Acronym thex
cantic tesseract

Cross sections
` ©`
wrt. oct
sqrt(2) = 1.414214
wrt. tut
3/sqrt(8) = 1.060660
Vertex figure
` ©`
Vertex layers
 Layer Symmetry Subsymmetries o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 x3x3o4o x3x3o .tut first x3x . o{6} first x . o4oedge first . x3o4ooct first 2 u3o3x . u3o . q u . x4o . u3o4o 3a x3o3u . x3o . Q x . u4o . x3x4o 3b u3x . o U . o4o 4a o3x3x .opposite tut o3x . Q o . x4q . u3o4o 4b x3u . o 5a o3u . q x . u4o . x3o4oopposite oct 5b U . o4o 6 x3x . oopposite {6} u . x4o 7 x . o4oopposite edge o3o3o *b3o o3o3o    . o3o . *b3o o . o    o . o3o *b3o 1 x3x3o *b3o x3x3o    .tut first x3x . *b3otut first x . o    oedge first . x3o *b3ooct first 2 u3o3x    . u3o . *b3x u . x    x . u3o *b3o 3a x3o3u    . x3o . *b3u x . u    u . x3x *b3x 3b U . o    o 4a o3x3x    .opposite tut o3x . *b3xopposite tut o . x    U . u3o *b3o 4b o . U    x 5a x . u    u . x3o *b3oopposite oct 5b U . o    o 6 u . x    x 7 x . o    oopposite edge
(Q=q+q, U=x+u)
Lace city
in approx. ASCII-art
 ``` ©   ``` ``` o4o x4o o4o -- x3o4o (oct) or o4o u4o o4o -- u3o4o (u-oct) x4o u4o x4q u4o x4o -- x3x4o (toe) o4o u4o o4o -- u3o4o (u-oct) o4o x4o o4o -- x3o4o (oct) where: d=3x ``` ``` o o x x o o o o u u o o x x u u xd dx u u x x o o u u o o o o x x o o ```
 ``` ©   ``` ```x o u x x u o x -- x3x3o (tut) u x d o o d x u -- u3o3x ((u,x)-co) x u o d d o u x -- x3o3u (inv. (u,x)-co) o x x u u x x o -- o3x3x (inv. tut) ```
```   x3o   u3o   x3x      	-- x3x3o (tut)

o3x         u3x   u3o   	-- u3o3x ((u,x)-co)

o3u   x3u         x3o	-- x3o3u (inv. (u,x)-co)

x3x   o3u   o3x   	-- o3x3x (inv. tut)
```
Coordinates (sqrt(2), 1/sqrt(2), 0, 0)   & all permutations, all changes of sign
Volume 77/6 = 12.833333
Surface 100 sqrt(2)/3 = 47.140452
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
 by cells: oct thah toe tut thex 8 0 0 16 titho 0 8 4 8
)
Dihedral angles
• at {3} between oct and tut:   120°
• at {6} between tut and tut:   120°
Confer
Grünbaumian relatives:
2thex
related CRFs:
octum   pex thex   pabex thex   pacprit
decompositions:
hex || thex
general polytopal classes:
lace simplices   partial Stott expansions
analogs:
truncated orthoplex tOn   truncated demihypercube tDn
External

Note that thex can be thought of as the external blend of 1 hex + 16 tetatuts + 8 octpies. This decomposition is described as the degenerate segmentoteron xx3ox3oo4oo&#x. – Alternatively it can be (subdimensioanlly degenerate) decomposed into 1 rit + 16 tetaltuts + 24 squascs + 8 octacoes according to ox3ox3xo4oo&#x.

Incidence matrix according to Dynkin symbol

```x3x3o4o

. . . . | 48 ♦  1  4 |  4  4 |  4 1
--------+----+-------+-------+-----
x . . . |  2 | 24  * |  4  0 |  4 0
. x . . |  2 |  * 96 |  1  2 |  2 1
--------+----+-------+-------+-----
x3x . . |  6 |  3  3 | 32  * |  2 0
. x3o . |  3 |  0  3 |  * 64 |  1 1
--------+----+-------+-------+-----
x3x3o . ♦ 12 |  6 12 |  4  4 | 16 *
. x3o4o ♦  6 |  0 12 |  0  8 |  * 8

snubbed forms: β3x3o4o, x3β3o4o, β3β3o4o
```

```x3x3o4/3o

. . .   . | 48 ♦  1  4 |  4  4 |  4 1
----------+----+-------+-------+-----
x . .   . |  2 | 24  * |  4  0 |  4 0
. x .   . |  2 |  * 96 |  1  2 |  2 1
----------+----+-------+-------+-----
x3x .   . |  6 |  3  3 | 32  * |  2 0
. x3o   . |  3 |  0  3 |  * 64 |  1 1
----------+----+-------+-------+-----
x3x3o   . ♦ 12 |  6 12 |  4  4 | 16 *
. x3o4/3o ♦  6 |  0 12 |  0  8 |  * 8
```

```x3x3/2o4o

. .   . . | 48 ♦  1  4 |  4  4 |  4 1
----------+----+-------+-------+-----
x .   . . |  2 | 24  * |  4  0 |  4 0
. x   . . |  2 |  * 96 |  1  2 |  2 1
----------+----+-------+-------+-----
x3x   . . |  6 |  3  3 | 32  * |  2 0
