Acronym tes, K-4.20
Name tesseract,
octachoron,
"hypercube",
4D measure-polytope4),
8-cell,
geochoron

Segmentochoron display
`     ©`
Cross sections
` ©`
Vertex figure
` ©`
Vertex layers
 Layer Symmetry Subsymmetries o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 o3o3o4x o3o3o .vertex first o3o . xedge first o . o4x{4} first . o3o4xcube first 2 o3o3q .vertex figure o3q . x q . o4x . o3o4xopposite cube 3 o3q3o . q3o . x o . o4xopposite {4} 4 q3o3o . o3o . xopposite edge 5 o3o3o .opposite vertex
Lace city
in approx. ASCII-art
 ``` ©   ``` ```x4o x4o x4o x4o ```
 ``` ©   ``` ```x x x x x x x x ```
```o3o  q3o  o3q  o3o

o3o  q3o  o3q  o3o
```
Coordinates (1/2, 1/2, 1/2, 1/2)   & all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
 by cells: cube tes 8
)
Dual hex
Confer
general duoprisms:
4,n-dip   4,2n-dip   n,n-dip   n,m-dip   2n,m-dip   2n,2m-dip   n/d,m/b-dip
compounds:
gico
Grünbaumian relatives:
2tes   6tes
general polytopal classes:
regular   hypercube
External

Augmenting cubpy onto the cube would lead to the ico (which would have an even larger symmetry)!

Incidence matrix according to Dynkin symbol

```o3o3o4x

. . . . | 16 ♦  4 |  6 | 4
--------+----+----+----+--
. . . x |  2 | 32 |  3 | 3
--------+----+----+----+--
. . o4x |  4 |  4 | 24 | 2
--------+----+----+----+--
. o3o4x ♦  8 | 12 |  6 | 8
```

```o3o3o4/3x

. . .   . | 16 ♦  4 |  6 | 4
----------+----+----+----+--
. . .   x |  2 | 32 |  3 | 3
----------+----+----+----+--
. . o4/3x |  4 |  4 | 24 | 2
----------+----+----+----+--
. o3o4/3x ♦  8 | 12 |  6 | 8
```

```o3o3/2o4x

. .   . . | 16 ♦  4 |  6 | 4
----------+----+----+----+--
. .   . x |  2 | 32 |  3 | 3
----------+----+----+----+--
. .   o4x |  4 |  4 | 24 | 2
----------+----+----+----+--
. o3/2o4x ♦  8 | 12 |  6 | 8
```

```o3o3/2o4/3x

. .   .   . | 16 ♦  4 |  6 | 4
------------+----+----+----+--
. .   .   x |  2 | 32 |  3 | 3
------------+----+----+----+--
. .   o4/3x |  4 |  4 | 24 | 2
------------+----+----+----+--
. o3/2o4/3x ♦  8 | 12 |  6 | 8
```

```o3/2o3o4x

.   . . . | 16 ♦  4 |  6 | 4
----------+----+----+----+--
.   . . x |  2 | 32 |  3 | 3
----------+----+----+----+--
.   . o4x |  4 |  4 | 24 | 2
----------+----+----+----+--
.   o3o4x ♦  8 | 12 |  6 | 8
```

```o3/2o3o4/3x

.   . .   . | 16 ♦  4 |  6 | 4
------------+----+----+----+--
.   . .   x |  2 | 32 |  3 | 3
------------+----+----+----+--
.   . o4/3x |  4 |  4 | 24 | 2
------------+----+----+----+--
.   o3o4/3x ♦  8 | 12 |  6 | 8
```

```o3/2o3/2o4x

.   .   . . | 16 ♦  4 |  6 | 4
------------+----+----+----+--
