. ox3oo ..    ..&#x ♦ 1 3 | 0  3  3 | 0  3  1 |  * 64 * * |  1  1  1 0 | 1 1 1
.. .x3.o3.o    ..    ♦ 0 4 | 0  0  6 | 0  0  4 |  *  * 8 * |  0  2  0 1 | 1 0 2
.. .x3.o .. *c3.o    ♦ 0 4 | 0  0  6 | 0  0  4 |  *  * * 8 |  0  0  2 1 | 0 1 2
---------------------+-----+---------+---------+-----------+------------+------
xo ox3oo ..    ..&#x ♦ 2 3 | 1  6  3 | 3  6  1 |  3  2 0 0 | 32  *  * * | 1 1 0
.. ox3oo3oo    ..&#x ♦ 1 4 | 0  4  6 | 0  6  4 |  0  4 1 0 |  * 16  * * | 1 0 1
.. ox3oo .. *c3oo&#x ♦ 1 4 | 0  4  6 | 0  6  4 |  0  4 0 1 |  *  * 16 * | 0 1 1
.. .x3.o3.o *c3.o    ♦ 0 8 | 0  0 24 | 0  0 32 |  0  0 8 8 |  *  *  * 1 | 0 0 2
---------------------+-----+---------+---------+-----------+------------+------
xo ox3oo3oo    ..&#x ♦ 2 4 | 1  8  6 | 4 12  4 |  6  8 1 0 |  4  2  0 0 | 8 * *
xo ox3oo .. *c3oo&#x ♦ 2 4 | 1  8  6 | 4 12  4 |  6  8 0 1 |  4  0  2 0 | * 8 *
.. ox3oo3oo *c3oo&#x ♦ 1 8 | 0  8 24 | 0 24 32 |  0 32 8 8 |  0  8  8 1 | * * 2
```

```xoo oxo3ooo3oox&#x   → height(1,2) = height(1,3) = sqrt(3/8) = 0.612372
height(2,3) = 1/sqrt(2) = 0.707107

o.. o..3o..3o..    | 2 * * ♦ 1 4 4 0  0 0 | 4  6 4  6 12 0  0  0 0 | 6 4 6 4 12 12 12 0 0 0 0 0 | 4 1 4 1 12 12 4  6 4 0 | 1 1 4 6 4 1
.o. .o.3.o.3.o.    | * 4 * ♦ 0 2 0 3  3 0 | 1  6 0  0  6 3  6  3 0 | 3 6 0 0  3 12  6 1 3 3 1 0 | 3 2 0 0  6  3 6  6 2 1 | 1 0 3 3 1 2
..o ..o3..o3..o    | * * 4 ♦ 0 0 2 0  3 3 | 0  0 1  6  6 0  3  6 3 | 0 0 3 6  3  6 12 0 1 3 3 1 | 0 0 3 2  3  6 2  6 6 1 | 0 1 1 3 3 2
-------------------+-------+--------------+------------------------+----------------------------+------------------------+------------
x.. ... ... ...    | 2 0 0 | 1 * * *  * * ♦ 4  0 4  0  0 0  0  0 0 | 6 0 6 0 12  0  0 0 0 0 0 0 | 4 0 4 0 12 12 0  0 0 0 | 1 1 4 6 4 0
oo. oo.3oo.3oo.&#x | 1 1 0 | * 8 * *  * * ♦ 1  3 0  0  3 0  0  0 0 | 3 3 0 0  3  6  3 0 0 0 0 0 | 3 1 0 0  6  3 3  3 1 0 | 1 0 3 3 1 1
o.o o.o3o.o3o.o&#x | 1 0 1 | * * 8 *  * * ♦ 0  0 1  3  3 0  0  0 0 | 0 0 3 3  3  3  6 0 0 0 0 0 | 0 0 3 1  3  6 1  3 3 0 | 0 1 1 3 3 1
... .x. ... ...    | 0 2 0 | * * * 6  * * | 0  2 0  0  0 2  2  0 0 | 1 4 0 0  0  4  0 1 2 1 0 0 | 2 2 0 0  2  0 4  2 0 1 | 1 0 2 1 0 2
.oo .oo3.oo3.oo&#x | 0 1 1 | * * * * 12 * | 0  0 0  0  2 0  2  2 0 | 0 0 0 0  1  4  4 0 1 2 1 0 | 0 0 0 0  2  2 2  4 2 1 | 0 0 1 2 1 2
... ... ... ..x    | 0 0 2 | * * * *  * 6 | 0  0 0  2  0 0  0  2 2 | 0 0 1 4  0  0  4 0 0 1 2 1 | 0 0 2 2  0  2 0  2 4 1 | 0 1 0 1 2 2
-------------------+-------+--------------+------------------------+----------------------------+------------------------+------------
