Acronym hex, K-4.2
tetracross4),
hemitesseract,
hemioctachoron,
16-cell,
aerochoron,
tetrahedral antiprism

Segmentochoron display
Cross sections
` ©`
Vertex figure
` ©`
Vertex layers
 Layer Symmetry Subsymmetries o3o3o4o o3o3o . o3o . o o . o4o . o3o4o 1 x3o3o4o x3o3o .tet first x3o . o{3} first x . o4oedge first . o3o4overtex first 2 o3o3x .opposite tet o3o . q o . x4o . x3o4overtex figure 3 o3x . oopposite {3} x . o4oopposite edge . o3o4oopposite vertex o3o3o *b3o o3o3o    . o3o . *b3o o . o    o . o3o *b3o 1 x3o3o *b3o x3o3o    .tet first x3o . *b3otet first x . o    oedge first . o3o *b3overtex first 2 o3o3x    .opposite tet o3o . *b3xopposite tet o . x    x . x3o *b3overtex figure 3 x . o    oopposite edge . o3o *b3oopposite vertex
Lace city
in approx. ASCII-art
 ``` ©   ``` ``` o4o o4o x4o o4o o4o ```
```    x3o
o3o     o3o
o3x
```
Coordinates
• as orthoplex (tetracross):   (1/sqrt(2), 0, 0, 0)   & all permutations, all changes of sign
• as hemitesseract:   (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8))   & all even permutations, all even changes of sign
• as "the other" (mirrored) hemitesseract:   (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), -1/sqrt(8))   & all even permutations, all even changes of sign
• (the compound of those 3 such oriented hexadecachora is {3,4,3}[3{3,3,4}]2{3,4,3}, inscribed in an ico)
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
 by cells: oct tet tho 4 8 hex 0 16
)
Dual tes
Confer
Grünbaumian relatives:
hex+8oct   2hex+8oct
general pyramid-antiprisms:
n-pyap
compounds:
general polytopal classes:
tetrahedrochora   regular   orthoplex
External

Incidence matrix according to Dynkin symbol

```x3o3o4o

. . . . | 8 ♦  6 | 12 |  8
--------+---+----+----+---
x . . . | 2 | 24 |  4 |  4
--------+---+----+----+---
x3o . . | 3 |  3 | 32 |  2
--------+---+----+----+---
x3o3o . ♦ 4 |  6 |  4 | 16
```

```x3o3o4/3o

. . .   . | 8 ♦  6 | 12 |  8
----------+---+----+----+---
x . .   . | 2 | 24 |  4 |  4
----------+---+----+----+---
x3o .   . | 3 |  3 | 32 |  2
----------+---+----+----+---
x3o3o   . ♦ 4 |  6 |  4 | 16
```

```x3o3/2o4o

. .   . . | 8 ♦  6 | 12 |  8
----------+---+----+----+---
x .   . . | 2 | 24 |  4 |  4
----------+---+----+----+---
x3o   . . | 3 |  3 | 32 |  2
----------+---+----+----+---
x3o3/2o . ♦ 4 |  6 |  4 | 16
```

```x3o3/2o4/3o

. .   .   . | 8 ♦  6 | 12 |  8
------------+---+----+----+---
x .   .   . | 2 | 24 |  4 |  4
------------+---+----+----+---
x3o   .   . | 3 |  3 | 32 |  2
------------+---+----+----+---
x3o3/2o   . ♦ 4 |  6 |  4 | 16
```

```x3/2o3o4o

.   . . . | 8 ♦  6 | 12 |  8
----------+---+----+----+---
x   . . . | 2 | 24 |  4 |  4
----------+---+----+----+---
x3/2o . . | 3 |  3 | 32 |  2
----------+---+----+----+---
x3/2o3o . ♦ 4 |  6 |  4 | 16
```

```x3/2o3o4/3o

.   . .   . | 8 ♦  6 | 12 |  8
------------+---+----+----+---
x   . .   . | 2 | 24 |  4 |  4
------------+---+----+----+---
x3/2o .   . | 3 |  3 | 32 |  2
------------+---+----+----+---
x3/2o3o   . ♦ 4 |  6 |  4 | 16
```