.   .   . x |  2 | 32 |  3 | 3
------------+----+----+----+--
.   .   o4x |  4 |  4 | 24 | 2
------------+----+----+----+--
.   o3/2o4x ♦  8 | 12 |  6 | 8
```

```o3/2o3/2o4/3x

.   .   .   . | 16 ♦  4 |  6 | 4
--------------+----+----+----+--
.   .   .   x |  2 | 32 |  3 | 3
--------------+----+----+----+--
.   .   o4/3x |  4 |  4 | 24 | 2
--------------+----+----+----+--
.   o3/2o4/3x ♦  8 | 12 |  6 | 8
```

```x4o x4o

. . . . | 16 ♦  2  2 | 1  4 1 | 2 2
--------+----+-------+--------+----
x . . . |  2 | 16  * | 1  2 0 | 2 1
. . x . |  2 |  * 16 | 0  2 1 | 1 2
--------+----+-------+--------+----
x4o . . |  4 |  4  0 | 4  * * | 2 0
x . x . |  4 |  2  2 | * 16 * | 1 1
. . x4o |  4 |  0  4 | *  * 4 | 0 2
--------+----+-------+--------+----
x4o x . ♦  8 |  8  4 | 2  4 0 | 4 *
x . x4o ♦  8 |  4  8 | 0  4 2 | * 4
```

```x4o x4/3o

. . .   . | 16 ♦  2  2 | 1  4 1 | 2 2
----------+----+-------+--------+----
x . .   . |  2 | 16  * | 1  2 0 | 2 1
. . x   . |  2 |  * 16 | 0  2 1 | 1 2
----------+----+-------+--------+----
x4o .   . |  4 |  4  0 | 4  * * | 2 0
x . x   . |  4 |  2  2 | * 16 * | 1 1
. . x4/3o |  4 |  0  4 | *  * 4 | 0 2
----------+----+-------+--------+----
x4o x   . ♦  8 |  8  4 | 2  4 0 | 4 *
x . x4/3o ♦  8 |  4  8 | 0  4 2 | * 4
```

```x4/3o x4/3o

.   . .   . | 16 ♦  2  2 | 1  4 1 | 2 2
------------+----+-------+--------+----
x   . .   . |  2 | 16  * | 1  2 0 | 2 1
.   . x   . |  2 |  * 16 | 0  2 1 | 1 2
------------+----+-------+--------+----
x4/3o .   . |  4 |  4  0 | 4  * * | 2 0
x   . x   . |  4 |  2  2 | * 16 * | 1 1
.   . x4/3o |  4 |  0  4 | *  * 4 | 0 2
------------+----+-------+--------+----
x4/3o x   . ♦  8 |  8  4 | 2  4 0 | 4 *
x   . x4/3o ♦  8 |  4  8 | 0  4 2 | * 4
```

```x o3o4x

. . . . | 16 ♦ 1  3 |  3  3 | 3 1
--------+----+------+-------+----
x . . . |  2 | 8  * |  3  0 | 3 0
. . . x |  2 | * 24 |  1  2 | 2 1
--------+----+------+-------+----
x . . x |  4 | 2  2 | 12  * | 2 0
. . o4x |  4 | 0  4 |  * 12 | 1 1
--------+----+------+-------+----
x . o4x ♦  8 | 4  8 |  4  2 | 6 *
. o3o4x ♦  8 | 0 12 |  0  6 | * 2
```

```x o3o4/3x

. . .   . | 16 ♦ 1  3 |  3  3 | 3 1
----------+----+------+-------+----
x . .   . |  2 | 8  * |  3  0 | 3 0
. . .   x |  2 | * 24 |  1  2 | 2 1
----------+----+------+-------+----
x . .   x |  4 | 2  2 | 12  * | 2 0
. . o4/3x |  4 | 0  4 |  * 12 | 1 1
----------+----+------+-------+----
x . o4/3x ♦  8 | 4  8 |  4  2 | 6 *
. o3o4/3x ♦  8 | 0 12 |  0  6 | * 2
```

```x o3/2o4x

. .   . . | 16 ♦ 1  3 |  3  3 | 3 1
----------+----+------+-------+----
x .   . . |  2 | 8  * |  3  0 | 3 0
