a3b3c

Acronym ...

Name a3b3c,
general variation of truncated octahedron

VRML
(change symmetry
to [3,3] )

⭳

Circumradius sqrt[(3a²+4b²+3c²+4ab+2ac+4bc)/8]

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			`. a4o` opposite {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)

Face vector 24, 36, 14

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
q3x3o
u3x3o
w3x3o
(-x)3x3o

co (a=c=x, b=o)
q3o3x
f3o3x
v3o3x
u3o3x

Confer

general polytopal classes:: Wythoffian polyhedra isogonal

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)
         for c = a also cf. b3a4o

. . . | 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

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