```x3/2o3/2o4o

.   .   . . | 8 ♦  6 | 12 |  8
------------+---+----+----+---
x   .   . . | 2 | 24 |  4 |  4
------------+---+----+----+---
x3/2o   . . | 3 |  3 | 32 |  2
------------+---+----+----+---
x3/2o3/2o . ♦ 4 |  6 |  4 | 16
```

```x3/2o3/2o4/3o

.   .   .   . | 8 ♦  6 | 12 |  8
--------------+---+----+----+---
x   .   .   . | 2 | 24 |  4 |  4
--------------+---+----+----+---
x3/2o   .   . | 3 |  3 | 32 |  2
--------------+---+----+----+---
x3/2o3/2o   . ♦ 4 |  6 |  4 | 16
```

```x3o3o *b3o

. . .    . | 8 ♦  6 | 12 | 4 4
-----------+---+----+----+----
x . .    . | 2 | 24 |  4 | 2 2
-----------+---+----+----+----
x3o .    . | 3 |  3 | 32 | 1 1
-----------+---+----+----+----
x3o3o    . ♦ 4 |  6 |  4 | 8 *
x3o . *b3o ♦ 4 |  6 |  4 | * 8
```

```x3o3o *b3/2o

. . .      . | 8 ♦  6 | 12 | 4 4
-------------+---+----+----+----
x . .      . | 2 | 24 |  4 | 2 2
-------------+---+----+----+----
x3o .      . | 3 |  3 | 32 | 1 1
-------------+---+----+----+----
x3o3o      . ♦ 4 |  6 |  4 | 8 *
x3o . *b3/2o ♦ 4 |  6 |  4 | * 8
```

```x3o3/2o *b3/2o

. .   .      . | 8 ♦  6 | 12 | 4 4
---------------+---+----+----+----
x .   .      . | 2 | 24 |  4 | 2 2
---------------+---+----+----+----
x3o   .      . | 3 |  3 | 32 | 1 1
---------------+---+----+----+----
x3o3/2o      . ♦ 4 |  6 |  4 | 8 *
x3o   . *b3/2o ♦ 4 |  6 |  4 | * 8
```

```x3/2o3o *b3o

.   . .    . | 8 ♦  6 | 12 | 4 4
-------------+---+----+----+----
x   . .    . | 2 | 24 |  4 | 2 2
-------------+---+----+----+----
x3/2o .    . | 3 |  3 | 32 | 1 1
-------------+---+----+----+----
x3/2o3o    . ♦ 4 |  6 |  4 | 8 *
x3/2o . *b3o ♦ 4 |  6 |  4 | * 8
```

```x3/2o3o *b3/2o

.   . .      . | 8 ♦  6 | 12 | 4 4
---------------+---+----+----+----
x   . .      . | 2 | 24 |  4 | 2 2
---------------+---+----+----+----
x3/2o .      . | 3 |  3 | 32 | 1 1
---------------+---+----+----+----
x3/2o3o      . ♦ 4 |  6 |  4 | 8 *
x3/2o . *b3/2o ♦ 4 |  6 |  4 | * 8
```

```x3/2o3/2o *b3/2o

.   .   .      . | 8 ♦  6 | 12 | 4 4
-----------------+---+----+----+----
x   .   .      . | 2 | 24 |  4 | 2 2
-----------------+---+----+----+----
x3/2o   .      . | 3 |  3 | 32 | 1 1
-----------------+---+----+----+----
x3/2o3/2o      . ♦ 4 |  6 |  4 | 8 *
x3/2o   . *b3/2o ♦ 4 |  6 |  4 | * 8
```

```s4o3o3o

demi( . . . . ) | 8 ♦  6 | 12 | 4 4
----------------+---+----+----+----
s4o . .   ♦ 2 | 24 |  4 | 2 2
----------------+---+----+----+----
sefa( s4o3o . ) | 3 |  3 | 32 | 1 1
----------------+---+----+----+----
s4o3o .   ♦ 4 |  6 |  4 | 8 *
sefa( s4o3o3o ) ♦ 4 |  6 |  4 | * 8
```