. x3/2o . |  3 |  0  3 |  * 64 |  1 1
----------+----+-------+-------+-----
x3x3/2o . ♦ 12 |  6 12 |  4  4 | 16 *
. x3/2o4o ♦  6 |  0 12 |  0  8 |  * 8
```

```x3x3/2o4/3o

. .   .   . | 48 ♦  1  4 |  4  4 |  4 1
------------+----+-------+-------+-----
x .   .   . |  2 | 24  * |  4  0 |  4 0
. x   .   . |  2 |  * 96 |  1  2 |  2 1
------------+----+-------+-------+-----
x3x   .   . |  6 |  3  3 | 32  * |  2 0
. x3/2o   . |  3 |  0  3 |  * 64 |  1 1
------------+----+-------+-------+-----
x3x3/2o   . ♦ 12 |  6 12 |  4  4 | 16 *
. x3/2o4/3o ♦  6 |  0 12 |  0  8 |  * 8
```

```x3x3o *b3o

. . .    . | 48 ♦  1  4 |  4  2  2 | 2 2 1
-----------+----+-------+----------+------
x . .    . |  2 | 24  * |  4  0  0 | 2 2 0
. x .    . |  2 |  * 96 |  1  1  1 | 1 1 1
-----------+----+-------+----------+------
x3x .    . |  6 |  3  3 | 32  *  * | 1 1 0
. x3o    . |  3 |  0  3 |  * 32  * | 1 0 1
. x . *b3o |  3 |  0  3 |  *  * 32 | 0 1 1
-----------+----+-------+----------+------
x3x3o    . ♦ 12 |  6 12 |  4  4  0 | 8 * *
x3x . *b3o ♦ 12 |  6 12 |  4  0  4 | * 8 *
. x3o *b3o ♦  6 |  0 12 |  0  4  4 | * * 8

snubbed forms: β3x3o *b3o, x3β3o *b3o, β3β3o *b3o
```

```x3x3o *b3/2o

. . .      . | 48 ♦  1  4 |  4  2  2 | 2 2 1
-------------+----+-------+----------+------
x . .      . |  2 | 24  * |  4  0  0 | 2 2 0
. x .      . |  2 |  * 96 |  1  1  1 | 1 1 1
-------------+----+-------+----------+------
x3x .      . |  6 |  3  3 | 32  *  * | 1 1 0
. x3o      . |  3 |  0  3 |  * 32  * | 1 0 1
. x . *b3/2o |  3 |  0  3 |  *  * 32 | 0 1 1
-------------+----+-------+----------+------
x3x3o      . ♦ 12 |  6 12 |  4  4  0 | 8 * *
x3x . *b3/2o ♦ 12 |  6 12 |  4  0  4 | * 8 *
. x3o *b3/2o ♦  6 |  0 12 |  0  4  4 | * * 8
```

```x3x3/2o *b3/2o

. .   .      . | 48 ♦  1  4 |  4  2  2 | 2 2 1
---------------+----+-------+----------+------
x .   .      . |  2 | 24  * |  4  0  0 | 2 2 0
. x   .      . |  2 |  * 96 |  1  1  1 | 1 1 1
---------------+----+-------+----------+------
x3x   .      . |  6 |  3  3 | 32  *  * | 1 1 0
. x3/2o      . |  3 |  0  3 |  * 32  * | 1 0 1
. x   . *b3/2o |  3 |  0  3 |  *  * 32 | 0 1 1
---------------+----+-------+----------+------
x3x3/2o      . ♦ 12 |  6 12 |  4  4  0 | 8 * *
x3x   . *b3/2o ♦ 12 |  6 12 |  4  0  4 | * 8 *
. x3/2o *b3/2o ♦  6 |  0 12 |  0  4  4 | * * 8
```

```s4o3x3o

demi( . . . . ) | 48 ♦  1  4 |  2  2  4 | 2 1 2
----------------+----+-------+----------+------
s4o . .   |  2 | 24  * |  0  0  4 | 2 0 2
demi( . . x . ) |  2 |  * 96 |  1  1  1 | 1 1 1
----------------+----+-------+----------+------
demi( . o3x . ) |  3 |  0  3 | 32  *  * | 1 1 0
demi( . . x3o ) |  3 |  0  3 |  * 32  * | 0 1 1
sefa( s4o3x . ) |  6 |  3  3 |  *  * 32 | 1 0 1
----------------+----+-------+----------+------
s4o3x .   ♦ 12 |  6 12 |  4  0  4 | 8 * *
demi( . o3x3o ) ♦  6 |  0 12 |  4  4  0 | * 8 *
sefa( s4o3x3o ) ♦ 12 |  6 12 |  0  4  4 | * * 8

starting figure: x4o3x3o
```

```xuxo3xoox3oxux&#xt   → all heights = 1/sqrt(2) = 0.707107
(tut || pseudo (u,x)-co || pseudo inv (u,x)-co || inv tut)

o...3o...3o...     | 12  *  *  * ♦ 1  2  2  0  0  0  0  0 0 | 2 1  2  2  1 0  0 0  0  0 0 0 | 1 2 1 1 0 0 0
.o..3.o..3.o..     |  * 12  *  * ♦ 0  0  2  2  1  0  0  0 0 | 0 0  2  1  2 1  2 0  0  0 0 0 | 0 1 2 1 1 0 0
..o.3..o.3..o.     |  *  * 12  * ♦ 0  0  0  0  1  2  2  0 0 | 0 0  2  0  0 0  2 1  2  1 0 0 | 0 1 2 0 1 1 0
...o3...o3...o     |  *  *  * 12 ♦ 0  0  0  0  0  0  2  2 1 | 0 0  0  0  0 0  2 0  1  2 1 2 | 0 0 1 0 2 1 1
-------------------+-------------+--------------------------+-------------------------------+--------------
x... .... ....     |  2  0  0  0 | 6  *  *  *  *  *  *  * * | 2 0  2  0  0 0  0 0  0  0 0 0 | 1 2 1 0 0 0 0