xo. ... ... ...&#x | 2 1 0 | 1 2 0 0  0 0 | 4  * *  *  * *  *  * * ♦ 3 0 0 0  3  0  0 0 0 0 0 0 | 3 0 0 0  6  3 0  0 0 0 | 1 0 3 3 1 0
... ox. ... ...&#x | 1 2 0 | 0 2 0 1  0 0 | * 12 *  *  * *  *  * * ♦ 1 2 0 0  0  2  0 0 0 0 0 0 | 2 1 0 0  2  0 2  1 0 0 | 1 0 2 1 0 1
x.o ... ... ...&#x | 2 0 1 | 1 0 2 0  0 0 | *  * 4  *  * *  *  * * ♦ 0 0 3 0  3  0  0 0 0 0 0 0 | 0 0 3 0  3  6 0  0 0 0 | 0 1 1 3 3 0
... ... ... o.x&#x | 1 0 2 | 0 0 2 0  0 1 | *  * * 12  * *  *  * * ♦ 0 0 1 2  0  0  2 0 0 0 0 0 | 0 0 2 1  0  2 0  1 2 0 | 0 1 0 1 2 1
ooo ooo3ooo3ooo&#x | 1 1 1 | 0 1 1 0  1 0 | *  * *  * 24 *  *  * * ♦ 0 0 0 0  1  2  2 0 0 0 0 0 | 0 0 0 0  2  2 1  2 1 0 | 0 0 1 2 1 1
... .x.3.o. ...    | 0 3 0 | 0 0 0 3  0 0 | *  * *  *  * 4  *  * * | 0 2 0 0  0  0  0 1 1 0 0 0 | 1 2 0 0  0  0 2  0 0 1 | 1 0 1 0 0 2
... .xo ... ...&#x | 0 2 1 | 0 0 0 1  2 0 | *  * *  *  * * 12  * * | 0 0 0 0  0  2  0 0 1 1 0 0 | 0 0 0 0  1  0 2  2 0 1 | 0 0 1 1 0 2
... ... ... .ox&#x | 0 1 2 | 0 0 0 0  2 1 | *  * *  *  * *  * 12 * | 0 0 0 0  0  0  2 0 0 1 1 0 | 0 0 0 0  0  1 0  2 2 1 | 0 0 0 1 1 2
... ... ..o3..x    | 0 0 3 | 0 0 0 0  0 3 | *  * *  *  * *  *  * 4 | 0 0 0 2  0  0  0 0 0 0 1 1 | 0 0 1 2  0  0 0  0 2 1 | 0 1 0 0 1 2
-------------------+-------+--------------+------------------------+----------------------------+------------------------+------------
xo. ox. ... ...&#x ♦ 2 2 0 | 1 4 0 1  0 0 | 2  2 0  0  0 0  0  0 0 | 6 * * *  *  *  * * * * * * | 2 0 0 0  2  0 0  0 0 0 | 1 0 2 1 0 0
... ox.3oo. ...&#x ♦ 1 3 0 | 0 3 0 3  0 0 | 0  3 0  0  0 1  0  0 0 | * 8 * *  *  *  * * * * * * | 1 1 0 0  0  0 1  0 0 0 | 1 0 1 0 0 1
x.o ... ... o.x&#x ♦ 2 0 2 | 1 0 4 0  0 1 | 0  0 2  2  0 0  0  0 0 | * * 6 *  *  *  * * * * * * | 0 0 2 0  0  2 0  0 0 0 | 0 1 0 1 2 0
... ... o.o3o.x&#x ♦ 1 0 3 | 0 0 3 0  0 3 | 0  0 0  3  0 0  0  0 1 | * * * 8  *  *  * * * * * * | 0 0 1 1  0  0 0  0 1 0 | 0 1 0 0 1 1
xoo ... ... ...&#x ♦ 2 1 1 | 1 2 2 0  1 0 | 1  0 1  0  2 0  0  0 0 | * * * * 12  *  * * * * * * | 0 0 0 0  2  2 0  0 0 0 | 0 0 1 2 1 0
... oxo ... ...&#x ♦ 1 2 1 | 0 2 1 1  2 0 | 0  1 0  0  2 0  1  0 0 | * * * *  * 24  * * * * * * | 0 0 0 0  1  0 1  1 0 0 | 0 0 1 1 0 1
... ... ... oox&#x ♦ 1 1 2 | 0 1 2 0  2 1 | 0  0 0  1  2 0  0  1 0 | * * * *  *  * 24 * * * * * | 0 0 0 0  0  1 0  1 1 0 | 0 0 0 1 1 1
... .x.3.o.3.o.    ♦ 0 4 0 | 0 0 0 6  0 0 | 0  0 0  0  0 4  0  0 0 | * * * *  *  *  * 1 * * * * | 0 2 0 0  0  0 0  0 0 1 | 1 0 0 0 0 2
... .xo3.oo ...&#x ♦ 0 3 1 | 0 0 0 3  3 0 | 0  0 0  0  0 1  3  0 0 | * * * *  *  *  * * 4 * * * | 0 0 0 0  0  0 2  0 0 1 | 0 0 1 0 0 2