. .   . x |  2 | * 24 |  1  2 | 2 1
----------+----+------+-------+----
x .   . x |  4 | 2  2 | 12  * | 2 0
. .   o4x |  4 | 0  4 |  * 12 | 1 1
----------+----+------+-------+----
x .   o4x ♦  8 | 4  8 |  4  2 | 6 *
. o3/2o4x ♦  8 | 0 12 |  0  6 | * 2
```

```x o3/2o4/3x

. .   .   . | 16 ♦ 1  3 |  3  3 | 3 1
------------+----+------+-------+----
x .   .   . |  2 | 8  * |  3  0 | 3 0
. .   .   x |  2 | * 24 |  1  2 | 2 1
------------+----+------+-------+----
x .   .   x |  4 | 2  2 | 12  * | 2 0
. .   o4/3x |  4 | 0  4 |  * 12 | 1 1
------------+----+------+-------+----
x .   o4/3x ♦  8 | 4  8 |  4  2 | 6 *
. o3/2o4/3x ♦  8 | 0 12 |  0  6 | * 2
```

```x x x4o

. . . . | 16 ♦ 1 1  2 | 1 2 2 1 | 2 1 1
--------+----+--------+---------+------
x . . . |  2 | 8 *  * | 1 2 0 0 | 2 1 0
. x . . |  2 | * 8  * | 1 0 2 0 | 2 0 1
. . x . |  2 | * * 16 | 0 1 1 1 | 1 1 1
--------+----+--------+---------+------
x x . . |  4 | 2 2  0 | 4 * * * | 2 0 0
x . x . |  4 | 2 0  2 | * 8 * * | 1 1 0
. x x . |  4 | 0 2  2 | * * 8 * | 1 0 1
. . x4o |  4 | 0 0  4 | * * * 4 | 0 1 1
--------+----+--------+---------+------
x x x . ♦  8 | 4 4  4 | 2 2 2 0 | 4 * *
x . x4o ♦  8 | 4 0  8 | 0 4 0 2 | * 2 *
. x x4o ♦  8 | 0 4  8 | 0 0 4 2 | * * 2
```

```x x x4/3o

. . .   . | 16 ♦ 1 1  2 | 1 2 2 1 | 2 1 1
----------+----+--------+---------+------
x . .   . |  2 | 8 *  * | 1 2 0 0 | 2 1 0
. x .   . |  2 | * 8  * | 1 0 2 0 | 2 0 1
. . x   . |  2 | * * 16 | 0 1 1 1 | 1 1 1
----------+----+--------+---------+------
x x .   . |  4 | 2 2  0 | 4 * * * | 2 0 0
x . x   . |  4 | 2 0  2 | * 8 * * | 1 1 0
. x x   . |  4 | 0 2  2 | * * 8 * | 1 0 1
. . x4/3o |  4 | 0 0  4 | * * * 4 | 0 1 1
----------+----+--------+---------+------
x x x   . ♦  8 | 4 4  4 | 2 2 2 0 | 4 * *
x . x4/3o ♦  8 | 4 0  8 | 0 4 0 2 | * 2 *
. x x4/3o ♦  8 | 0 4  8 | 0 0 4 2 | * * 2
```

```x x x x

. . . . | 16 ♦ 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1
--------+----+---------+-------------+--------
x . . . |  2 | 8 * * * | 1 1 1 0 0 0 | 1 1 1 0
. x . . |  2 | * 8 * * | 1 0 0 1 1 0 | 1 1 0 1
. . x . |  2 | * * 8 * | 0 1 0 1 0 1 | 1 0 1 1
. . . x |  2 | * * * 8 | 0 0 1 0 1 1 | 0 1 1 1
--------+----+---------+-------------+--------
x x . . |  4 | 2 2 0 0 | 4 * * * * * | 1 1 0 0
x . x . |  4 | 2 0 2 0 | * 4 * * * * | 1 0 1 0
x . . x |  4 | 2 0 0 2 | * * 4 * * * | 0 1 1 0
. x x . |  4 | 0 2 2 0 | * * * 4 * * | 1 0 0 1
. x . x |  4 | 0 2 0 2 | * * * * 4 * | 0 1 0 1
. . x x |  4 | 0 0 2 2 | * * * * * 4 | 0 0 1 1
--------+----+---------+-------------+--------
x x x . ♦  8 | 4 4 4 0 | 2 2 0 2 0 0 | 2 * * *