.... x... ....     |  2  0  0  0 | * 12  *  *  *  *  *  * * | 1 1  0  1  0 0  0 0  0  0 0 0 | 1 1 0 1 0 0 0
oo..3oo..3oo..&#x  |  1  1  0  0 | *  * 24  *  *  *  *  * * | 0 0  1  1  1 0  0 0  0  0 0 0 | 0 1 1 1 0 0 0
.... .... .x..     |  0  2  0  0 | *  *  * 12  *  *  *  * * | 0 0  0  0  1 1  1 0  0  0 0 0 | 0 0 1 1 1 0 0
.oo.3.oo.3.oo.&#x  |  0  1  1  0 | *  *  *  * 12  *  *  * * | 0 0  2  0  0 0  2 0  0  0 0 0 | 0 1 2 0 1 0 0
..x. .... ....     |  0  0  2  0 | *  *  *  *  * 12  *  * * | 0 0  1  0  0 0  0 1  1  0 0 0 | 0 1 1 0 0 1 0
..oo3..oo3..oo&#x  |  0  0  1  1 | *  *  *  *  *  * 24  * * | 0 0  0  0  0 0  1 0  1  1 0 0 | 0 0 1 0 1 1 0
.... ...x ....     |  0  0  0  2 | *  *  *  *  *  *  * 12 * | 0 0  0  0  0 0  0 0  0  1 1 1 | 0 0 0 0 1 1 1
.... .... ...x     |  0  0  0  2 | *  *  *  *  *  *  *  * 6 | 0 0  0  0  0 0  2 0  0  0 0 2 | 0 0 1 0 2 0 1
-------------------+-------------+--------------------------+-------------------------------+--------------
x...3x... ....     |  6  0  0  0 | 3  3  0  0  0  0  0  0 0 | 4 *  *  *  * *  * *  *  * * * | 1 1 0 0 0 0 0
.... x...3o...     |  3  0  0  0 | 0  3  0  0  0  0  0  0 0 | * 4  *  *  * *  * *  *  * * * | 1 0 0 1 0 0 0
xux. .... ....&#xt |  2  2  2  0 | 1  0  2  0  2  1  0  0 0 | * * 12  *  * *  * *  *  * * * | 0 1 1 0 0 0 0
.... xo.. ....&#x  |  2  1  0  0 | 0  1  2  0  0  0  0  0 0 | * *  * 12  * *  * *  *  * * * | 0 1 0 1 0 0 0
.... .... ox..&#x  |  1  2  0  0 | 0  0  2  1  0  0  0  0 0 | * *  *  * 12 *  * *  *  * * * | 0 0 1 1 0 0 0
.... .o..3.x..     |  0  3  0  0 | 0  0  0  3  0  0  0  0 0 | * *  *  *  * 4  * *  *  * * * | 0 0 0 1 1 0 0
.... .... .xux&#xt |  0  2  2  2 | 0  0  0  1  2  0  2  0 1 | * *  *  *  * * 12 *  *  * * * | 0 0 1 0 1 0 0
..x.3..o. ....     |  0  0  3  0 | 0  0  0  0  0  3  0  0 0 | * *  *  *  * *  * 4  *  * * * | 0 1 0 0 0 1 0
..xo .... ....&#x  |  0  0  2  1 | 0  0  0  0  0  1  2  0 0 | * *  *  *  * *  * * 12  * * * | 0 0 1 0 0 1 0
.... ..ox ....&#x  |  0  0  1  2 | 0  0  0  0  0  0  2  1 0 | * *  *  *  * *  * *  * 12 * * | 0 0 0 0 1 1 0
...o3...x ....     |  0  0  0  3 | 0  0  0  0  0  0  0  3 0 | * *  *  *  * *  * *  *  * 4 * | 0 0 0 0 0 1 1
.... ...x3...x     |  0  0  0  6 | 0  0  0  0  0  0  0  3 3 | * *  *  *  * *  * *  *  * * 4 | 0 0 0 0 1 0 1
-------------------+-------------+--------------------------+-------------------------------+--------------
x...3x...3o...     ♦ 12  0  0  0 | 6 12  0  0  0  0  0  0 0 | 4 4  0  0  0 0  0 0  0  0 0 0 | 1 * * * * * *
xux.3xoo. ....&#xt ♦  6  3  3  0 | 3  3  6  0  3  3  0  0 0 | 1 0  3  3  0 0  0 1  0  0 0 0 | * 4 * * * * *
xuxo .... oxux&#xt ♦  2  4  4  2 | 1  0  4  2  4  2  4  0 1 | 0 0  2  0  2 0  2 0  2  0 0 0 | * * 6 * * * *
.... xo..3ox..&#x  ♦  3  3  0  0 | 0  3  6  3  0  0  0  0 0 | 0 1  0  3  3 1  0 0  0  0 0 0 | * * * 4 * * *
.... .oox3.xux&#xt ♦  0  3  3  6 | 0  0  0  3  3  0  6  3 3 | 0 0  0  0  0 1  3 0  0  3 0 1 | * * * * 4 * *
..xo3..ox ....&#x  ♦  0  0  3  3 | 0  0  0  0  0  3  6  3 0 | 0 0  0  0  0 0  0 1  3  3 1 0 | * * * * * 4 *
...o3...x3...x     ♦  0  0  0 12 | 0  0  0  0  0  0  0 12 6 | 0 0  0  0  0 0  0 0  0  0 4 4 | * * * * * * 1
```
```or
o...3o...3o...      & | 24  * ♦  1  2  2  0  0 | 2 1  2  2  1 0 | 1 2 1 1
.o..3.o..3.o..      & |  * 24 ♦  0  0  2  2  1 | 0 0  4  1  2 1 | 0 2 2 1
----------------------+-------+----------------+----------------+--------
x... .... ....      & |  2  0 | 12  *  *  *  * | 2 0  2  0  0 0 | 1 2 1 0
.... x... ....      & |  2  0 |  * 24  *  *  * | 1 1  0  1  0 0 | 1 1 0 1
oo..3oo..3oo..&#x   & |  1  1 |  *  * 48  *  * | 0 0  1  1  1 0 | 0 1 1 1