```s2s4o3o

demi( . . . . ) | 8 ♦  3  3 |  9 3 | 3 1 4
----------------+---+-------+------+------
s2s . .   ♦ 2 | 12  * |  4 0 | 2 0 2
. s4o .   ♦ 2 |  * 12 |  2 2 | 1 1 2
----------------+---+-------+------+------
sefa( s2s4o . ) | 3 |  2  1 | 24 * | 1 0 1
sefa( . s4o3o ) | 3 |  0  3 |  * 8 | 0 1 1
----------------+---+-------+------+------
s2s4o .   ♦ 4 |  4  2 |  4 0 | 6 * *
. s4o3o   ♦ 4 |  0  6 |  0 4 | * 2 *
sefa( s2s4o3o ) ♦ 4 |  3  3 |  3 1 | * * 8
```

```s4o2s4o

demi( . . . . ) | 8 ♦ 1  4 1 |  6  6 | 2 2 4
----------------+---+--------+-------+------
s4o . .   ♦ 2 | 4  * * |  4  0 | 2 0 2
s 2 s .   ♦ 2 | * 16 * |  2  2 | 1 1 2
. . s4o   ♦ 2 | *  * 4 |  0  4 | 0 2 2
----------------+---+--------+-------+------
sefa( s4o2s . ) | 3 | 1  2 0 | 16  * | 1 1 0
sefa( s 2 s4o ) | 3 | 0  2 1 |  * 16 | 0 1 1
----------------+---+--------+-------+------
s4o2s .   ♦ 4 | 2  4 0 |  4  0 | 4 * *
s 2 s4o   ♦ 4 | 0  4 2 |  0  4 | * 4 *
sefa( s4o2s4o ) ♦ 4 | 1  4 1 |  2  2 | * * 8

or
demi( . . . . )    | 8 ♦ 2  4 | 12 | 4 4
-------------------+---+------+----+----
s4o . .    & ♦ 2 | 8  * |  4 | 2 2
s 2 s .      ♦ 2 | * 16 |  4 | 2 2
-------------------+---+------+----+----
sefa( s4o2s . )  & | 3 | 1  2 | 32 | 1 1
-------------------+---+------+----+----
s4o2s .    & ♦ 4 | 2  4 |  4 | 8 *
sefa( s4o2s4o )    ♦ 4 | 2  4 |  4 | * 8
```

```s2s2s4o

demi( . . . . ) | 8 ♦ 1 2 2 1 |  6 3 3 | 2 1 1 4
----------------+---+---------+--------+--------
s2s . .   ♦ 2 | 4 * * * |  4 0 0 | 2 0 0 2
s 2 s .   ♦ 2 | * 8 * * |  2 2 0 | 1 1 0 2
. s2s .   ♦ 2 | * * 8 * |  2 0 2 | 1 0 1 2
. . s4o   ♦ 2 | * * * 4 |  0 2 2 | 0 1 1 2
----------------+---+---------+--------+--------
sefa( s2s2s . ) | 3 | 1 1 1 0 | 16 * * | 1 0 0 1
sefa( s 2 s4o ) | 3 | 0 2 0 1 |  * 8 * | 0 1 0 1
sefa( . s2s4o ) | 3 | 0 0 2 1 |  * * 8 | 0 0 1 1
----------------+---+---------+--------+--------
s2s2s .   ♦ 4 | 2 2 2 0 |  4 0 0 | 4 * * *
s 2 s4o   ♦ 4 | 0 4 0 2 |  0 4 0 | * 2 * *
. s2s4o   ♦ 4 | 0 0 4 2 |  0 0 4 | * * 2 *
sefa( s2s2s4o ) ♦ 4 | 1 2 2 1 |  2 1 1 | * * * 8
```