x x . x ♦  8 | 4 4 0 4 | 2 0 2 0 2 0 | * 2 * *
x . x x ♦  8 | 4 0 4 4 | 0 2 2 0 0 2 | * * 2 *
. x x x ♦  8 | 0 4 4 4 | 0 0 0 2 2 2 | * * * 2
```

```oo3oo4xx&#x   → height = 1
(cube || cube)

o.3o.4o.    | 8 * ♦  3 1  0 | 3  3 0 | 1 3 0
.o3.o4.o    | * 8 ♦  0 1  3 | 0  3 3 | 0 3 1
------------+-----+---------+--------+------
.. .. x.    | 2 0 | 12 *  * | 2  1 0 | 1 2 0
oo3oo4oo&#x | 1 1 |  * 8  * | 0  3 0 | 0 3 0
.. .. .x    | 0 2 |  * * 12 | 0  1 2 | 0 2 1
------------+-----+---------+--------+------
.. o.4x.    | 4 0 |  4 0  0 | 6  * * | 1 1 0
.. .. xx&#x | 2 2 |  1 2  1 | * 12 * | 0 2 0
.. .o4.x    | 0 4 |  0 0  4 | *  * 6 | 0 1 1
------------+-----+---------+--------+------
o.3o.4x.    ♦ 8 0 | 12 0  0 | 6  0 0 | 1 * *
.. oo4xx&#x ♦ 4 4 |  4 4  4 | 1  4 1 | * 6 *
.o3.o4.x    ♦ 0 8 |  0 0 12 | 0  0 6 | * * 1
```

```xx xx4oo&#x   → height = 1
(cube || cube)

o. o.4o.    | 8 * ♦ 1 2 1 0 0 | 2 1 1 2 0 0 | 1 2 1 0
.o .o4.o    | * 8 ♦ 0 0 1 1 2 | 0 0 1 2 2 1 | 0 2 1 1
------------+-----+-----------+-------------+--------
x. .. ..    | 2 0 | 4 * * * * | 2 0 1 0 0 0 | 1 2 0 0
.. x. ..    | 2 0 | * 8 * * * | 1 1 0 1 0 0 | 1 1 1 0
oo oo4oo&#x | 1 1 | * * 8 * * | 0 0 1 2 0 0 | 0 2 1 0
.x .. ..    | 0 2 | * * * 4 * | 0 0 1 0 2 0 | 0 2 0 1
.. .x ..    | 0 2 | * * * * 8 | 0 0 0 1 1 1 | 0 1 1 1
------------+-----+-----------+-------------+--------
x. x. ..    | 4 0 | 2 2 0 0 0 | 4 * * * * * | 1 1 0 0
.. x.4o.    | 4 0 | 0 4 0 0 0 | * 2 * * * * | 1 0 1 0
xx .. ..&#x | 2 2 | 1 0 2 1 0 | * * 4 * * * | 0 2 0 0
.. xx ..&#x | 2 2 | 0 1 2 0 1 | * * * 8 * * | 0 0 2 0
.x .x ..    | 0 4 | 0 0 0 2 2 | * * * * 4 * | 0 1 0 1
.. .x4.o    | 0 4 | 0 0 0 0 4 | * * * * * 2 | 0 0 1 1
------------+-----+-----------+-------------+--------
x. x.4o.    ♦ 8 0 | 4 8 0 0 0 | 4 2 0 0 0 0 | 1 * * *
xx xx ..&#x ♦ 4 4 | 2 2 4 2 2 | 1 0 4 0 1 0 | * 4 * *
.. xx4oo&#x ♦ 4 4 | 0 4 4 0 4 | 0 1 0 4 0 1 | * * 2 *
.x .x4.o    ♦ 0 8 | 0 0 0 4 8 | 0 0 0 0 4 2 | * * * 1
```

```xx xx xx&#x   → height = 1
(cube || cube)

o. o. o.    | 8 * ♦ 1 1 1 1 0 0 0 | 1 1 1 1 1 1 0 0 0 | 1 1 1 1 0
.o .o .o    | * 8 ♦ 0 0 0 1 1 1 1 | 0 0 0 1 1 1 1 1 1 | 0 1 1 1 1
------------+-----+---------------+-------------------+----------
x. .. ..    | 2 0 | 4 * * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
.. x. ..    | 2 0 | * 4 * * * * * | 1 0 1 0 1 0 0 0 0 | 1 1 0 1 0
.. .. x.    | 2 0 | * * 4 * * * * | 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0
oo oo oo&#x | 1 1 | * * * 8 * * * | 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0
