Acronym sirco
TOCID symbol rCO
Name small rhombicuboctahedron,
expanded octahedron,
expanded cube
 
 © ©    ©
Circumradius sqrt[5+2 sqrt(2)]/2 = 1.398966
Vertex figure [3,43] = xq&#q
Snub derivation
Vertex layers
LayerSymmetrySubsymmetries
 o3o4oo3o .o . o. o4o
1x3o4xx3o .
{3} first
x . x
{4} first
. o4x
{4} first
2x3q .o . w. x4x
3w3o .w . x. x4x
4o3w .q . w. o4x
opposite {4}
5q3x .w . x 
6o3x .
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
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral members: socco   sroh – other edge facetings)
Dual sit
Dihedral angles
  • between {3} and {4}:   arccos(-sqrt(2/3)) = 144.735610°
  • between {4} and {4}:   135°
Confer
related Johnson solids:
squacu   escu   op   squobcu   esquigybcu  
variations:
a3b4c   q3o4x   f3o4x   u3o4x   x3o4q  
compounds:
rasseri  
unit-edged relatives:
mono-lower-sirco   patex sirco  
general polytopal classes:
partial Stott expansions  
External
links
hedrondude   wikipedia   polytopewiki   WikiChoron   mathworld   Polyedergarten   quickfur

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.

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.


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, 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

© 2004-2020
top of page