girco

Acronym girco

TOCID symbol tCO

Name great rhombicuboctahedron (but not querco),
truncated cuboctahedron,
omnitruncated octahedron,
omnitruncated cube

VRML

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Circumradius sqrt[13+6 sqrt(2)]/2 = 2.317611

Inradius
wrt. {4} (3+sqrt(2))/2 = 2.207107

Inradius
wrt. {6} sqrt[9+6 sqrt(2)]/2 = 2.090770

Inradius
wrt. {8} sqrt(2)+1/2 = 1.914214

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		`x3U .`	`W . x`	`. u4x`
6a		`w3u .`	`w . X`	`. x4x` opposite {8}
6b		`w3u .`	`W . x`	`. x4x` opposite {8}
7		`w3x .`	`U . w`
8		`x3x .` opposite {6}	`x . X`
9			`u . w`
10			`x . x` opposite{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

Volume 2[11+7 sqrt(2)] = 41.798990

Surface 12[2+sqrt(2)+sqrt(3)] = 61.755172

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

Face vector 48, 72, 26

Confer

decompositions:: sirco || girco
variations:: a3b4c x3x4q q3x4x w3x4x
general polytopal classes:: Wythoffian polyhedra partial Stott expansions
analogs:: omnitruncated hypercube otC_n

External
links

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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.

When looking more into classes of isogonal variants, then this polyhedron also could be addressed as a truncated cuboctahedron. However true truncation would not produce squares there. In fact, as shown in the near-right picture, it rather would produce x3t4q instead, where the relative size of t depends on the truncational depth in an inverse ratio.

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

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

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, s3s4s'

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

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