.x .. ..    | 0 2 | * * * * 4 * * | 0 0 0 1 0 0 1 1 0 | 0 1 1 0 1
.. .x ..    | 0 2 | * * * * * 4 * | 0 0 0 0 1 0 1 0 1 | 0 1 0 1 1
.. .. .x    | 0 2 | * * * * * * 4 | 0 0 0 0 0 1 0 1 1 | 0 0 1 1 1
------------+-----+---------------+-------------------+----------
x. x. ..    | 4 0 | 2 2 0 0 0 0 0 | 2 * * * * * * * * | 1 1 0 0 0
x. .. x.    | 4 0 | 2 0 2 0 0 0 0 | * 2 * * * * * * * | 1 0 1 0 0
.. x. x.    | 4 0 | 0 2 2 0 0 0 0 | * * 2 * * * * * * | 1 0 0 1 0
xx .. ..&#x | 2 2 | 1 0 0 2 1 0 0 | * * * 4 * * * * * | 0 1 1 0 0
.. xx ..&#x | 2 2 | 0 1 0 2 0 1 0 | * * * * 4 * * * * | 0 1 0 1 0
.. .. xx&#x | 2 2 | 0 0 1 2 0 0 1 | * * * * * 4 * * * | 0 0 1 1 0
.x .x ..    | 0 4 | 0 0 0 0 2 2 0 | * * * * * * 2 * * | 0 1 0 0 1
.x .. .x    | 0 4 | 0 0 0 0 2 0 2 | * * * * * * * 2 * | 0 0 1 0 1
.. .x .x    | 0 4 | 0 0 0 0 0 2 2 | * * * * * * * * 2 | 0 0 0 1 1
------------+-----+---------------+-------------------+----------
x. x. x.    ♦ 8 0 | 4 4 4 0 0 0 0 | 2 2 2 0 0 0 0 0 0 | 1 * * * *
xx xx ..&#x ♦ 4 4 | 2 2 0 4 2 2 0 | 1 0 0 2 2 0 1 0 0 | * 2 * * *
xx .. xx&#x ♦ 4 4 | 2 0 2 4 2 0 2 | 0 1 0 2 0 2 0 1 0 | * * 2 * *
.. xx xx&#x ♦ 4 4 | 0 2 2 4 0 2 2 | 0 0 1 0 2 2 0 0 1 | * * * 2 *
.x .x .x    ♦ 0 8 | 0 0 0 0 4 4 4 | 0 0 0 0 0 0 2 2 2 | * * * * 1
```

```oqo xxx4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
({4} || pseudo (q,x,x)-cube || {4})

o.. o..4o..     | 4 * * ♦ 2 2 0 0 0 | 1 2 1 0 0 0 | 2 2 0
.o. .o.4.o.     | * 8 * ♦ 0 1 2 1 0 | 0 2 1 1 2 0 | 1 2 1
..o ..o4..o     | * * 4 ♦ 0 0 0 2 2 | 0 0 1 0 4 1 | 0 2 2
----------------+-------+-----------+-------------+------
... x.. ...     | 2 0 0 | 4 * * * * | 1 2 0 0 0 0 | 2 1 0
oo. oo.4oo.&#x  | 1 1 0 | * 8 * * * | 0 2 1 0 0 0 | 1 2 0
... .x. ...     | 0 2 0 | * * 8 * * | 0 1 0 1 1 0 | 1 1 1
.oo .oo4.oo&#x  | 0 1 1 | * * * 8 * | 0 0 1 0 2 0 | 0 2 1
... ..x ...     | 0 0 2 | * * * * 4 | 0 0 0 0 2 1 | 0 1 2
----------------+-------+-----------+-------------+------
... x..4o..     | 4 0 0 | 4 0 0 0 0 | 1 * * * * * | 2 0 0
... xx. ...&#x  | 2 2 0 | 1 2 1 0 0 | * 8 * * * * | 1 1 0
oqo ... ...&#xt | 1 2 1 | 0 2 0 2 0 | * * 4 * * * | 0 2 0
... .x.4.o.     | 0 4 0 | 0 0 4 0 0 | * * * 2 * * | 1 0 1
... .xx ...&#x  | 0 2 2 | 0 0 1 2 1 | * * * * 8 * | 0 1 1
... ..x4..o     | 0 0 4 | 0 0 0 0 4 | * * * * * 1 | 0 0 2
----------------+-------+-----------+-------------+------
... xx.4oo.&#x  ♦ 4 4 0 | 4 4 4 0 0 | 1 4 0 1 0 0 | 2 * *
oqo xxx ...&#xt ♦ 2 4 2 | 1 4 2 4 1 | 0 2 2 0 2 0 | * 4 *
... .xx4.oo&#x  ♦ 0 4 4 | 0 0 4 4 4 | 0 0 0 1 4 1 | * * 2

or
o.. o..4o..      & | 8 * ♦ 2  2 0 | 1  4 1 0 | 2 2
