a3b4c

Acronym ...

Name a3b4c,
general variation of great rhombicuboctahedron

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

Vertex layers

Layer	Symmetry	Subsymmetries
	`o3o4o`	`o3o .`	`. o4o`
1	`a3b4c`	`a3b .` (a,b)-{6} first	`. b4c` (b,c)-{8} first
2		`a3B .`	`. C4c`
3		`C3B .`	`. a4A`
4		`b3X .` (layers 4 & 5 interchange for c > aq)	`. a4A`
5		`X3b .`	`. C4c`
6		`B3C .`	`. b4c` opposite (b,c)-{8}
7		`B3a .`
8		`b3a .` opposite (b,a)-{6}

(A=c+bq, B=b+cq, C=a+b, X=a+b+cq)

Lace tower
and approx. ASCII-art

          b   a   b                  
        o---o---o---o           - a3b
    c / B c | a | c B \ c           height = c/sqrt(3)
    o-------o---o-------o       - a3B
 b /  B  b /  C  \ b  B  \ b        height = bq/sqrt(3)
  o-------o-------o-------o     - C3B
c | b  c/  \a b a/  \c  b | c       height = c/sqrt(3) resp. height = aq/sqrt(3)
  o---o     o---o     o---o     - b3X + X3b [comp. only if c=aq, else 2 layers: heigth = |c-aq|/sqrt(3)]
a | C a\  /c  B  c\  /a C | a       height = aq/sqrt(3) resp. height = c/sqrt(3)
  o-----o-----------o-----o     - B3C
 b \  a |b    B    b| a  / b        height = bq/sqrt(3)
    o---o-----------o---o       - B3a
    c \   \c     c/   / c           height = c/sqrt(3)
        o---o---o---o           - b3a
          a   b   a

      c b   c   b c               
      o---o---o---o          - b4c
 a  / C a | c | a C \  a          height = a/sqrt(2)
c o-------o---o-------o c    - C4c
b | a b /   A   \ b a | b         height = b/sqrt(2)
A o---o-----------o---o A    - a4A
c | a | c   A   c | a | c         height = c
A o---o-----------o---o A    - a4A
b | C b \   c   / b C | b         height = b/sqrt(2)
c o-------o---o-------o c    - C4c
 a  \   a |   | a   /  a          height = a/sqrt(2)
   c  o---o---o---o  c       - b4c
        b   c   b

Coordinates (c/2, (bq+c)/2, (aq+bq+c)/2) & all permutations & all changes of sign

General of army (is itself convex)

Colonel of regiment (is itself locally convex)

Face vector 48, 72, 26

Especially

girco (a=b=c=x)
x3x4q
q3x4x
q3q4x
w3x4x

toe (a=b=x, c=o)
x3f4o
x3u4o
x3w4o
u3x4o
(-x)3x4o

sirco (a=c=x, b=o)
reco (a=x, b=o, c=q)
q3o4x
f3o4x
u3o4x

tic (a=o, b=c=x)
o3x4q
o3x4u
o3q4x

Confer

general polytopal classes:: Wythoffian polyhedra isogonal

Incidence matrix according to Dynkin symbol

a3b4c   (a ≠ 0, b ≠ 0, c ≠ 0 : general girco-variant)

. . . | 48 |  1  1  1 | 1  1 1
------+----+----------+-------
a . . |  2 | 24  *  * | 1  1 0  a
. b . |  2 |  * 24  * | 1  0 1  b
. . c |  2 |  *  * 24 | 0  1 1  c
------+----+----------+-------
a3b . |  6 |  3  3  0 | 8  * *
a . c |  4 |  2  0  2 | * 12 *
. b4c |  8 |  0  4  4 | *  * 6

