Acronym	sirco (alt.: esquobcu)
TOCID symbol	rCO
Name	small rhombicuboctahedron, expanded octahedron, expanded cube, elongated square-orthobicupola
VRML	⭳ ©
Circumradius	sqrt[5+2 sqrt(2)]/2 = 1.398966
Inradius wrt. {3}	[3+sqrt(2)]/sqrt(12) = 1.274274
Inradius wrt. {4}	(1+sqrt(2))/2 = 1.207107
Vertex figure	[3,43] = xq&#q
Snub derivation / VRML	⭳
Vertex layers	Layer Symmetry Subsymmetries   o3o4o o3o . o . o . o4o 1 x3o4x x3o . {3} first x . x {4} first . o4x {4} first 2 x3q . o . w . x4x 3 w3o . w . x . x4x 4 o3w . q . w . o4x opposite {4} 5 q3x . w . x   6 o3x . opposite {3} o . w 7   x . x opposite {4}
Lace city in approx. ASCII-art	x x x w w x x w w x x x
Coordinates	((1+sqrt(2))/2, 1/2, 1/2)   & all permutations, all changes of sign
Volume	[12+10 sqrt(2)]/3 = 8.714045
Surface	18+2 sqrt(3) = 21.464102
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – other uniform polyhedral members: socco   sroh – other edge facetings)
Dual	sladid (old: sit)
Dihedral angles	between {3} and {4}:   arccos[-sqrt(2/3)] = 144.735610° between {4} and {4}:   135°
Face vector	24, 48, 26
Confer	related Johnson solids: squacu   escu   op   squobcu   esquigybcu   variations: a3b4c   q3o4x   f3o4x   u3o4x   x3o4q (reco)   xfF fFx Fxf&#zx   xfV fVx Vxf&#zq   xwX wXx Xxw&#zh   compounds: rasseri   unit-edged relatives: mono-lower-sirco   patex sirco   ambification: resirco   general polytopal classes: Wythoffian polyhedra   partial Stott expansions   analogs: rhombated hypercube rbCn   maximal expanded hypercube eCn
External links

As abstract polytope sirco is isomorphic to querco, thereby replacing prograde triangles by retrograde ones.

The right image shows where the rhombi part of its name comes from: in fact it also can be seen as a rectified version of the rhombi-dodecahedron.

When looking more into classes of isogonal variants, then this polyhedron also could be addressed as a rectified cuboctahedron. However true rectification would not produce squares there. In fact it rather would produce reco (x3o4q) instead.

This polyhedron allows for 2 square-prismatic edge facetings tepcus and topfu.

There also is a gyrated axial stack too: squacu + op + gyro squacu = esquigybcu (J37), one of the Johnson solids. In fact, both have the same vertex figure all over. But sirco features full cubical symmetry, while esquigybcu features 4-fold antiprismatic symmetry only, thereby dividing the according vertex set into 2 classes.

The snub derivational process features 2 operations, first the mere alternated faceting, here wrt. a to be alternated edge of girco as its starting figure, which then would result in xwX wXx Xxw&#zh (where X = x+2q = q+w), a pyritohedral gyrated sirco variant, and a secondary edge rescaling to all unit sizes only.

Incidence matrix according to Dynkin symbol

x3o4x

. . . | 24 |  2  2 | 1  2 1
------+----+-------+-------
x . . |  2 | 24  * | 1  1 0
. . x |  2 |  * 24 | 0  1 1
------+----+-------+-------
x3o . |  3 |  3  0 | 8  * *
x . x |  4 |  2  2 | * 12 *
. o4x |  4 |  0  4 | *  * 6

snubbed forms: β3o4x, x3o4s, x3o4s (as mere faceting), β3o4β

x4/3o3/2x

.   .   . | 24 |  2  2 | 1  2 1
----------+----+-------+-------
x   .   . |  2 | 24  * | 1  1 0
.   .   x |  2 |  * 24 | 0  1 1
----------+----+-------+-------
x4/3o   . |  4 |  4  0 | 6  * *
x   .   x |  4 |  2  2 | * 12 *
.   o3/2x |  3 |  0  3 | *  * 8

s3s4x

demi( . . . ) | 24 |  1  2  1 | 1 1  2
--------------+----+----------+-------
demi( . . x ) |  2 | 12  *  * | 0 1  1
sefa( s3s . ) |  2 |  * 24  * | 1 0  1
sefa( . s4x ) |  2 |  *  * 12 | 0 1  1
--------------+----+----------+-------
      s3s .   ♦  3 |  0  3  0 | 8 *  *
      . s4x   ♦  4 |  2  0  2 | * 6  *
sefa( s3s4x ) |  4 |  1  2  1 | * * 12