.o. .o.4.o.        | * 8 ♦ 0  2 2 | 0  4 1 1 | 2 2
-------------------+-----+--------+----------+----
... x.. ...      & | 2 0 | 8  * * | 1  2 0 0 | 2 1
oo. oo.4oo.&#x   & | 1 1 | * 16 * | 0  2 1 0 | 1 2
... .x. ...        | 0 2 | *  * 8 | 0  2 0 1 | 2 1
-------------------+-----+--------+----------+----
... x..4o..      & | 4 0 | 4  0 0 | 2  * * * | 2 0
... xx. ...&#x   & | 2 2 | 1  2 1 | * 16 * * | 1 1
oqo ... ...&#xt    | 2 2 | 0  4 0 | *  * 4 * | 0 2
... .x.4.o.        | 0 4 | 0  0 4 | *  * * 2 | 2 0
-------------------+-----+--------+----------+----
... xx.4oo.&#x   & ♦ 4 4 | 4  4 4 | 1  4 0 1 | 4 *
oqo xxx ...&#xt    ♦ 4 4 | 2  8 2 | 0  4 2 0 | * 4
```

```oqo xxx xxx&#xt   → both heights = 1/sqrt(2) = 0.707107
({4} || pseudo (q,x,x)-cube || {4})

o.. o.. o..     | 4 * * ♦ 1 1 2 0 0 0 0 0 | 1 2 2 1 0 0 0 0 | 2 1 1 0
.o. .o. .o.     | * 8 * ♦ 0 0 1 1 1 1 0 0 | 0 1 1 1 1 1 1 0 | 1 1 1 1
..o ..o ..o     | * * 4 ♦ 0 0 0 0 0 2 1 1 | 0 0 0 1 0 2 2 1 | 0 1 1 2
----------------+-------+-----------------+-----------------+--------
... x.. ...     | 2 0 0 | 2 * * * * * * * | 1 2 0 0 0 0 0 0 | 2 1 0 0
... ... x..     | 2 0 0 | * 2 * * * * * * | 1 0 2 0 0 0 0 0 | 2 0 1 0
oo. oo. oo.&#x  | 1 1 0 | * * 8 * * * * * | 0 1 1 1 0 0 0 0 | 1 1 1 0
... .x. ...     | 0 2 0 | * * * 4 * * * * | 0 1 0 0 1 1 0 0 | 1 1 0 1
... ... .x.     | 0 2 0 | * * * * 4 * * * | 0 0 1 0 1 0 1 0 | 1 0 1 1
.oo .oo .oo&#x  | 0 1 1 | * * * * * 8 * * | 0 0 0 1 0 1 1 0 | 0 1 1 1
... ..x ...     | 0 0 2 | * * * * * * 2 * | 0 0 0 0 0 2 0 1 | 0 1 0 2
... ... ..x     | 0 0 2 | * * * * * * * 2 | 0 0 0 0 0 0 2 1 | 0 0 1 2
----------------+-------+-----------------+-----------------+--------
... x.. x..     | 4 0 0 | 2 2 0 0 0 0 0 0 | 1 * * * * * * * | 2 0 0 0
... xx. ...&#x  | 2 2 0 | 1 0 2 1 0 0 0 0 | * 4 * * * * * * | 1 1 0 0
... ... xx.&#x  | 2 2 0 | 0 1 2 0 1 0 0 0 | * * 4 * * * * * | 1 0 1 0
oqo ... ...&#xt | 1 2 1 | 0 0 2 0 0 2 0 0 | * * * 4 * * * * | 0 1 1 0
... .x. .x.     | 0 4 0 | 0 0 0 2 2 0 0 0 | * * * * 2 * * * | 1 0 0 1
... .xx ...&#x  | 0 2 2 | 0 0 0 1 0 2 1 0 | * * * * * 4 * * | 0 1 0 1
... ... .xx&#x  | 0 2 2 | 0 0 0 0 1 2 0 1 | * * * * * * 4 * | 0 0 1 1
... ..x ..x     | 0 0 4 | 0 0 0 0 0 0 2 2 | * * * * * * * 1 | 0 0 0 2
----------------+-------+-----------------+-----------------+--------
... xx. xx.&#x  ♦ 4 4 0 | 2 2 4 2 2 0 0 0 | 1 2 2 0 1 0 0 0 | 2 * * *
oqo xxx ...&#xt ♦ 2 4 2 | 1 0 4 2 0 4 1 0 | 0 2 0 2 0 2 0 0 | * 2 * *
oqo ... xxx&#xt ♦ 2 4 2 | 0 1 4 0 2 4 0 1 | 0 0 2 2 0 0 2 0 | * * 2 *
... .xx .xx&#x  ♦ 0 4 4 | 0 0 0 2 2 4 2 2 | 0 0 0 0 1 2 2 1 | * * * 2

or