... .xo ... .ox&#x ♦ 0 2 2 | 0 0 0 1  4 1 | 0  0 0  0  0 0  2  2 0 | * * * *  *  *  * * * 6 * * | 0 0 0 0  0  0 0  2 0 1 | 0 0 0 1 0 2
... ... .oo3.ox&#x ♦ 0 1 3 | 0 0 0 0  3 3 | 0  0 0  0  0 0  0  3 1 | * * * *  *  *  * * * * 4 * | 0 0 0 0  0  0 0  0 2 1 | 0 0 0 0 1 2
... ..o3..o3..x    ♦ 0 0 4 | 0 0 0 0  0 6 | 0  0 0  0  0 0  0  0 4 | * * * *  *  *  * * * * * 1 | 0 0 0 2  0  0 0  0 0 1 | 0 1 0 0 0 2
-------------------+-------+--------------+------------------------+----------------------------+------------------------+------------
xo. ox.3oo. ...&#x ♦ 2 3 0 | 1 6 0 3  0 0 | 3  6 0  0  0 1  0  0 0 | 3 2 0 0  0  0  0 0 0 0 0 0 | 4 * * *  *  * *  * * * | 1 0 1 0 0 0
... ox.3oo.3oo.&#x ♦ 1 4 0 | 0 4 0 6  0 0 | 0  6 0  0  0 4  0  0 0 | 0 4 0 0  0  0  0 1 0 0 0 0 | * 2 * *  *  * *  * * * | 1 0 0 0 0 1
x.o ... o.o3o.x&#x ♦ 2 0 3 | 1 0 6 0  0 3 | 0  0 3  6  0 0  0  0 1 | 0 0 3 2  0  0  0 0 0 0 0 0 | * * 4 *  *  * *  * * * | 0 1 0 0 1 0
... o.o3o.o3o.x&#x ♦ 1 0 4 | 0 0 4 0  0 6 | 0  0 0  6  0 0  0  0 4 | 0 0 0 4  0  0  0 0 0 0 0 1 | * * * 2  *  * *  * * * | 0 1 0 0 0 1
xoo oxo ... ...&#x ♦ 2 2 1 | 1 4 2 1  2 0 | 2  2 1  0  4 0  1  0 0 | 1 0 0 0  2  2  0 0 0 0 0 0 | * * * * 12  * *  * * * | 0 0 1 1 0 0
xoo ... ... oox&#x ♦ 2 1 2 | 1 2 4 0  2 1 | 1  0 2  2  4 0  0  1 0 | 0 0 1 0  2  0  2 0 0 0 0 0 | * * * *  * 12 *  * * * | 0 0 0 1 1 0
... oxo3ooo ...&#x ♦ 1 3 1 | 0 3 1 3  3 0 | 0  3 0  0  3 1  3  0 0 | 0 1 0 0  0  3  0 0 1 0 0 0 | * * * *  *  * 8  * * * | 0 0 1 0 0 1
... oxo ... oox&#x ♦ 1 2 2 | 0 2 2 1  4 1 | 0  1 0  1  4 0  2  2 0 | 0 0 0 0  0  2  2 0 0 1 0 0 | * * * *  *  * * 12 * * | 0 0 0 1 0 1
... ... ooo3oox&#x ♦ 1 1 3 | 0 1 3 0  3 3 | 0  0 0  3  3 0  0  3 1 | 0 0 0 1  0  0  3 0 0 0 1 0 | * * * *  *  * *  * 8 * | 0 0 0 0 1 1
... .xo3.oo3.ox&#x ♦ 0 4 4 | 0 0 0 6 12 6 | 0  0 0  0  0 4 12 12 4 | 0 0 0 0  0  0  0 1 4 6 4 1 | * * * *  *  * *  * * 1 | 0 0 0 0 0 2
-------------------+-------+--------------+------------------------+----------------------------+------------------------+------------
xo. ox.3oo.3oo.&#x ♦ 2 4 0 | 1 8 0 6  0 0 | 4 12 0  0  0 4  0  0 0 | 6 8 0 0  0  0  0 1 0 0 0 0 | 4 2 0 0  0  0 0  0 0 0 | 1 * * * * *
x.o o.o3o.o3o.x&#x ♦ 2 0 4 | 1 0 8 0  0 6 | 0  0 4 12  0 0  0  0 4 | 0 0 6 8  0  0  0 0 0 0 0 1 | 0 0 4 2  0  0 0  0 0 0 | * 1 * * * *
xoo oxo3ooo ...&#x ♦ 2 3 1 | 1 6 2 3  3 0 | 3  6 1  0  6 1  3  0 0 | 3 2 0 0  3  6  0 0 1 0 0 0 | 1 0 0 0  3  0 2  0 0 0 | * * 4 * * *