```s2s2s2s

demi( . . . .  ) | 8 ♦ 1 1 1 1 1 1 | 3 3 3 3 | 1 1 1 1 4
-----------------+---+-------------+---------+----------
s2s . .    ♦ 2 | 4 * * * * * | 2 2 0 0 | 1 1 0 0 2
s 2 s .    ♦ 2 | * 4 * * * * | 2 0 2 0 | 1 0 1 0 2
s . . s2*a ♦ 2 | * * 4 * * * | 0 2 2 0 | 0 1 1 0 2
. s2s .    ♦ 2 | * * * 4 * * | 2 0 0 2 | 1 0 0 1 2
. s 2 s    ♦ 2 | * * * * 4 * | 0 2 0 2 | 0 1 0 1 2
. . s2s    ♦ 2 | * * * * * 4 | 0 0 2 2 | 0 0 1 1 2
-----------------+---+-------------+---------+----------
sefa( s2s2s .  ) | 3 | 1 1 0 1 0 0 | 8 * * * | 1 0 0 0 1
sefa( s2s 2 s  ) | 3 | 1 0 1 0 1 0 | * 8 * * | 0 1 0 0 1
sefa( s 2 s2s  ) | 3 | 0 1 1 0 0 1 | * * 8 * | 0 0 1 0 1
sefa( . s2s2s  ) | 3 | 0 0 0 1 1 1 | * * * 8 | 0 0 0 1 1
-----------------+---+-------------+---------+----------
s2s2s .    ♦ 4 | 2 2 0 2 0 0 | 4 0 0 0 | 2 * * * *
s2s 2 s    ♦ 4 | 2 0 2 0 2 0 | 0 4 0 0 | * 2 * * *
s 2 s2s    ♦ 4 | 0 2 2 0 0 2 | 0 0 4 0 | * * 2 * *
. s2s2s    ♦ 4 | 0 0 0 2 2 2 | 0 0 0 4 | * * * 2 *
sefa( s2s2s2s  ) ♦ 4 | 1 1 1 1 1 1 | 1 1 1 1 | * * * * 8
```

```xo3oo3ox&#x   → height = 1/sqrt(2) = 0.707107
(tet || dual tet)

o.3o.3o.    | 4 * ♦ 3  3 0 | 3  6  3 0 | 1 3 3 1 0
.o3.o3.o    | * 4 ♦ 0  3 3 | 0  3  6 3 | 0 1 3 3 1
------------+-----+--------+-----------+----------
x. .. ..    | 2 0 | 6  * * | 2  2  0 0 | 1 2 1 0 0
oo3oo3oo&#x | 1 1 | * 12 * | 0  2  2 0 | 0 1 2 1 0
.. .. .x    | 0 2 | *  * 6 | 0  0  2 2 | 0 0 1 2 1
------------+-----+--------+-----------+----------
x.3o. ..    | 3 0 | 3  0 0 | 4  *  * * | 1 1 0 0 0
xo .. ..&#x | 2 1 | 1  2 0 | * 12  * * | 0 1 1 0 0
.. .. ox&#x | 1 2 | 0  2 1 | *  * 12 * | 0 0 1 1 0
.. .o3.x    | 0 3 | 0  0 3 | *  *  * 4 | 0 0 0 1 1
------------+-----+--------+-----------+----------
x.3o.3o.    ♦ 4 0 | 6  0 0 | 4  0  0 0 | 1 * * * *
xo3oo ..&#x ♦ 3 1 | 3  3 0 | 1  3  0 0 | * 4 * * *
xo .. ox&#x ♦ 2 2 | 1  4 1 | 0  2  2 0 | * * 6 * *
.. oo3ox&#x ♦ 1 3 | 0  3 3 | 0  0  3 1 | * * * 4 *
.o3.o3.x    ♦ 0 4 | 0  0 6 | 0  0  0 4 | * * * * 1

or
o.3o.3o.    & | 8 ♦  3  3 | 3  9 | 1 4 3
--------------+---+-------+------+------
x. .. ..    & | 2 | 12  * | 2  2 | 1 2 1
oo3oo3oo&#x   | 2 |  * 12 | 0  4 | 0 2 2
--------------+---+-------+------+------
x.3o. ..    & | 3 |  3  0 | 8  * | 1 1 0
xo .. ..&#x & | 3 |  1  2 | * 24 | 0 1 1
--------------+---+-------+------+------
x.3o.3o.    & ♦ 4 |  6  0 | 4  0 | 2 * *
xo3oo ..&#x & ♦ 4 |  3  3 | 1  3 | * 8 *
xo .. ox&#x   ♦ 4 |  2  4 | 0  4 | * * 6
```