o.. o.. o..      & | 8 * ♦ 1 1  2 0 0 | 1 2 2 1 0 | 2 1 1
.o. .o. .o.        | * 8 ♦ 0 0  2 1 1 | 0 2 2 1 1 | 2 1 1
-------------------+-----+------------+-----------+------
... x.. ...      & | 2 0 | 4 *  * * * | 1 2 0 0 0 | 2 1 0
... ... x..      & | 2 0 | * 4  * * * | 1 0 2 0 0 | 2 0 1
oo. oo. oo.&#x   & | 1 1 | * * 16 * * | 0 1 1 1 0 | 1 1 1
... .x. ...        | 0 2 | * *  * 4 * | 0 2 0 0 1 | 2 1 0
... ... .x.        | 0 2 | * *  * * 4 | 0 0 2 0 1 | 2 0 1
-------------------+-----+------------+-----------+------
... x.. x..      & | 4 0 | 2 2  0 0 0 | 2 * * * * | 2 0 0
... xx. ...&#x   & | 2 2 | 1 0  2 1 0 | * 8 * * * | 1 1 0
... ... xx.&#x   & | 2 2 | 0 1  2 0 1 | * * 8 * * | 1 0 1
oqo ... ...&#xt    | 2 2 | 0 0  4 0 0 | * * * 4 * | 0 1 1
... .x. .x.        | 0 4 | 0 0  0 2 2 | * * * * 2 | 2 0 0
-------------------+-----+------------+-----------+------
... xx. xx.&#x   & ♦ 4 4 | 2 2  4 2 2 | 1 2 2 0 1 | 4 * *
oqo xxx ...&#xt    ♦ 4 4 | 2 0  8 2 0 | 0 4 0 2 0 | * 2 *
oqo ... xxx&#xt    ♦ 4 4 | 0 2  8 0 2 | 0 0 4 2 0 | * * 2
```

```xxxx oqoo3ooqo&#xt   → all heights = 1/sqrt(3) = 0.577350
(line || pseudo (q,x)-trip || pseudo inv (q,x)-trip || line)

o... o...3o...     | 2 * * * ♦ 1 3 0  0 0 0 0 | 3 3 0 0 0 | 3 1 0
.o.. .o..3.o..     | * 6 * * ♦ 0 1 1  2 0 0 0 | 1 2 2 1 0 | 2 1 1
..o. ..o.3..o.     | * * 6 * ♦ 0 0 0  2 1 1 0 | 0 1 2 2 1 | 1 1 2
...o ...o3...o     | * * * 2 ♦ 0 0 0  0 0 3 1 | 0 0 0 3 3 | 0 1 3
-------------------+---------+----------------+-----------+------
x... .... ....     | 2 0 0 0 | 1 * *  * * * * | 3 0 0 0 0 | 3 0 0
oo.. oo..3oo..&#x  | 1 1 0 0 | * 6 *  * * * * | 1 2 0 0 0 | 2 1 0
.x.. .... ....     | 0 2 0 0 | * * 3  * * * * | 1 0 2 0 0 | 2 0 1
.oo. .oo.3.oo.&#x  | 0 1 1 0 | * * * 12 * * * | 0 1 1 1 0 | 1 1 1
..x. .... ....     | 0 0 2 0 | * * *  * 3 * * | 0 0 2 0 1 | 1 0 2
..oo ..oo3..oo&#x  | 0 0 1 1 | * * *  * * 6 * | 0 0 0 2 1 | 0 1 2
...x .... ....     | 0 0 0 2 | * * *  * * * 1 | 0 0 0 0 3 | 0 0 3
-------------------+---------+----------------+-----------+------
xx.. .... ....&#x  | 2 2 0 0 | 1 2 1  0 0 0 0 | 3 * * * * | 2 0 0
.... oqo. ....&#xt | 1 2 1 0 | 0 2 0  2 0 0 0 | * 6 * * * | 1 1 0
.xx. .... ....&#x  | 0 2 2 0 | 0 0 1  2 1 0 0 | * * 6 * * | 1 0 1
.... .... .oqo&#xt | 0 1 2 1 | 0 0 0  2 0 2 0 | * * * 6 * | 0 1 1
..xx .... ....&#x  | 0 0 2 2 | 0 0 0  0 1 2 1 | * * * * 3 | 0 0 2
-------------------+---------+----------------+-----------+------
xxx. oqo. ....&#xt ♦ 2 4 2 0 | 1 4 2  4 1 0 0 | 2 2 2 0 0 | 3 * *
.... oqoo3ooqo&#xt ♦ 1 3 3 1 | 0 3 0  6 0 3 0 | 0 3 0 3 0 | * 2 *