starting figure: x3x4x

xxxx4oxxo&#xt   → outer heights = 1/sqrt(2) = 0.707107
                  inner height  = 1
({4} || pseudo {8} || pseudo {8} || {4})

o...4o...    | 4 * * * | 2 2 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0
.o..4.o..    | * 8 * * | 0 1 1 1 1 0 0 0 0 | 0 1 1 1 1 0 0 0
..o.4..o.    | * * 8 * | 0 0 0 0 1 1 1 1 0 | 0 0 0 1 1 1 1 0
...o4...o    | * * * 4 | 0 0 0 0 0 0 0 2 2 | 0 0 0 0 0 2 1 1
-------------+---------+-------------------+----------------
x... ....    | 2 0 0 0 | 4 * * * * * * * * | 1 1 0 0 0 0 0 0
oo..4oo..&#x | 1 1 0 0 | * 8 * * * * * * * | 0 1 1 0 0 0 0 0
.x.. ....    | 0 2 0 0 | * * 4 * * * * * * | 0 1 0 1 0 0 0 0
.... .x..    | 0 2 0 0 | * * * 4 * * * * * | 0 0 1 0 1 0 0 0
.oo.4.oo.&#x | 0 1 1 0 | * * * * 8 * * * * | 0 0 0 1 1 0 0 0
..x. ....    | 0 0 2 0 | * * * * * 4 * * * | 0 0 0 1 0 1 0 0
.... ..x.    | 0 0 2 0 | * * * * * * 4 * * | 0 0 0 0 1 0 1 0
..oo4..oo&#x | 0 0 1 1 | * * * * * * * 8 * | 0 0 0 0 0 1 1 0
...x ....    | 0 0 0 2 | * * * * * * * * 4 | 0 0 0 0 0 1 0 1
-------------+---------+-------------------+----------------
x...4o...    | 4 0 0 0 | 4 0 0 0 0 0 0 0 0 | 1 * * * * * * *
xx.. ....&#x | 2 2 0 0 | 1 2 1 0 0 0 0 0 0 | * 4 * * * * * *
.... ox..&#x | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | * * 4 * * * * *
.xx. ....&#x | 0 2 2 0 | 0 0 1 0 2 1 0 0 0 | * * * 4 * * * *
.... .xx.&#x | 0 2 2 0 | 0 0 0 1 2 0 1 0 0 | * * * * 4 * * *
..xx ....&#x | 0 0 2 2 | 0 0 0 0 0 1 0 2 1 | * * * * * 4 * *
.... ..xo&#x | 0 0 2 1 | 0 0 0 0 0 0 1 2 0 | * * * * * * 4 *
...x4...o    | 0 0 0 4 | 0 0 0 0 0 0 0 0 4 | * * * * * * * 1

or
o...4o...    & | 8  * | 2  2 0 0 0 | 1 2 1 0 0
.o..4.o..    & | * 16 | 0  1 1 1 1 | 0 1 1 1 1
---------------+------+------------+----------
x... ....    & | 2  0 | 8  * * * * | 1 1 0 0 0
oo..4oo..&#x & | 1  1 | * 16 * * * | 0 1 1 0 0
.x.. ....    & | 0  2 | *  * 8 * * | 0 1 0 1 0
.... .x..    & | 0  2 | *  * * 8 * | 0 0 1 0 1
.oo.4.oo.&#x & | 0  2 | *  * * * 8 | 0 0 0 1 1
---------------+------+------------+----------
x...4o...    & | 4  0 | 4  0 0 0 0 | 2 * * * *
xx.. ....&#x & | 2  2 | 1  2 1 0 0 | * 8 * * *
.... ox..&#x & | 1  2 | 0  2 0 1 0 | * * 8 * *
.xx. ....&#x   | 0  4 | 0  0 2 0 2 | * * * 4 *
.... .xx.&#x   | 0  4 | 0  0 0 2 2 | * * * * 4

xxwoqo3oqowxx&#xt   → height(1,2) = height(2,3) = height(4,5) = height(5,6) = 1/sqrt(3) = 0.577350
                      height(3,4) = [sqrt(2)-1]/sqrt(3) = 0.239146