```oxo3ooo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o..3o..4o..    | 1 * * ♦ 6  0 0 | 12 0  0 | 8 0
.o.3.o.4.o.    | * 6 * ♦ 1  4 1 |  4 4  4 | 4 4
..o3..o4..o    | * * 1 ♦ 0  0 6 |  0 0 12 | 0 8
---------------+-------+--------+---------+----
oo.3oo.4oo.&#x | 1 1 0 | 6  * * |  4 0  0 | 4 0
.x. ... ...    | 0 2 0 | * 12 * |  1 2  1 | 2 2
.oo3.oo4.oo&#x | 0 1 1 | *  * 6 |  0 0  4 | 0 4
---------------+-------+--------+---------+----
ox. ... ...&#x | 1 2 0 | 2  1 0 | 12 *  * | 2 0
.x.3.o. ...    | 0 3 0 | 0  3 0 |  * 8  * | 1 1
.xo ... ...&#x | 0 2 1 | 0  1 2 |  * * 12 | 0 2
---------------+-------+--------+---------+----
ox.3oo. ...&#x ♦ 1 3 0 | 3  3 0 |  3 1  0 | 8 *
.xo3.oo ...&#x ♦ 0 3 1 | 0  3 3 |  0 1  3 | * 8

or
o..3o..4o..    & | 2 * ♦  6  0 | 12 0 |  8
.o.3.o.4.o.      | * 6 ♦  2  4 |  8 4 |  8
-----------------+-----+-------+------+---
oo.3oo.4oo.&#x & | 1 1 | 12  * |  4 0 |  4
.x. ... ...      | 0 2 |  * 12 |  2 2 |  4
-----------------+-----+-------+------+---
ox. ... ...&#x & | 1 2 |  2  1 | 24 * |  2
.x.3.o. ...      | 0 3 |  0  3 |  * 8 |  2
-----------------+-----+-------+------+---
ox.3oo. ...&#x & ♦ 1 3 |  3  3 |  3 1 | 16
```

```ooo3oxo3ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)

o..3o..3o..    | 1 * * ♦ 6  0 0 | 12 0 0  0 | 4 4 0 0
.o.3.o.3.o.    | * 6 * ♦ 1  4 1 |  4 2 2  4 | 2 2 2 2
..o3..o3..o    | * * 1 ♦ 0  0 6 |  0 0 0 12 | 0 0 4 4
---------------+-------+--------+-----------+--------
oo.3oo.3oo.&#x | 1 1 0 | 6  * * |  4 0 0  0 | 2 2 0 0
... .x. ...    | 0 2 0 | * 12 * |  1 1 1  1 | 1 1 1 1
.oo3.oo3.oo&#x | 0 1 1 | *  * 6 |  0 0 0  4 | 0 0 2 2
---------------+-------+--------+-----------+--------
... ox. ...&#x | 1 2 0 | 2  1 0 | 12 * *  * | 1 1 0 0
.o.3.x. ...    | 0 3 0 | 0  3 0 |  * 4 *  * | 1 0 1 0
... .x.3.o.    | 0 3 0 | 0  3 0 |  * * 4  * | 0 1 0 1
... .xo ...&#x | 0 2 1 | 0  1 2 |  * * * 12 | 0 0 1 1
---------------+-------+--------+-----------+--------
oo.3ox. ...&#x ♦ 1 3 0 | 3  3 0 |  3 1 0  0 | 4 * * *
... ox.3oo.&#x ♦ 1 3 0 | 3  3 0 |  3 0 1  0 | * 4 * *
.oo3.xo ...&#x ♦ 0 3 1 | 0  3 3 |  0 1 0  3 | * * 4 *
... .xo3.oo&#x ♦ 0 3 1 | 0  3 3 |  0 0 1  3 | * * * 4

or
o..3o..3o..    & | 2 * ♦  6  0 | 12 0 0 | 4 4
.o.3.o.3.o.      | * 6 ♦  2  4 |  8 2 2 | 4 4
-----------------+-----+-------+--------+----
oo.3oo.3oo.&#x & | 1 1 | 12  * |  4 0 0 | 2 2
... .x. ...      | 0 2 |  * 12 |  2 1 1 | 2 2
-----------------+-----+-------+--------+----
... ox. ...&#x & | 1 2 |  2  1 | 24 * * | 1 1
.o.3.x. ...      | 0 3 |  0  3 |  * 4 * | 2 0
... .x.3.o.      | 0 3 |  0  3 |  * * 4 | 0 2
-----------------+-----+-------+--------+----
oo.3ox. ...&#x & ♦ 1 3 |  3  3 |  3 1 0 | 8 *
... ox.3oo.&#x & ♦ 1 3 |  3  3 |  3 0 1 | * 8
```