.... .... .x..      & |  0  2 |  *  *  * 24  * | 0 0  1  0  1 1 | 0 1 1 1
.oo.3.oo.3.oo.&#x     |  0  2 |  *  *  *  * 12 | 0 0  4  0  0 0 | 0 2 2 0
----------------------+-------+----------------+----------------+--------
x...3x... ....      & |  6  0 |  3  3  0  0  0 | 8 *  *  *  * * | 1 1 0 0
.... x...3o...      & |  3  0 |  0  3  0  0  0 | * 8  *  *  * * | 1 0 0 1
xux. .... ....&#xt  & |  2  4 |  1  0  2  1  2 | * * 24  *  * * | 0 1 1 0
.... xo.. ....&#x   & |  2  1 |  0  1  2  0  0 | * *  * 24  * * | 0 1 0 1
.... .... ox..&#x   & |  1  2 |  0  0  2  1  0 | * *  *  * 24 * | 0 0 1 1
.... .o..3.x..      & |  0  3 |  0  0  0  3  0 | * *  *  *  * 8 | 0 1 0 1
----------------------+-------+----------------+----------------+--------
x...3x...3o...      & ♦ 12  0 |  6 12  0  0  0 | 4 4  0  0  0 0 | 2 * * *
xux.3xoo. ....&#xt  & ♦  6  6 |  3  3  6  3  3 | 1 0  3  3  0 1 | * 8 * *
xuxo .... oxux&#xt    ♦  4  8 |  2  0  8  4  4 | 0 0  4  0  4 0 | * * 6 *
.... xo..3ox..&#x   & ♦  3  3 |  0  3  6  3  0 | 0 1  0  3  3 1 | * * * 8
```

```xuxux3ooxoo4ooooo&#xt   → all heights = 1/sqrt(2) = 0.707107
(oct || pseudo u-oct || pseudo toe || pseudo u-oct || oct)

o....3o....4o....     | 6 *  * * * ♦  4 1  0  0  0  0 0  0 | 4  4  0 0  0  0 0 | 1 4 0 0 0
.o...3.o...4.o...     | * 6  * * * ♦  0 1  4  0  0  0 0  0 | 0  4  4 0  0  0 0 | 0 4 1 0 0
..o..3..o..4..o..     | * * 24 * * ♦  0 0  1  1  2  1 0  0 | 0  1  2 2  1  2 0 | 0 2 1 2 0
...o.3...o.4...o.     | * *  * 6 * ♦  0 0  0  0  0  4 1  0 | 0  0  0 0  4  4 0 | 0 0 1 4 0
....o3....o4....o     | * *  * * 6 ♦  0 0  0  0  0  0 1  4 | 0  0  0 0  4  0 4 | 0 0 0 4 1
----------------------+------------+-----------------------+-------------------+----------
x.... ..... .....     | 2 0  0 0 0 | 12 *  *  *  *  * *  * | 2  1  0 0  0  0 0 | 1 2 0 0 0
oo...3oo...4oo...&#x  | 1 1  0 0 0 |  * 6  *  *  *  * *  * | 0  4  0 0  0  0 0 | 0 4 0 0 0
.oo..3.oo..4.oo..&#x  | 0 1  1 0 0 |  * * 24  *  *  * *  * | 0  1  2 0  0  0 0 | 0 2 1 0 0
..x.. ..... .....     | 0 0  2 0 0 |  * *  * 12  *  * *  * | 0  1  0 2  1  0 0 | 0 2 0 2 0
..... ..x.. .....     | 0 0  2 0 0 |  * *  *  * 24  * *  * | 0  0  1 1  0  1 0 | 0 1 1 1 0
..oo.3..oo.4..oo.&#x  | 0 0  1 1 0 |  * *  *  *  * 24 *  * | 0  0  0 0  1  2 0 | 0 0 1 2 0
...oo3...oo4...oo&#x  | 0 0  0 1 1 |  * *  *  *  *  * 6  * | 0  0  0 0  4  0 0 | 0 0 0 4 0
....x ..... .....     | 0 0  0 0 2 |  * *  *  *  *  * * 12 | 0  0  0 0  1  0 2 | 0 0 0 2 1
----------------------+------------+-----------------------+-------------------+----------
x....3o.... .....     | 3 0  0 0 0 |  3 0  0  0  0  0 0  0 | 8  *  * *  *  * * | 1 1 0 0 0
xux.. ..... .....&#xt | 2 2  2 0 0 |  1 2  2  1  0  0 0  0 | * 12  * *  *  * * | 0 2 0 0 0
..... .ox.. .....&#x  | 0 1  2 0 0 |  0 0  2  0  1  0 0  0 | *  * 24 *  *  * * | 0 1 1 0 0
..x..3..x.. .....     | 0 0  6 0 0 |  0 0  0  3  3  0 0  0 | *  *  * 8  *  * * | 0 1 0 1 0
..xux ..... .....&#xt | 0 0  2 2 2 |  0 0  0  1  0  2 2  1 | *  *  * * 12  * * | 0 0 0 2 0
..... ..xo. .....&#x  | 0 0  2 1 0 |  0 0  0  0  1  2 0  0 | *  *  * *  * 24 * | 0 0 1 1 0
....x3....o .....     | 0 0  0 0 3 |  0 0  0  0  0  0 0  3 | *  *  * *  *  * 8 | 0 0 0 1 1
----------------------+------------+-----------------------+-------------------+----------
x....3o....4o....     ♦ 6 0  0 0 0 | 12 0  0  0  0  0 0  0 | 8  0  0 0  0  0 0 | 1 * * * *
xux..3oox.. .....&#xt ♦ 3 3  6 0 0 |  3 3  6  3  3  0 0  0 | 1  3  3 1  0  0 0 | * 8 * * *
..... .oxo.4.ooo.&#xt ♦ 0 1  4 1 0 |  0 0  4  0  4  4 0  0 | 0  0  4 0  0  4 0 | * * 6 * *
..xux3..xoo .....&#xt ♦ 0 0  6 3 3 |  0 0  0  3  3  6 3  3 | 0  0  0 1  3  3 1 | * * * 8 *
....x3....o4....o     ♦ 0 0  0 0 6 |  0 0  0  0  0  0 0 12 | 0  0  0 0  0  0 8 | * * * * 1