o.....3o.....     | 3 * * * * * | 2 2 0 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0 0
.o....3.o....     | * 6 * * * * | 0 1 1 1 1 0 0 0 0 0 | 0 1 1 1 1 0 0 0 0
..o...3..o...     | * * 3 * * * | 0 0 0 2 0 2 0 0 0 0 | 0 0 1 0 2 1 0 0 0
...o..3...o..     | * * * 3 * * | 0 0 0 0 2 0 2 0 0 0 | 0 0 0 1 2 0 1 0 0
....o.3....o.     | * * * * 6 * | 0 0 0 0 0 1 1 1 1 0 | 0 0 0 0 1 1 1 1 0
.....o3.....o     | * * * * * 3 | 0 0 0 0 0 0 0 0 2 2 | 0 0 0 0 0 0 1 2 1
------------------+-------------+---------------------+------------------
x..... ......     | 2 0 0 0 0 0 | 3 * * * * * * * * * | 1 1 0 0 0 0 0 0 0
oo....3oo....&#x  | 1 1 0 0 0 0 | * 6 * * * * * * * * | 0 1 1 0 0 0 0 0 0
.x.... ......     | 0 2 0 0 0 0 | * * 3 * * * * * * * | 0 1 0 1 0 0 0 0 0
.oo...3.oo...&#x  | 0 1 1 0 0 0 | * * * 6 * * * * * * | 0 0 1 0 1 0 0 0 0
.o.o..3.o.o..&#x  | 0 1 0 1 0 0 | * * * * 6 * * * * * | 0 0 0 1 1 0 0 0 0
..o.o.3..o.o.&#x  | 0 0 1 0 1 0 | * * * * * 6 * * * * | 0 0 0 0 1 1 0 0 0
...oo.3...oo.&#x  | 0 0 0 1 1 0 | * * * * * * 6 * * * | 0 0 0 0 1 0 1 0 0
...... ....x.     | 0 0 0 0 2 0 | * * * * * * * 3 * * | 0 0 0 0 0 1 0 1 0
....oo3....oo&#x  | 0 0 0 0 1 1 | * * * * * * * * 6 * | 0 0 0 0 0 0 1 1 0
...... .....x     | 0 0 0 0 0 2 | * * * * * * * * * 3 | 0 0 0 0 0 0 0 1 1
------------------+-------------+---------------------+------------------
x.....3o.....     | 3 0 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * *
xx.... ......&#x  | 2 2 0 0 0 0 | 1 2 1 0 0 0 0 0 0 0 | * 3 * * * * * * *
...... oqo...&#xt | 1 2 1 0 0 0 | 0 2 0 2 0 0 0 0 0 0 | * * 3 * * * * * *
.x.o.. ......&#x  | 0 2 0 1 0 0 | 0 0 1 0 2 0 0 0 0 0 | * * * 3 * * * * *
.oooo.3.oooo.&#xr | 0 1 1 1 1 0 | 0 0 0 1 1 1 1 0 0 0 | * * * * 6 * * * *
...... ..o.x.&#x  | 0 0 1 0 2 0 | 0 0 0 0 0 2 0 1 0 0 | * * * * * 3 * * *
...oqo ......&#xt | 0 0 0 1 2 1 | 0 0 0 0 0 0 2 0 2 0 | * * * * * * 3 * *
...... ....xx&#x  | 0 0 0 0 2 2 | 0 0 0 0 0 0 0 1 2 1 | * * * * * * * 3 *
.....o3.....x     | 0 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * 1

or
o.....3o.....     & | 6  * * | 2  2 0  0  0 | 1 2 1 0 0
.o....3.o....     & | * 12 * | 0  1 1  1  1 | 0 1 1 1 1
..o...3..o...     & | *  * 6 | 0  0 0  2  2 | 0 0 1 1 2
--------------------+--------+--------------+----------
x..... ......     & | 2  0 0 | 6  * *  *  * | 1 1 0 0 0
oo....3oo....&#x  & | 1  1 0 | * 12 *  *  * | 0 1 1 0 0
.x.... ......     & | 0  2 0 | *  * 6  *  * | 0 1 0 1 0
.oo...3.oo...&#x  & | 0  1 1 | *  * * 12  * | 0 0 1 0 1
.o.o..3.o.o..&#x  & | 0  1 1 | *  * *  * 12 | 0 0 0 1 1
--------------------+--------+--------------+----------
x.....3o.....     & | 3  0 0 | 3  0 0  0  0 | 2 * * * *
xx.... ......&#x  & | 2  2 0 | 1  2 1  0  0 | * 6 * * *
...... oqo...&#xt & | 1  2 1 | 0  2 0  2  0 | * * 6 * *
.x.o.. ......&#x  & | 0  2 1 | 0  0 1  0  2 | * * * 6 *
.oooo.3.oooo.&#xr   | 0  2 2 | 0  0 0  2  2 | * * * * 6

qo3xx3oq&#zx   → height = 0
(tegum sum of 2 mutually inverted (q,x)-tuts)