```xox oxo4ooo&#xt   → both heights = 1/2
(line || ortho pseudo {4} || line)

o.. o..4o..    | 2 * * ♦ 1 4 1 0 0 0 | 4 4 4 0 0 | 4 4 0
.o. .o.4.o.    | * 4 * ♦ 0 2 0 2 2 0 | 1 4 2 1 4 | 2 4 2
..o ..o4..o    | * * 2 ♦ 0 0 1 0 4 1 | 0 0 4 4 4 | 0 4 4
---------------+-------+-------------+-----------+------
x.. ... ...    | 2 0 0 | 1 * * * * * | 4 0 0 0 0 | 4 0 0
oo. oo.4oo.&#x | 1 1 0 | * 8 * * * * | 1 2 1 0 0 | 2 2 0
o.o o.o4o.o&#x | 1 0 1 | * * 2 * * * | 0 0 4 0 0 | 0 4 0
... .x. ...    | 0 2 0 | * * * 4 * * | 0 2 0 0 2 | 1 2 1
.oo .oo4.oo&#x | 0 1 1 | * * * * 8 * | 0 0 1 1 2 | 0 2 2
..x ... ...    | 0 0 2 | * * * * * 1 | 0 0 0 4 0 | 0 0 4
---------------+-------+-------------+-----------+------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 4 * * * * | 2 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 8 * * * | 1 1 0
ooo ooo4ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * 8 * * | 0 2 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 2 1 | * * * 4 * | 0 0 2
... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * 8 | 0 1 1
---------------+-------+-------------+-----------+------
xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 | 2 2 0 0 0 | 4 * *
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 1 2 0 1 | * 8 *
.ox .xo ...&#x ♦ 0 2 2 | 0 0 0 1 4 1 | 0 0 0 2 2 | * * 4

or
o.. o..4o..    & | 4 * ♦ 1  4 1 0 | 4  4 4 | 4 4
.o. .o.4.o.      | * 4 ♦ 0  4 0 2 | 2  8 2 | 4 4
-----------------+-----+----------+--------+----
x.. ... ...    & | 2 0 | 2  * * * | 4  0 0 | 4 0
oo. oo.4oo.&#x & | 1 1 | * 16 * * | 1  2 1 | 2 2
o.o o.o4o.o&#x   | 2 0 | *  * 2 * | 0  0 4 | 0 4
... .x. ...      | 0 2 | *  * * 4 | 0  4 0 | 2 2
-----------------+-----+----------+--------+----
xo. ... ...&#x & | 2 1 | 1  2 0 0 | 8  * * | 2 0
... ox. ...&#x & | 1 2 | 0  2 0 1 | * 16 * | 1 1
ooo ooo4ooo&#x   | 2 1 | 0  2 1 0 | *  * 8 | 0 2
-----------------+-----+----------+--------+----
xo. ox. ...&#x & ♦ 2 2 | 1  4 0 1 | 2  2 0 | 8 *
... oxo ...&#x   ♦ 2 2 | 0  4 1 1 | 0  2 2 | * 8
```

```xox oxo oxo&#xt   → both heights = 1/2
(line || ortho pseudo {4} || line)