.xxx .... .oqo&#xt ♦ 0 2 4 2 | 0 0 1  4 2 4 1 | 0 0 2 2 2 | * * 3

or
o... o...3o...      & | 4  * ♦ 1  3 0  0 | 3  3 0 | 3 1
.o.. .o..3.o..      & | * 12 ♦ 0  1 1  2 | 1  3 2 | 3 1
----------------------+------+-----------+--------+----
x... .... ....      & | 2  0 | 2  * *  * | 3  0 0 | 3 0
oo.. oo..3oo..&#x   & | 1  1 | * 12 *  * | 1  2 0 | 2 1
.x.. .... ....      & | 0  2 | *  * 6  * | 1  0 2 | 3 0
.oo. .oo.3.oo.&#x     | 0  2 | *  * * 12 | 0  2 1 | 2 1
----------------------+------+-----------+--------+----
xx.. .... ....&#x   & | 2  2 | 1  2 1  0 | 6  * * | 2 0
.... oqo. ....&#xt  & | 1  3 | 0  2 0  2 | * 12 * | 1 1
.xx. .... ....&#x     | 0  4 | 0  0 2  2 | *  * 6 | 2 0
----------------------+------+-----------+--------+----
xxx. oqo. ....&#xt  & ♦ 2  6 | 1  4 3  4 | 2  2 2 | 6 *
.... oqoo3ooqo&#xt    ♦ 2  6 | 0  6 0  6 | 0  6 0 | * 2
```

```oqooo3ooqoo3oooqo&#xt   → all heights = 1/2
(pt || pseudo q-tet || pseudo q-oct || pseudo dual q-tet || pt)

o....3o....3o....     | 1 * * * * ♦ 4  0  0 0 | 6  0 0 | 4 0
.o...3.o...3.o...     | * 4 * * * ♦ 1  3  0 0 | 3  3 0 | 3 1
..o..3..o..3..o..     | * * 6 * * ♦ 0  2  2 0 | 1  4 1 | 2 2
...o.3...o.3...o.     | * * * 4 * ♦ 0  0  3 1 | 0  3 3 | 1 3
....o3....o3....o     | * * * * 1 ♦ 0  0  0 4 | 0  0 6 | 0 4
----------------------+-----------+-----------+--------+----
oo...3oo...3oo...&#x  | 1 1 0 0 0 | 4  *  * * | 3  0 0 | 3 0
.oo..3.oo..3.oo..&#x  | 0 1 1 0 0 | * 12  * * | 1  2 0 | 2 1
..oo.3..oo.3..oo.&#x  | 0 0 1 1 0 | *  * 12 * | 0  2 1 | 1 2
...oo3...oo3...oo&#x  | 0 0 0 1 1 | *  *  * 4 | 0  0 3 | 0 3
----------------------+-----------+-----------+--------+----
oqo.. ..... .....&#xt | 1 2 1 0 0 | 2  2  0 0 | 6  * * | 2 0
..... .oqo. .....&#xt | 0 1 2 1 0 | 0  2  2 0 | * 12 * | 1 1
..... ..... ..oqo&#xt | 0 0 1 2 1 | 0  0  2 2 | *  * 6 | 0 2
----------------------+-----------+-----------+--------+----
oqoo.3ooqo. .....&#xt ♦ 1 3 3 1 0 | 3  6  3 0 | 3  3 0 | 4 *
..... .oqoo3.ooqo&#xt ♦ 0 1 3 3 1 | 0  3  6 3 | 0  3 3 | * 4

or
o....3o....3o....      & | 2 * * ♦ 4  0 |  6  0 | 4
.o...3.o...3.o...      & | * 8 * ♦ 1  3 |  3  3 | 4
..o..3..o..3..o..        | * * 6 ♦ 0  4 |  2  4 | 4
-------------------------+-------+------+-------+--
oo...3oo...3oo...&#x   & | 1 1 0 | 8  * |  3  0 | 3
.oo..3.oo..3.oo..&#x   & | 0 1 1 | * 24 |  1  2 | 3
-------------------------+-------+------+-------+--
oqo.. ..... .....&#xt  & | 1 2 1 | 2  2 | 12  * | 2
..... .oqo. .....&#xt    | 0 2 2 | 0  4 |  * 12 | 2
-------------------------+-------+------+-------+--
oqoo.3ooqo. .....&#xt  & ♦ 1 4 3 | 3  9 |  3  3 | 8
```