Acronym ...
Name a3b3c,
general variation of truncated octahedron

Vertex layers
 Layer Symmetry Subsymmetries o3o4o o3o . . o4o 1 b3a4o(c=a) b3a .(b,a)-{6} first . a4o{4} first 2 A3a . . A4o 3 a3A . . b4Q 4 a3b .opposite (a,b)-{6} . A4o 5 . a4oopposite {4} o3o3o o3o . o . o 1 a3b3c a3b .(a,b)-{6} first a . c(a,c)-{4} first 2 a3A . C . A 3 C3c . b . X(layers 3 & 4 interchange for a > c) 4 b3c .opposite (b,c)-{6} X . b 5 A . C 6 c . a
(A=b+c, B=a+c, C=b+a, X=a+b+c, Q=aq)
Lace tower
and approx. ASCII-art
```    a b C a b
o---o---o        - a3b
c / A c \ a \ c          height = c sqrt(2/3)
a o-------o---o A    - a3A
b \   b /  C  \ b        height = b sqrt(2/3)
C o---o-------o c  - C3c
a \ c \ a   / a        height = a sqrt(2/3)
b o---o---o c    - b3c
c A b
```
```      a  c  a
o-----o        - a c
b /   A   \ b          height = b/sqrt(2)
C o---------o C    - C A
a / \ c b c / \ a        height = min(a,c)/sqrt(2)
b o---o-----o---o b  - b X + X b [comp. only if a=c, else 2 layers: heigth = |a-c|/sqrt(2)]
c \ / a C a \ / c        height = min(a,c)/sqrt(2)
A o---------o A    - A C
b \   a   / b          height = b/sqrt(2)
o-----o        - c a
c     c
```
```case c=a:

a  a  a
o-----o        - a4o
b /   A   \ b          height = b/sqrt(2)
A o---------o A    - A4o
a /Q\ a b a /Q\ a        height = a/sqrt(2)
b o---o-----o---o b  - b4Q
a \ / a A a \ / a        height = a/sqrt(2)
A o---------o A    - A4o
b \   a   / b          height = b/sqrt(2)
o-----o        - a4o
a     a
```
Coordinates ((a-c)/sqrt(8), (a+c)/sqrt(8), (A+C)/sqrt(8))     & all permutations & even changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Especially
 toe (a=b=c=x) f3x3f u3x3u w3x3w x3u3x x3x3u x3(-x)3x tut (a=b=x, c=o) x3q3o x3v3o x3u3o x3w3o v3x3o u3x3o w3x3o (-x)3x3o co (a=c=x, b=o) q3o3x f3o3x v3o3x u3o3x
Confer a3b4c   a3b5c

Within case a=c the symbol a3b3c = a3b3a has an additional symmetry, which then allows to rewrite it as b3a4o.

Incidence matrix according to Dynkin symbol

```a3b3c   (a ≠ 0, b ≠ 0, c ≠ 0 : general toe-variant)

. . . | 24 |  1  1  1 | 1 1 1
------+----+----------+------
a . . |  2 | 12  *  * | 1 1 0
. b . |  2 |  * 12  * | 1 0 1
. . c |  2 |  *  * 12 | 0 1 1
------+----+----------+------
a3b . |  6 |  3  3  0 | 4 * *
a . c |  4 |  2  0  2 | * 6 *
. b3c |  6 |  0  3  3 | * * 4
```

```a3b3o   (a ≠ 0, b ≠ 0, c = 0 : general tut-variant)

. . . | 12 | 1  2 | 2 1
------+----+------+----
a . . |  2 | 6  * | 2 0
. b . |  2 | * 12 | 1 1
------+----+------+----
a3b . |  6 | 3  3 | 4 *
. b3o |  3 | 0  3 | * 4
```

```a3o3c   (a ≠ 0, b = 0, c ≠ 0 : general co-variant)

. . . | 12 |  2  2 | 1 2 1
------+----+-------+------
a . . |  2 | 12  * | 1 1 0
. . c |  2 |  * 12 | 0 1 1
------+----+-------+------
a3o . |  3 |  3  0 | 4 * *
a . c |  4 |  2  2 | * 6 *
. o3c |  3 |  0  3 | * * 4
```