o.. o.. o..    | 2 * * ♦ 1 4 1 0 0 0 0 | 4 2 2 4 0 0 0 | 2 2 2 2 0 0
.o. .o. .o.    | * 4 * ♦ 0 2 0 1 1 2 0 | 1 2 2 2 1 2 2 | 1 1 2 2 1 1
..o ..o ..o    | * * 2 ♦ 0 0 1 0 0 4 1 | 0 0 0 4 4 2 2 | 0 0 2 2 2 2
---------------+-------+---------------+---------------+------------
x.. ... ...    | 2 0 0 | 1 * * * * * * | 4 0 0 0 0 0 0 | 2 2 0 0 0 0
oo. oo. oo.&#x | 1 1 0 | * 8 * * * * * | 1 1 1 1 0 0 0 | 1 1 1 1 0 0
o.o o.o o.o&#x | 1 0 1 | * * 2 * * * * | 0 0 0 4 0 0 0 | 0 0 2 2 0 0
... .x. ...    | 0 2 0 | * * * 2 * * * | 0 2 0 0 0 2 0 | 1 0 2 0 1 0
... ... .x.    | 0 2 0 | * * * * 2 * * | 0 0 2 0 0 0 2 | 0 1 0 2 0 1
.oo .oo .oo&#x | 0 1 1 | * * * * * 8 * | 0 0 0 1 1 1 1 | 0 0 1 1 1 1
..x ... ...    | 0 0 2 | * * * * * * 1 | 0 0 0 0 4 0 0 | 0 0 0 0 2 2
---------------+-------+---------------+---------------+------------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 0 | 4 * * * * * * | 1 1 0 0 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 0 | * 4 * * * * * | 1 0 1 0 0 0
... ... ox.&#x | 1 2 0 | 0 2 0 0 1 0 0 | * * 4 * * * * | 0 1 0 1 0 0
ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * 8 * * * | 0 0 1 1 0 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 0 2 1 | * * * * 4 * * | 0 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0 0 0 1 0 2 0 | * * * * * 4 * | 0 0 1 0 1 0
... ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * 4 | 0 0 0 1 0 1
---------------+-------+---------------+---------------+------------
xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 0 | 2 2 0 0 0 0 0 | 2 * * * * *
xo. ... ox.&#x ♦ 2 2 0 | 1 4 0 0 1 0 0 | 2 0 2 0 0 0 0 | * 2 * * * *
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 0 2 0 | 0 1 0 2 0 1 0 | * * 4 * * *
... ... oxo&#x ♦ 1 2 1 | 0 2 1 0 1 2 0 | 0 0 1 2 0 0 1 | * * * 4 * *
.ox .xo ...&#x ♦ 0 2 2 | 0 0 0 1 0 4 1 | 0 0 0 0 2 2 0 | * * * * 2 *
.ox ... .xo&#x ♦ 0 2 2 | 0 0 0 0 1 4 1 | 0 0 0 0 2 0 2 | * * * * * 2

or
o.. o.. o..    & | 4 * ♦ 1  4 1 0 0 | 4 2 2 4 | 2 2 2 2
.o. .o. .o.      | * 4 ♦ 0  4 0 1 1 | 2 4 4 2 | 2 2 2 2
-----------------+-----+------------+---------+--------
x.. ... ...    & | 2 0 | 2  * * * * | 4 0 0 0 | 2 2 0 0
oo. oo. oo.&#x & | 1 1 | * 16 * * * | 1 1 1 1 | 1 1 1 1
o.o o.o o.o&#x   | 2 0 | *  * 2 * * | 0 0 0 4 | 0 0 2 2
... .x. ...      | 0 2 | *  * * 2 * | 0 4 0 0 | 2 0 2 0
... ... .x.      | 0 2 | *  * * * 2 | 0 0 4 0 | 0 2 0 2
-----------------+-----+------------+---------+--------
xo. ... ...&#x & | 2 1 | 1  2 0 0 0 | 8 * * * | 1 1 0 0
... ox. ...&#x & | 1 2 | 0  2 0 1 0 | * 8 * * | 1 0 1 0
... ... ox.&#x & | 1 2 | 0  2 0 0 1 | * * 8 * | 0 1 0 1
ooo ooo ooo&#x   | 2 1 | 0  2 1 0 0 | * * * 8 | 0 0 1 1
-----------------+-----+------------+---------+--------
xo. ox. ...&#x & ♦ 2 2 | 1  4 0 1 0 | 2 2 0 0 | 4 * * *
xo. ... ox.&#x & ♦ 2 2 | 1  4 0 0 1 | 2 0 2 0 | * 4 * *
... oxo ...&#x   ♦ 2 2 | 0  4 1 1 0 | 0 2 0 2 | * * 4 *
... ... oxo&#x   ♦ 2 2 | 0  4 1 0 1 | 0 0 2 2 | * * * 4
```