o.3o.3o.     | 12  * |  2  2  0 | 1 1  2 0
.o3.o3.o     |  * 12 |  0  2  2 | 0 1  2 1
-------------+-------+----------+---------
.. x. ..     |  2  0 | 12  *  * | 1 0  1 0
oo3oo3oo&#x  |  1  1 |  * 24  * | 0 1  1 0
.. .x ..     |  0  2 |  *  * 12 | 0 0  1 1
-------------+-------+----------+---------
.. x.3o.     |  3  0 |  3  0  0 | 4 *  * *
qo .. oq&#zx |  2  2 |  0  4  0 | * 6  * *
.. xx ..&#x  |  2  2 |  1  2  1 | * * 12 *
.o3.x ..     |  0  3 |  0  0  3 | * *  * 4

or
o.3o.3o.     & | 24 |  2  2 | 1 1  2
---------------+----+-------+-------
.. x. ..     & |  2 | 24  * | 1 0  1
oo3oo3oo&#x    |  2 |  * 24 | 0 1  1
---------------+----+-------+-------
.. x.3o.     & |  3 |  3  0 | 8 *  *
qo .. oq&#zx   |  4 |  0  4 | * 6  *
.. xx ..&#x    |  4 |  2  2 | * * 12

wx xx4ox&#zx   → height = 0
(tegum sum of (w,x,x)-cube and ortho op)

o. o.4o.    | 8  * | 2  2 0 0 0 | 1 2 1 0 0
.o .o4.o    | * 16 | 0  1 1 1 1 | 0 1 1 1 1
------------+------+------------+----------
.. x. ..    | 2  0 | 8  * * * * | 1 1 0 0 0
oo oo4oo&#x | 1  1 | * 16 * * * | 0 1 1 0 0
.x .. ..    | 0  2 | *  * 8 * * | 0 0 0 1 1
.. .x ..    | 0  2 | *  * * 8 * | 0 1 0 1 0
.. .. .x    | 0  2 | *  * * * 8 | 0 0 1 0 1
------------+------+------------+----------
.. x.4o.    | 4  0 | 4  0 0 0 0 | 2 * * * *
.. xx ..&#x | 2  2 | 1  2 0 1 0 | * 8 * * *
.. .. ox&#x | 1  2 | 0  2 0 0 1 | * * 8 * *
.x .x ..    | 0  4 | 0  0 2 2 0 | * * * 4 *
.x .. .x    | 0  4 | 0  0 2 0 2 | * * * * 4

wxx2xwx2xxw&#zx   → all heights = 0
(tegum sum of 3 mutually orthogonal (w,x,x)-cubes)

o..2o..2o..    | 8 * * | 1 1 1 1 0 0 0 0 0 | 1 1 1 1 0 0 0
.o.2.o.2.o.    | * 8 * | 0 0 1 0 1 1 1 0 0 | 0 1 0 1 1 1 0
..o2..o2..o    | * * 8 | 0 0 0 1 0 0 1 1 1 | 0 0 1 1 0 1 1
---------------+-------+-------------------+--------------
... x.. ...    | 2 0 0 | 4 * * * * * * * * | 1 0 1 0 0 0 0
... ... x..    | 2 0 0 | * 4 * * * * * * * | 1 1 0 0 0 0 0
oo.2oo.2oo.&#x | 1 1 0 | * * 8 * * * * * * | 0 1 0 1 0 0 0
o.o2o.o2o.o&#x | 1 0 1 | * * * 8 * * * * * | 0 0 1 1 0 0 0
.x. ... ...    | 0 2 0 | * * * * 4 * * * * | 0 0 0 0 1 1 0
... ... .x.    | 0 2 0 | * * * * * 4 * * * | 0 1 0 0 1 0 0
.oo2.oo2.oo&#x | 0 1 1 | * * * * * * 8 * * | 0 0 0 1 0 1 0
..x ... ...    | 0 0 2 | * * * * * * * 4 * | 0 0 0 0 0 1 1
... ..x ...    | 0 0 2 | * * * * * * * * 4 | 0 0 1 0 0 0 1
---------------+-------+-------------------+--------------
... x..2x..    | 4 0 0 | 2 2 0 0 0 0 0 0 0 | 2 * * * * * *
... ... xx.&#x | 2 2 0 | 0 1 2 0 0 1 0 0 0 | * 4 * * * * *
... x.x ...&#x | 2 0 2 | 1 0 0 2 0 0 0 0 1 | * * 4 * * * *
ooo2ooo2ooo&#x | 1 1 1 | 0 0 1 1 0 0 1 0 0 | * * * 8 * * *
.x. ... .x.    | 0 4 0 | 0 0 0 0 2 2 0 0 0 | * * * * 2 * *
.xx ... ...&#x | 0 2 2 | 0 0 0 0 1 0 2 1 0 | * * * * * 4 *
..x2..x ...    | 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * 2