```
```or
o....3o....4o....      & | 12  *  * ♦  4  1  0  0  0 |  4  4  0 0 | 1  4 0
.o...3.o...4.o...      & |  * 12  * ♦  0  1  4  0  0 |  0  4  4 0 | 0  4 1
..o..3..o..4..o..        |  *  * 24 ♦  0  0  2  1  2 |  0  2  4 2 | 0  4 1
-------------------------+----------+----------------+------------+-------
x.... ..... .....      & |  2  0  0 | 24  *  *  *  * |  2  1  0 0 | 1  2 0
oo...3oo...4oo...&#x   & |  1  1  0 |  * 12  *  *  * |  0  4  0 0 | 0  4 0
.oo..3.oo..4.oo..&#x   & |  0  1  1 |  *  * 48  *  * |  0  1  2 0 | 0  2 1
..x.. ..... .....        |  0  0  2 |  *  *  * 12  * |  0  2  0 2 | 0  4 0
..... ..x.. .....        |  0  0  2 |  *  *  *  * 24 |  0  0  2 1 | 0  2 1
-------------------------+----------+----------------+------------+-------
x....3o.... .....      & |  3  0  0 |  3  0  0  0  0 | 16  *  * * | 1  1 0
xux.. ..... .....&#xt  & |  2  2  2 |  1  2  2  1  0 |  * 24  * * | 0  2 0
..... .ox.. .....&#x   & |  0  1  2 |  0  0  2  0  1 |  *  * 48 * | 0  1 1
..x..3..x.. .....        |  0  0  6 |  0  0  0  3  3 |  *  *  * 8 | 0  2 0
-------------------------+----------+----------------+------------+-------
x....3o....4o....      & ♦  6  0  0 | 12  0  0  0  0 |  8  0  0 0 | 2  * *
xux..3oox.. .....&#xt  & ♦  3  3  6 |  3  3  6  3  3 |  1  3  3 1 | * 16 *
..... .oxo.4.ooo.&#xt    ♦  0  2  4 |  0  0  8  0  4 |  0  0  8 0 | *  * 6
```

```ooxoo3xuxux3ooxoo&#xt   → all heights = 1/sqrt(2) = 0.707107
(oct || pseudo u-oct || pseudo toe || pseudo u-oct || oct)

o....3o....3o....     | 6 *  * * * ♦  4 1  0  0  0  0  0 0  0 | 2 2  4  0  0 0 0  0  0  0 0 0 | 1 2 2 0 0 0 0
.o...3.o...3.o...     | * 6  * * * ♦  0 1  4  0  0  0  0 0  0 | 0 0  4  2  2 0 0  0  0  0 0 0 | 0 2 2 1 0 0 0
..o..3..o..3..o..     | * * 24 * * ♦  0 0  1  1  1  1  1 0  0 | 0 0  1  1  1 1 1  1  1  1 0 0 | 0 1 1 1 1 1 0
...o.3...o.3...o.     | * *  * 6 * ♦  0 0  0  0  0  0  4 1  0 | 0 0  0  0  0 0 0  2  4  2 0 0 | 0 0 0 1 2 2 0
....o3....o3....o     | * *  * * 6 ♦  0 0  0  0  0  0  0 1  4 | 0 0  0  0  0 0 0  0  4  0 2 2 | 0 0 0 0 2 2 1
----------------------+------------+--------------------------+-------------------------------+--------------
..... x.... .....     | 2 0  0 0 0 | 12 *  *  *  *  *  * *  * | 1 1  1  0  0 0 0  0  0  0 0 0 | 1 1 1 0 0 0 0
oo...3oo...3oo...&#x  | 1 1  0 0 0 |  * 6  *  *  *  *  * *  * | 0 0  4  0  0 0 0  0  0  0 0 0 | 0 2 2 0 0 0 0
.oo..3.oo..3.oo..&#x  | 0 1  1 0 0 |  * * 24  *  *  *  * *  * | 0 0  1  1  1 0 0  0  0  0 0 0 | 0 1 1 1 0 0 0
..x.. ..... .....     | 0 0  2 0 0 |  * *  * 12  *  *  * *  * | 0 0  0  1  0 1 0  1  0  0 0 0 | 0 1 0 1 1 0 0
..... ..x.. .....     | 0 0  2 0 0 |  * *  *  * 12  *  * *  * | 0 0  1  0  0 1 1  0  1  0 0 0 | 0 1 1 0 1 1 0
..... ..... ..x..     | 0 0  2 0 0 |  * *  *  *  * 12  * *  * | 0 0  0  0  1 0 1  0  0  1 0 0 | 0 0 1 1 0 1 0
..oo.3..oo.3..oo.&#x  | 0 0  1 1 0 |  * *  *  *  *  * 24 *  * | 0 0  0  0  0 0 0  1  1  1 0 0 | 0 0 0 1 1 1 0
...oo3...oo3...oo&#x  | 0 0  0 1 1 |  * *  *  *  *  *  * 6  * | 0 0  0  0  0 0 0  0  4  0 0 0 | 0 0 0 0 2 2 0
..... ....x .....     | 0 0  0 0 2 |  * *  *  *  *  *  * * 12 | 0 0  0  0  0 0 0  0  1  0 1 1 | 0 0 0 0 1 1 1
----------------------+------------+--------------------------+-------------------------------+--------------
o....3x.... .....     | 3 0  0 0 0 |  3 0  0  0  0  0  0 0  0 | 4 *  *  *  * * *  *  *  * * * | 1 1 0 0 0 0 0
..... x....3o....     | 3 0  0 0 0 |  3 0  0  0  0  0  0 0  0 | * 4  *  *  * * *  *  *  * * * | 1 0 1 0 0 0 0
..... xux.. .....&#xt | 2 2  2 0 0 |  1 2  2  0  1  0  0 0  0 | * * 12  *  * * *  *  *  * * * | 0 1 1 0 0 0 0