```xoo3oox oqo&#xt   → both heights = 1/sqrt(6) = 0.408248
({3} || ortho q-line || dual {3})

o..3o.. o..     | 3 * * ♦ 2 2 2 0 0 | 1 4 2 1  4 0 0 | 2 4 2 0
.o.3.o. .o.     | * 2 * ♦ 0 3 0 3 0 | 0 3 0 0  6 3 0 | 1 3 3 1
..o3..o ..o     | * * 3 ♦ 0 0 2 2 2 | 0 0 1 2  4 4 1 | 0 2 4 2
----------------+-------+-----------+----------------+--------
x.. ... ...     | 2 0 0 | 3 * * * * | 1 2 1 0  0 0 0 | 2 2 0 0
oo.3oo. oo.&#x  | 1 1 0 | * 6 * * * | 0 2 0 0  2 0 0 | 1 2 1 0
o.o3o.o o.o&#x  | 1 0 1 | * * 6 * * | 0 0 1 1  2 0 0 | 0 2 2 0
.oo3.oo .oo&#x  | 0 1 1 | * * * 6 * | 0 0 0 0  2 2 0 | 0 1 2 1
... ..x ...     | 0 0 2 | * * * * 3 | 0 0 0 1  0 2 1 | 0 0 2 2
----------------+-------+-----------+----------------+--------
x..3o.. ...     | 3 0 0 | 3 0 0 0 0 | 1 * * *  * * * | 2 0 0 0
xo. ... ...&#x  | 2 1 0 | 1 2 0 0 0 | * 6 * *  * * * | 1 1 0 0
x.o ... ...&#x  | 2 0 1 | 1 0 2 0 0 | * * 3 *  * * * | 0 2 0 0
... o.x ...&#x  | 1 0 2 | 0 0 2 0 1 | * * * 3  * * * | 0 0 2 0
ooo3ooo ooo&#xt | 1 1 1 | 0 1 1 1 0 | * * * * 12 * * | 0 1 1 0
... .ox ...&#x  | 0 1 2 | 0 0 0 2 1 | * * * *  * 6 * | 0 0 1 1
..o3..x ...     | 0 0 3 | 0 0 0 0 3 | * * * *  * * 1 | 0 0 0 2
----------------+-------+-----------+----------------+--------
xo.3oo. ...&#x  ♦ 3 1 0 | 3 3 0 0 0 | 1 3 0 0  0 0 0 | 2 * * *
xoo ... ...&#xt ♦ 2 1 1 | 1 2 2 1 0 | 0 1 1 0  2 0 0 | * 6 * *
... oox ...&#xt ♦ 1 1 2 | 0 1 2 2 1 | 0 0 0 1  2 1 0 | * * 6 *
.oo3.ox ...&#x  ♦ 0 1 3 | 0 0 0 3 3 | 0 0 0 0  0 3 1 | * * * 2

or
o..3o.. o..      & | 6 * ♦ 2  2 2 | 1  4 3  4 | 2  6
.o.3.o. .o.        | * 2 ♦ 0  6 0 | 0  6 0  6 | 2  6
-------------------+-----+--------+-----------+-----
x.. ... ...      & | 2 0 | 6  * * | 1  2 1  0 | 2  2
oo.3oo. oo.&#x   & | 1 1 | * 12 * | 0  2 0  2 | 1  3
o.o3o.o o.o&#x     | 2 0 | *  * 6 | 0  0 2  2 | 0  4
-------------------+-----+--------+-----------+-----
x..3o.. ...      & | 3 0 | 3  0 0 | 2  * *  * | 2  0
xo. ... ...&#x   & | 2 1 | 1  2 0 | * 12 *  * | 1  1
x.o ... ...&#x   & | 3 0 | 1  0 2 | *  * 6  * | 0  2
ooo3ooo ooo&#xt    | 2 1 | 0  2 1 | *  * * 12 | 0  2
-------------------+-----+--------+-----------+-----
xo.3oo. ...&#x   & ♦ 3 1 | 3  3 0 | 1  3 0  0 | 4  *
xoo ... ...&#xt  & ♦ 3 1 | 1  3 2 | 0  1 1  2 | * 12
```

```oxoo3ooox&#xr   → all heights = sqrt(2/3) = 0.816497
(pt || ({3} || inv {3}) || pt)

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