xoo oxo ... oox&#x ♦ 2 2 2 | 1 4 4 1  4 1 | 2  2 2  2  8 0  2  2 0 | 1 0 1 0  4  4  4 0 0 1 0 0 | 0 0 0 0  2  2 0  2 0 0 | * * * 6 * *
xoo ... ooo3oox&#x ♦ 2 1 3 | 1 2 6 0  3 3 | 1  0 3  6  6 0  0  3 1 | 0 0 3 2  3  0  6 0 0 0 1 0 | 0 0 1 0  0  3 0  0 2 0 | * * * * 4 *
... oxo3ooo3oox&#x ♦ 1 4 4 | 0 4 4 6 12 6 | 0  6 0  6 12 4 12 12 4 | 0 4 0 4  0 12 12 1 4 6 4 1 | 0 1 0 1  0  0 4  6 4 1 | * * * * * 2
```

```oxo oox oxo oox xoo&#x   → height(1,2) = height(1,3) = 1/2
height(2,3) = 0
(line || tegum sum of {4} and fully perp {4})

o.. o.. o.. o.. o..      | 2 * ♦ 1  8 0  0 |  8 8 16  0 | 8 32 16  0 | 16 32 0 | 16 1
.o. .o. .o. .o. .o.    & | * 8 ♦ 0  2 2  4 |  4 1  8 12 | 2 24  4  8 | 16 12 1 |  8 2
-------------------------+-----+-----------+------------+------------+---------+-----
... ... ... ... x..      | 2 0 | 1  * *  * ♦  0 8  0  0 | 8  0 16  0 |  0 32 0 | 16 0
oo. oo. oo. oo. oo.&#x & | 1 1 | * 16 *  * ♦  2 1  4  0 | 2 12  4  0 |  8 12 0 |  8 1
.x. ... ... ... ...    & | 0 2 | *  * 8  * |  2 0  0  4 | 1  8  0  4 |  8  4 1 |  4 2
.oo .oo .oo .oo .oo&#x   | 0 2 | *  * * 16 |  0 0  2  4 | 0  8  1  4 |  8  4 1 |  4 2
-------------------------+-----+-----------+------------+------------+---------+-----
ox. ... ... ... ...&#x & | 1 2 | 0  2 1  0 | 16 *  *  * ♦ 1  4  0  0 |  4  4 0 |  4 1
... ... ... ... xo.&#x & | 2 1 | 1  2 0  0 |  * 8  *  * ♦ 2  0  4  0 |  0 12 0 |  8 0
ooo ooo ooo ooo ooo&#x   | 1 2 | 0  2 0  1 |  * * 32  * ♦ 0  4  1  0 |  4  4 0 |  4 1
.xo ... ... ... ...&#x & | 0 3 | 0  0 1  2 |  * *  * 32 | 0  2  0  2 |  4  1 1 |  2 2
-------------------------+-----+-----------+------------+------------+---------+-----
ox. ... ... ... xo.&#x & ♦ 2 2 | 1  4 1  0 |  2 2  0  0 | 8  *  *  * |  0  4 0 |  4 0
oxo ... ... ... ...&#x & ♦ 1 3 | 0  3 1  2 |  1 0  2  1 | * 64  *  * |  2  1 0 |  2 1
... ... ... ... xoo&#x   ♦ 2 2 | 1  4 0  1 |  0 2  2  0 | *  * 16  * |  0  4 0 |  4 0
.xo .ox ... ... ...&#x & ♦ 0 4 | 0  0 2  4 |  0 0  0  4 | *  *  * 16 |  2  0 1 |  1 2
-------------------------+-----+-----------+------------+------------+---------+-----
oxo oox ... ... ...&#x & ♦ 1 4 | 0  4 2  4 |  2 0  4  4 | 0  4  0  1 | 32  * * |  1 1
oxo ... ... ... xoo&#x & ♦ 2 3 | 1  6 1  2 |  2 3  4  1 | 1  2  2  0 |  * 32 * |  2 0
.xo .ox .xo .ox ...&#x   ♦ 0 8 | 0  0 8 16 |  0 0  0 32 | 0  0  0 16 |  *  * 1 |  0 2
-------------------------+-----+-----------+------------+------------+---------+-----
oxo oox ... ... xoo&#x & ♦ 2 4 | 1  8 2  4 |  4 4  8  4 | 2  8  4  1 |  2  4 0 | 16 *
oxo oox oxo oox ...&#x   ♦ 1 8 | 0  8 8 16 |  8 0 16 32 | 0 32  0 16 | 16  0 1 |  * 2
```