.ox.. ..... .....&#x  | 0 1  2 0 0 |  0 0  2  1  0  0  0 0  0 | * *  * 12  * * *  *  *  * * * | 0 1 0 1 0 0 0
..... ..... .ox..&#x  | 0 1  2 0 0 |  0 0  2  0  0  1  0 0  0 | * *  *  * 12 * *  *  *  * * * | 0 0 1 1 0 0 0
..x..3..x.. .....     | 0 0  6 0 0 |  0 0  0  3  3  0  0 0  0 | * *  *  *  * 4 *  *  *  * * * | 0 1 0 0 1 0 0
..... ..x..3..x..     | 0 0  6 0 0 |  0 0  0  0  3  3  0 0  0 | * *  *  *  * * 4  *  *  * * * | 0 0 1 0 0 1 0
..xo. ..... .....&#x  | 0 0  2 1 0 |  0 0  0  1  0  0  2 0  0 | * *  *  *  * * * 12  *  * * * | 0 0 0 1 1 0 0
..... ..xux .....&#xt | 0 0  2 2 2 |  0 0  0  0  1  0  2 2  1 | * *  *  *  * * *  * 12  * * * | 0 0 0 0 1 1 0
..... ..... ..xo.&#x  | 0 0  2 1 0 |  0 0  0  0  0  1  2 0  0 | * *  *  *  * * *  *  * 12 * * | 0 0 0 1 0 1 0
....o3....x .....     | 0 0  0 0 3 |  0 0  0  0  0  0  0 0  3 | * *  *  *  * * *  *  *  * 4 * | 0 0 0 0 1 0 1
..... ....x3....o     | 0 0  0 0 3 |  0 0  0  0  0  0  0 0  3 | * *  *  *  * * *  *  *  * * 4 | 0 0 0 0 0 1 1
----------------------+------------+--------------------------+-------------------------------+--------------
o....3x....3o....     ♦ 6 0  0 0 0 | 12 0  0  0  0  0  0 0  0 | 4 4  0  0  0 0 0  0  0  0 0 0 | 1 * * * * * *
oox..3xux.. .....&#xt ♦ 3 3  6 0 0 |  3 3  6  3  3  0  0 0  0 | 1 0  3  3  0 1 0  0  0  0 0 0 | * 4 * * * * *
..... xux..3oox..&#xt ♦ 3 3  6 0 0 |  3 3  6  0  3  3  0 0  0 | 0 1  3  0  3 0 1  0  0  0 0 0 | * * 4 * * * *
.oxo. ..... .oxo.&#xt ♦ 0 1  4 1 0 |  0 0  4  2  0  2  4 0  0 | 0 0  0  2  2 0 0  2  0  2 0 0 | * * * 6 * * *
..xoo3..xux .....&#xt ♦ 0 0  6 3 3 |  0 0  0  3  3  0  6 3  3 | 0 0  0  0  0 1 0  3  3  0 1 0 | * * * * 4 * *
..... ..xux3..xoo&#xt ♦ 0 0  6 3 3 |  0 0  0  0  3  3  6 3  3 | 0 0  0  0  0 0 1  0  3  3 0 1 | * * * * * 4 *
....o3....x3....o     ♦ 0 0  0 0 6 |  0 0  0  0  0  0  0 0 12 | 0 0  0  0  0 0 0  0  0  0 4 4 | * * * * * * 1
```
```or
o....3o....3o....      & | 12  *  * ♦  4  1  0  0  0  0 | 2 2  4  0  0 0 0 | 1 2 2 0
.o...3.o...3.o...      & |  * 12  * ♦  0  1  4  0  0  0 | 0 0  4  2  2 0 0 | 0 2 2 1
..o..3..o..3..o..        |  *  * 24 ♦  0  0  2  1  1  1 | 0 0  2  2  2 1 1 | 0 2 2 1
-------------------------+----------+-------------------+------------------+--------
..... x.... .....      & |  2  0  0 | 24  *  *  *  *  * | 1 1  1  0  0 0 0 | 1 1 1 0
oo...3oo...3oo...&#x   & |  1  1  0 |  * 12  *  *  *  * | 0 0  4  0  0 0 0 | 0 2 2 0
.oo..3.oo..3.oo..&#x   & |  0  1  1 |  *  * 48  *  *  * | 0 0  1  1  1 0 0 | 0 1 1 1
..x.. ..... .....        |  0  0  2 |  *  *  * 12  *  * | 0 0  0  2  0 1 0 | 0 2 0 1
..... ..x.. .....        |  0  0  2 |  *  *  *  * 12  * | 0 0  2  0  0 1 1 | 0 2 2 0
..... ..... ..x..        |  0  0  2 |  *  *  *  *  * 12 | 0 0  0  0  2 0 1 | 0 0 2 1
-------------------------+----------+-------------------+------------------+--------
o....3x.... .....      & |  3  0  0 |  3  0  0  0  0  0 | 8 *  *  *  * * * | 1 1 0 0
..... x....3o....      & |  3  0  0 |  3  0  0  0  0  0 | * 8  *  *  * * * | 1 0 1 0
..... xux.. .....&#xt  & |  2  2  2 |  1  2  2  0  1  0 | * * 24  *  * * * | 0 1 1 0
.ox.. ..... .....&#x   & |  0  1  2 |  0  0  2  1  0  0 | * *  * 24  * * * | 0 1 0 1
..... ..... .ox..&#x   & |  0  1  2 |  0  0  2  0  0  1 | * *  *  * 24 * * | 0 0 1 1
..x..3..x.. .....        |  0  0  6 |  0  0  0  3  3  0 | * *  *  *  * 4 * | 0 2 0 0
..... ..x..3..x..        |  0  0  6 |  0  0  0  0  3  3 | * *  *  *  * * 4 | 0 0 2 0
-------------------------+----------+-------------------+------------------+--------
o....3x....3o....      & ♦  6  0  0 | 12  0  0  0  0  0 | 4 4  0  0  0 0 0 | 2 * * *
oox..3xux.. .....&#xt  & ♦  3  3  6 |  3  3  6  3  3  0 | 1 0  3  3  0 1 0 | * 8 * *
