Acronym girco
TOCID symbol tCO
Name great rhombicuboctahedron (but not querco),
truncated cuboctahedron,
omnitruncated octahedron,
omnitruncated cube

` © ©`
Vertex figure [4,6,8]
Vertex layers
 Layer Symmetry Subsymmetries o3o4o o3o . o . o . o4o 1 x3x4x x3x .{6} first x . x{4} first . x4x{8} first 2 x3w . u . w . u4x 3 u3w . x . X . x4w 4 U3x . U . w . x4w 5a x3U . w . X . u4x 5b W . x 6a w3u . w . X . x4xopposite {8} 6b W . x 7 w3x . U . w 8 x3x .opposite {6} x . X 9 u . w 10 x . xopposite{4}
`(X=x+q+q, W=u+w, U=x+w)`
Lace city
in approx. ASCII-art
```  x w  w x
x   X  X   x
w X      X w

w X      X w
x   X  X   x
x w  w x
```
Coordinates ((1+2 sqrt(2))/2, (1+sqrt(2))/2, 1/2)   & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – no other uniform polyhedral members)
Dihedral angles
• between {4} and {6}:   arccos(-sqrt(2/3)) = 144.735610°
• between {4} and {8}:   135°
• between {6} and {8}:   arccos(-1/sqrt(3)) = 125.264390°
Dual m3m4m
Confer
decompositions:
sirco || girco
variations:
a3b4c   x3x4q   q3x4x   w3x4x
general polytopal classes:
partial Stott expansions
External

The naming great rhombicuboctahedron derives from the fact that it has faces in the face planes of a rhombidodecahedron, of an cube, and of a octahedron. As there are 2 such archimedean figures the additional qualifier great versus small is being applied. Note that the respective scalings of those 3 constituents has to be adopted accordingly each. The other naming truncated cuboctahedron derives from the right picture. Obviously any truncation applied to the cuboctahedron will never result in a uniform figure, rather in isogonal variants thereof only. Sure, an afterwards applied edge resizement could fix that problem.

As abstract polytope girco is isomorphic to quitco, thereby replacing octagons by octagrams.

Note that girco can be thought of as the external blend of 1 sirco + 8 tricues + 6 squacues + 12 cubes, cf. the Steward toroid K4 \ 8Q3(E4). This decomposition is also described as the degenerate segmentochoron xx3ox4xx&#xt.