Bb3aa3bB&#zc   → height = 0,
                 where: B = b+cq = b+c sqrt(2) (pseudo),
                 same as general girco-variant a3b4c)

o.3o.3o.     & | 48 |  1  1  1 | 1 1  1
---------------+----+----------+-------
.. a. ..     & |  2 | 24  *  * | 1 0  1  a
.. .. b.     & |  2 |  * 24  * | 1 1  0  b
oo3oo3oo&#c    |  2 |  *  * 24 | 0 1  1  c
---------------+----+----------+-------
.. a.3b.     & |  6 |  3  3  0 | 8 *  *
Bb .. bB&#zc   |  8 |  0  4  4 | * 6  *
.. aa ..&#c    |  4 |  2  0  2 | * * 12

a3b4o   (a ≠ 0, b ≠ 0, c = 0 : general toe-variant)

. . . | 24 |  1  2 | 2 1
------+----+-------+----
a . . |  2 | 12  * | 2 0  a
. b . |  2 |  * 24 | 1 1  b
------+----+-------+----
a3b . |  6 |  3  3 | 8 *
. b4o |  4 |  0  4 | * 6

b3a3b   (same as general toe-variant a3b4o)

. . .    | 24 |  2  1 | 2 1
---------+----+-------+----
b . .  & |  2 | 24  * | 1 1  b
. a .    |  2 |  * 12 | 2 0  a
---------+----+-------+----
b3a .  & |  6 |  3  3 | 8 *
b . b    |  4 |  4  0 | * 6

a3o4c   (a ≠ 0, b = 0, c ≠ 0 : general sirco-variant)

. . . | 24 |  2  2 | 1  2 1
------+----+-------+-------
a . . |  2 | 24  * | 1  1 0  a
. . c |  2 |  * 24 | 0  1 1  c
------+----+-------+-------
a3o . |  3 |  3  0 | 8  * *
a . c |  4 |  2  2 | * 12 *
. o4c |  4 |  0  4 | *  * 6

Co3aa3oC&#zc   → height = 0,
                 where: C = cq = c sqrt(2) (pseudo),
                 same as general sirco-variant a3o4c)

o.3o.3o.     & | 24 |  2  2 | 1 1  2
---------------+----+-------+-------
.. a. ..     & |  2 | 24  * | 1 0  1  a
oo3oo3oo&#c    |  2 |  * 24 | 0 1  1  c
---------------+----+-------+-------
.. a.3o.     & |  3 |  3  0 | 8 *  *
Co .. oC&#zc   |  4 |  0  4 | * 6  *
.. aa ..&#c    |  4 |  2  2 | * * 12

o3b4c   (a = 0, b ≠ 0, c ≠ 0 : general tic-variant)

. . . | 24 |  2  1 | 1 2
------+----+-------+----
. b . |  2 | 24  * | 1 1  b
. . c |  2 |  * 12 | 0 2  c
------+----+-------+----
o3b . |  3 |  3  0 | 8 *
. b4c |  8 |  4  4 | * 6

Bb3oo3bB&#zc   → height = 0,
                 where: B = b+cq = b+c sqrt(2) (pseudo),
                 same as general tic-variant o3b4c)

o.3o.3o.     & | 24 |  2  1 | 1 2
---------------+----+-------+----
.. .. b.     & |  2 | 24  * | 1 1  b
oo3oo3oo&#c    |  2 |  * 12 | 0 2  c
---------------+----+-------+----
.. o.3b.     & |  3 |  3  0 | 8 *
Bb .. bB&#zc   |  8 |  4  4 | * 6

o3o4c   (a = b = 0, c ≠ 0 : c-scaled cube o3o4c)

. . . | 8 |  3 | 3
------+---+----+--
. . c | 2 | 12 | 2  c
------+---+----+--
. o4c | 4 |  4 | 6

Co3oo3oC&#zc   → height = 0,
                 where: C = cq = c sqrt(2) (pseudo),
                 same as c-scaled cube o3o4c)

o.3o.3o.     & | 8 |  3 | 3
---------------+---+----+--
oo3oo3oo&#c    | 2 | 12 | 2  c
---------------+---+----+--
Co .. oC&#zc   | 4 |  4 | 6

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