..... xux..3oox..&#xt  & ♦  3  3  6 |  3  3  6  0  3  3 | 0 1  3  0  3 0 1 | * * 8 *
.oxo. ..... .oxo.&#xt    ♦  0  2  4 |  0  0  8  2  0  2 | 0 0  0  4  4 0 0 | * * * 6
```

```Qqo xux3oox4ooo&#zxt (where Q=2q)

o.. o..3o..4o..     | 12  *  * ♦  4  1  0  0  0 |  4  4  0 0 | 1  4 0
.o. .o.3.o.4.o.     |  * 12  * ♦  0  1  4  0  0 |  0  4  4 0 | 0  4 1
..o ..o3..o4..o     |  *  * 24 ♦  0  0  2  1  2 |  0  2  4 2 | 0  4 1
--------------------+----------+----------------+------------+-------
... x.. ... ...     |  2  0  0 | 24  *  *  *  * |  2  1  0 0 | 1  2 0
oo. oo.3oo.4oo.&#x  |  1  1  0 |  * 12  *  *  * |  0  4  0 0 | 0  4 0
.oo .oo3.oo4.oo&#x  |  0  1  1 |  *  * 48  *  * |  0  1  2 0 | 0  2 1
... ..x ... ...     |  0  0  2 |  *  *  * 12  * |  0  2  0 2 | 0  4 0
... ... ..x ...     |  0  0  2 |  *  *  *  * 24 |  0  0  2 1 | 0  2 1
--------------------+----------+----------------+------------+-------
... x..3o.. ...     |  3  0  0 |  3  0  0  0  0 | 16  *  * * | 1  1 0
... xux ... ...&#xt |  2  2  2 |  1  2  2  1  0 |  * 24  * * | 0  2 0
... ... .ox ...&#x  |  0  1  2 |  0  0  2  0  1 |  *  * 48 * | 0  1 1
... ..x3..x ...     |  0  0  6 |  0  0  0  3  3 |  *  *  * 8 | 0  2 0
--------------------+----------+----------------+------------+-------
... x..3o..4o..     ♦  6  0  0 | 12  0  0  0  0 |  8  0  0 0 | 2  * *
... xux3oox ...&#xt ♦  3  3  6 |  3  3  6  3  3 |  1  3  3 1 | * 16 *
.qo ... .ox4.oo&#zx ♦  0  2  4 |  0  0  8  0  4 |  0  0  8 0 | *  * 6
```

```ooxxuuxd ooxxuudx QqQoqooo qQoQoqoo &#zx (where Q=2q, d=3x)

o....... o....... o....... o.......      | 4 * * * * * * * ♦ 1  4  0 0 0 0 0 0 0  0 0 0 0 0 0 0 |  4 2 2 0 0 0 0 0 0  0  0 | 2 2 1 0 0
.o...... .o...... .o...... .o......      | * 4 * * * * * * ♦ 1  0  4 0 0 0 0 0 0  0 0 0 0 0 0 0 |  4 0 0 2 2 0 0 0 0  0  0 | 2 2 0 1 0
..o..... ..o..... ..o..... ..o.....      | * * 8 * * * * * ♦ 0  2  0 1 1 1 0 0 0  0 0 0 0 0 0 0 |  2 2 2 0 0 1 1 0 0  0  0 | 2 2 1 0 0
...o.... ...o.... ...o.... ...o....      | * * * 8 * * * * ♦ 0  0  2 0 0 0 1 1 1  0 0 0 0 0 0 0 |  2 0 0 2 2 0 0 1 1  0  0 | 2 2 0 1 0
....o... ....o... ....o... ....o...      | * * * * 8 * * * ♦ 0  0  0 0 0 1 0 0 0  2 1 1 0 0 0 0 |  2 0 0 0 0 1 1 0 0  2  2 | 2 2 0 0 1
.....o.. .....o.. .....o.. .....o..      | * * * * * 8 * * ♦ 0  0  0 0 0 0 0 0 1  2 0 0 1 1 0 0 |  2 0 0 0 0 0 0 1 1  2  2 | 2 2 0 0 1
......o. ......o. ......o. ......o.      | * * * * * * 4 * ♦ 0  0  0 0 0 0 0 0 0  0 2 0 2 0 1 0 |  0 0 0 0 0 2 0 2 0  4  0 | 4 0 0 0 1
.......o .......o .......o .......o      | * * * * * * * 4 ♦ 0  0  0 0 0 0 0 0 0  0 0 2 0 2 0 1 |  0 0 0 0 0 0 2 0 2  0  4 | 0 4 0 0 1
-----------------------------------------+-----------------+------------------------------------+--------------------------+----------
oo...... oo...... oo...... oo...... &#x  | 1 1 0 0 0 0 0 0 | 4  *  * * * * * * *  * * * * * * * |  4 0 0 0 0 0 0 0 0  0  0 | 2 2 0 0 0
o.o..... o.o..... o.o..... o.o..... &#x  | 1 0 1 0 0 0 0 0 | * 16  * * * * * * *  * * * * * * * |  1 1 1 0 0 0 0 0 0  0  0 | 1 1 1 0 0
.o.o.... .o.o.... .o.o.... .o.o.... &#x  | 0 1 0 1 0 0 0 0 | *  * 16 * * * * * *  * * * * * * * |  1 0 0 1 1 0 0 0 0  0  0 | 1 1 0 1 0
..x..... ........ ........ ........      | 0 0 2 0 0 0 0 0 | *  *  * 4 * * * * *  * * * * * * * |  0 2 0 0 0 1 0 0 0  0  0 | 2 0 1 0 0
........ ..x..... ........ ........      | 0 0 2 0 0 0 0 0 | *  *  * * 4 * * * *  * * * * * * * |  0 0 2 0 0 0 1 0 0  0  0 | 0 2 1 0 0
..o.o... ..o.o... ..o.o... ..o.o... &#x  | 0 0 1 0 1 0 0 0 | *  *  * * * 8 * * *  * * * * * * * |  2 0 0 0 0 1 1 0 0  0  0 | 2 2 0 0 0
...x.... ........ ........ ........      | 0 0 0 2 0 0 0 0 | *  *  * * * * 4 * *  * * * * * * * |  0 0 0 2 0 0 0 1 0  0  0 | 2 0 0 1 0
