Acronym sirco
TOCID symbol rCO
Name small rhombicuboctahedron
 
 © ©
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   WikiChoron   mathworld   Polyedergarten   quickfur

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

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


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

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