Incidence matrix according to Dynkin symbol

```x3x4x

. . . | 48 |  1  1  1 | 1  1 1
------+----+----------+-------
x . . |  2 | 24  *  * | 1  1 0
. x . |  2 |  * 24  * | 1  0 1
. . x |  2 |  *  * 24 | 0  1 1
------+----+----------+-------
x3x . |  6 |  3  3  0 | 8  * *
x . x |  4 |  2  0  2 | * 12 *
. x4x |  8 |  0  4  4 | *  * 6

snubbed forms: β3x4x, x3β4x, x3x4s, s3s4x (or as mere faceting xwX wXx Xxw&#zh), β3x4β, x3β4β, s3s4s, β3β4β
```

```xxwwxx4xuxxux&#xt   → height(1,2) = height(2,3) = height(4,5) = height(5,6) = 1/sqrt(2) = 0.707107
height(3,4) = 1
({8} || pseudo (x,u)-{8} || pseudo (w,x)-{8} || pseudo (w,x)-{8} || pseudo (x,u)-{8} || {8})

o.....4o.....     | 8 * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 0 0
.o....4.o....     | * 8 * * * * | 0 0 1 1 1 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0 0 0
..o...4..o...     | * * 8 * * * | 0 0 0 0 1 1 1 0 0 0 0 0 0 | 0 0 1 1 1 0 0 0
...o..4...o..     | * * * 8 * * | 0 0 0 0 0 0 1 1 1 0 0 0 0 | 0 0 0 1 1 1 0 0
....o.4....o.     | * * * * 8 * | 0 0 0 0 0 0 0 0 1 1 1 0 0 | 0 0 0 1 0 1 1 0
.....o4.....o     | * * * * * 8 | 0 0 0 0 0 0 0 0 0 0 1 1 1 | 0 0 0 0 0 1 1 1
------------------+-------------+---------------------------+----------------
x..... ......     | 2 0 0 0 0 0 | 4 * * * * * * * * * * * * | 1 1 0 0 0 0 0 0
...... x.....     | 2 0 0 0 0 0 | * 4 * * * * * * * * * * * | 1 0 1 0 0 0 0 0
oo....4oo....&#x  | 1 1 0 0 0 0 | * * 8 * * * * * * * * * * | 0 1 1 0 0 0 0 0
.x.... ......     | 0 2 0 0 0 0 | * * * 4 * * * * * * * * * | 0 1 0 1 0 0 0 0
.oo...4.oo...&#x  | 0 1 1 0 0 0 | * * * * 8 * * * * * * * * | 0 0 1 1 0 0 0 0
...... ..x...     | 0 0 2 0 0 0 | * * * * * 4 * * * * * * * | 0 0 1 0 1 0 0 0
..oo..4..oo..&#x  | 0 0 1 1 0 0 | * * * * * * 8 * * * * * * | 0 0 0 1 1 0 0 0
...... ...x..     | 0 0 0 2 0 0 | * * * * * * * 4 * * * * * | 0 0 0 0 1 1 0 0
...oo.4...oo.&#x  | 0 0 0 1 1 0 | * * * * * * * * 8 * * * * | 0 0 0 1 0 1 0 0
....x. ......     | 0 0 0 0 2 0 | * * * * * * * * * 4 * * * | 0 0 0 1 0 0 1 0
....oo4....oo&#x  | 0 0 0 0 1 1 | * * * * * * * * * * 8 * * | 0 0 0 0 0 1 1 0
.....x ......     | 0 0 0 0 0 2 | * * * * * * * * * * * 4 * | 0 0 0 0 0 0 1 1
...... .....x     | 0 0 0 0 0 2 | * * * * * * * * * * * * 4 | 0 0 0 0 0 1 0 1
------------------+-------------+---------------------------+----------------
x.....4x.....     | 8 0 0 0 0 0 | 4 4 0 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * *
xx.... ......&#x  | 2 2 0 0 0 0 | 1 0 2 1 0 0 0 0 0 0 0 0 0 | * 4 * * * * * *
...... xux...&#xt | 2 2 2 0 0 0 | 0 1 2 0 2 1 0 0 0 0 0 0 0 | * * 4 * * * * *
.xwwx. ......&#xt | 0 2 2 2 2 0 | 0 0 0 1 2 0 2 0 2 1 0 0 0 | * * * 4 * * * *
...... ..xx..&#x  | 0 0 2 2 0 0 | 0 0 0 0 0 1 2 1 0 0 0 0 0 | * * * * 4 * * *
...... ...xux&#xt | 0 0 0 2 2 2 | 0 0 0 0 0 0 0 1 2 0 2 0 1 | * * * * * 4 * *
....xx ......&#x  | 0 0 0 0 2 2 | 0 0 0 0 0 0 0 0 0 1 2 1 0 | * * * * * * 4 *
.....x4.....x     | 0 0 0 0 0 8 | 0 0 0 0 0 0 0 0 0 0 0 4 4 | * * * * * * * 1
```
```or
o.....4o.....      & | 16  *  * | 1 1  1 0  0 0 0 | 1 1 1 0 0
.o....4.o....      & |  * 16  * | 0 0  1 1  1 0 0 | 0 1 1 1 0