........ ...x.... ........ ........      | 0 0 0 2 0 0 0 0 | *  *  * * * * * 4 *  * * * * * * * |  0 0 0 0 2 0 0 0 1  0  0 | 0 2 0 1 0
...o.o.. ...o.o.. ...o.o.. ...o.o.. &#x  | 0 0 0 1 0 1 0 0 | *  *  * * * * * * 8  * * * * * * * |  2 0 0 0 0 0 0 1 1  0  0 | 2 2 0 0 0
....oo.. ....oo.. ....oo.. ....oo.. &#x  | 0 0 0 0 1 1 0 0 | *  *  * * * * * * * 16 * * * * * * |  1 0 0 0 0 0 0 0 0  1  1 | 1 1 0 0 1
....o.o. ....o.o. ....o.o. ....o.o. &#x  | 0 0 0 0 1 0 1 0 | *  *  * * * * * * *  * 8 * * * * * |  0 0 0 0 0 1 0 0 0  2  0 | 2 0 0 0 1
....o..o ....o..o ....o..o ....o..o &#x  | 0 0 0 0 1 0 0 1 | *  *  * * * * * * *  * * 8 * * * * |  0 0 0 0 0 0 1 0 0  0  2 | 0 2 0 0 1
.....oo. .....oo. .....oo. .....oo. &#x  | 0 0 0 0 0 1 1 0 | *  *  * * * * * * *  * * * 8 * * * |  0 0 0 0 0 0 0 1 0  2  0 | 2 0 0 0 1
.....o.o .....o.o .....o.o .....o.o &#x  | 0 0 0 0 0 1 0 1 | *  *  * * * * * * *  * * * * 8 * * |  0 0 0 0 0 0 0 0 1  0  2 | 0 2 0 0 1
......x. ........ ........ ........      | 0 0 0 0 0 0 2 0 | *  *  * * * * * * *  * * * * * 2 * |  0 0 0 0 0 2 0 2 0  0  0 | 4 0 0 0 0
........ .......x ........ ........      | 0 0 0 0 0 0 0 2 | *  *  * * * * * * *  * * * * * * 2 |  0 0 0 0 0 0 2 0 2  0  0 | 0 4 0 0 0
-----------------------------------------+-----------------+------------------------------------+--------------------------+----------
oooooo.. oooooo.. oooooo.. oooooo.. &#x  | 1 1 1 1 1 1 0 0 | 1  1  1 0 0 1 0 0 1  1 0 0 0 0 0 0 | 16 * * * * * * * *  *  * | 1 1 0 0 0
o.x..... ........ ........ ........ &#x  | 1 0 2 0 0 0 0 0 | 0  2  0 1 0 0 0 0 0  0 0 0 0 0 0 0 |  * 8 * * * * * * *  *  * | 1 0 1 0 0
........ o.x..... ........ ........ &#x  | 1 0 2 0 0 0 0 0 | 0  2  0 0 1 0 0 0 0  0 0 0 0 0 0 0 |  * * 8 * * * * * *  *  * | 0 1 1 0 0
.o.x.... ........ ........ ........ &#x  | 0 1 0 2 0 0 0 0 | 0  0  2 0 0 0 1 0 0  0 0 0 0 0 0 0 |  * * * 8 * * * * *  *  * | 1 0 0 1 0
........ .o.x.... ........ ........ &#x  | 0 1 0 2 0 0 0 0 | 0  0  2 0 0 0 0 1 0  0 0 0 0 0 0 0 |  * * * * 8 * * * *  *  * | 0 1 0 1 0
..x.u.x. ........ ........ ........ &#xt | 0 0 2 0 2 0 2 0 | 0  0  0 1 0 2 0 0 0  0 2 0 0 0 1 0 |  * * * * * 4 * * *  *  * | 2 0 0 0 0
........ ..x.u..x ........ ........ &#xt | 0 0 2 0 2 0 0 2 | 0  0  0 0 1 2 0 0 0  0 0 2 0 0 0 1 |  * * * * * * 4 * *  *  * | 0 2 0 0 0
...x.ux. ........ ........ ........ &#xt | 0 0 0 2 0 2 2 0 | 0  0  0 0 0 0 1 0 2  0 0 0 2 0 1 0 |  * * * * * * * 4 *  *  * | 2 0 0 0 0
........ ...x.u.x ........ ........ &#xt | 0 0 0 2 0 2 0 2 | 0  0  0 0 0 0 0 1 2  0 0 0 0 2 0 1 |  * * * * * * * * 4  *  * | 0 2 0 0 0
....ooo. ....ooo. ....ooo. ....ooo. &#x  | 0 0 0 0 1 1 1 0 | 0  0  0 0 0 0 0 0 0  1 1 0 1 0 0 0 |  * * * * * * * * * 16  * | 1 0 0 0 1
....oo.o ....oo.o ....oo.o ....oo.o &#x  | 0 0 0 0 1 1 0 1 | 0  0  0 0 0 0 0 0 0  1 0 1 0 1 0 0 |  * * * * * * * * *  * 16 | 0 1 0 0 1
-----------------------------------------+-----------------+------------------------------------+--------------------------+----------
ooxxuux. ........ ........ ........ &#xr ♦ 1 1 2 2 2 2 2 0 | 1  2  2 1 0 2 1 0 2  2 2 0 2 0 1 0 |  2 1 0 1 0 1 0 1 0  2  0 | 8 * * * *
........ ooxxuu.x ........ ........ &#xr ♦ 1 1 2 2 2 2 0 2 | 1  2  2 0 1 2 0 1 2  2 0 2 0 2 0 1 |  2 0 1 0 1 0 1 0 1  0  2 | * 8 * * *
o.x..... o.x..... ........ q.o..... &#zx ♦ 2 0 4 0 0 0 0 0 | 0  8  0 2 2 0 0 0 0  0 0 0 0 0 0 0 |  0 4 4 0 0 0 0 0 0  0  0 | * * 2 * *
.o.x.... .o.x.... .q.o.... ........ &#zx ♦ 0 2 0 4 0 0 0 0 | 0  0  8 0 0 0 2 2 0  0 0 0 0 0 0 0 |  0 0 0 4 4 0 0 0 0  0  0 | * * * 2 *
........ ........ ....qooo ....oqoo &#xt ♦ 0 0 0 0 2 2 1 1 | 0  0  0 0 0 0 0 0 0  4 2 2 2 2 0 0 |  0 0 0 0 0 0 0 0 0  4  4 | * * * * 4
```