..o...4..o...      & |  *  * 16 | 0 0  0 0  1 1 1 | 0 0 1 1 1
---------------------+----------+-----------------+----------
x..... ......      & |  2  0  0 | 8 *  * *  * * * | 1 1 0 0 0
...... x.....      & |  2  0  0 | * 8  * *  * * * | 1 0 1 0 0
oo....4oo....&#x   & |  1  1  0 | * * 16 *  * * * | 0 1 1 0 0
.x.... ......      & |  0  2  0 | * *  * 8  * * * | 0 1 0 1 0
.oo...4.oo...&#x   & |  0  1  1 | * *  * * 16 * * | 0 0 1 1 0
...... ..x...      & |  0  0  2 | * *  * *  * 8 * | 0 0 1 0 1
..oo..4..oo..&#x     |  0  0  2 | * *  * *  * * 8 | 0 0 0 1 1
---------------------+----------+-----------------+----------
x.....4x.....      & |  8  0  0 | 4 4  0 0  0 0 0 | 2 * * * *
xx.... ......&#x   & |  2  2  0 | 1 0  2 1  0 0 0 | * 8 * * *
...... xux...&#xt  & |  2  2  2 | 0 1  2 0  2 1 0 | * * 8 * *
.xwwx. ......&#xt    |  0  4  4 | 0 0  0 2  4 0 2 | * * * 4 *
...... ..xx..&#x     |  0  0  4 | 0 0  0 0  0 2 2 | * * * * 4
```

```wx3xx3xw&#zx   → height = 0
(tegum sum of 2 mutually inverse (w,x,x)-toes)

o.3o.3o.     | 24  * |  1  1  1  0  0 | 1 1  1 0
.o3.o3.o     |  * 24 |  0  0  1  1  1 | 0 1  1 1
-------------+-------+----------------+---------
.. x. ..     |  2  0 | 12  *  *  *  * | 1 0  1 0
.. .. x.     |  2  0 |  * 12  *  *  * | 1 1  0 0
oo3oo3oo&#x  |  1  1 |  *  * 24  *  * | 0 1  1 0
.x .. ..     |  0  2 |  *  *  * 12  * | 0 1  0 1
.. .x ..     |  0  2 |  *  *  *  * 12 | 0 0  1 1
-------------+-------+----------------+---------
.. x.3x.     |  6  0 |  3  3  0  0  0 | 4 *  * *
wx .. xw&#zx |  4  4 |  0  2  4  2  0 | * 6  * *
.. xx ..&#x  |  2  2 |  1  0  2  0  1 | * * 12 *
.x3.x ..     |  0  6 |  0  0  0  3  3 | * *  * 4
```
```or
o.3o.3o.     & | 48 |  1  1  1 | 1 1  1
---------------+----+----------+-------
.. x. ..     & |  2 | 24  *  * | 1 0  1
.. .. x.     & |  2 |  * 24  * | 1 1  0
oo3oo3oo&#x    |  2 |  *  * 24 | 0 1  1
---------------+----+----------+-------
.. x.3x.     & |  6 |  3  3  0 | 8 *  *
wx .. xw&#zx   |  8 |  0  4  4 | * 6  *
.. xx ..&#x    |  4 |  2  0  2 | * * 12
```

```xwX wxx4xux&#zxt   → height = 0, X=x+q+q = 3.828427

o.. o..4o..     | 16  *  * | 1 1  1 0  0 0 0 | 1 1 1 0 0
.o. .o.4.o.     |  * 16  * | 0 0  1 1  1 0 0 | 0 1 1 1 0
..o ..o4..o     |  *  * 16 | 0 0  0 0  1 1 1 | 0 0 1 1 1
----------------+----------+-----------------+----------
x.. ... ...     |  2  0  0 | 8 *  * *  * * * | 1 1 0 0 0
... ... x..     |  2  0  0 | * 8  * *  * * * | 1 0 1 0 0
oo. oo.4oo.&#x  |  1  1  0 | * * 16 *  * * * | 0 1 1 0 0
... .x. ...     |  0  2  0 | * *  * 8  * * * | 0 1 0 1 0
.oo .oo4.oo&#x  |  0  1  1 | * *  * * 16 * * | 0 0 1 1 0
... ..x ...     |  0  0  2 | * *  * *  * 8 * | 0 0 0 1 1
... ... ..x     |  0  0  2 | * *  * *  * * 8 | 0 0 1 0 1
----------------+----------+-----------------+----------
x.. ... x..     |  4  0  0 | 2 2  0 0  0 0 0 | 4 * * * *
xw. wx. ...&#zx |  4  4  0 | 2 0  4 2  0 0 0 | * 4 * * *
... ... xux&#xt |  2  2  2 | 0 1  2 0  2 0 1 | * * 8 * *
... .xx ...&#x  |  0  2  2 | 0 0  0 1  2 1 0 | * * * 8 *
... ..x4..x     |  0  0  8 | 0 0  0 0  0 4 4 | * * * * 2
```