Acronym co   (alt.: ratit – old: rotet)
TOCID symbol CO, rTT
Name cuboctahedron,
rhombitetratetrahedron,
rectified cube,
rectified octahedron,
cantellated tetrahedron,
expanded tetrahedron,
trigonal gyrobicupola,
vertex figure of octet
 
 © ©
Circumradius 1
Edge radius sqrt(3)/2 = 0.866025
Inradius at {3}:   sqrt(2/3) = 0.816497
at {4}:   1/sqrt(2) = 0.707107
Vertex figure [(3,4)2] = x q
Snub derivation
Vertex layers
LayerSymmetrySubsymmetries
 o3o4oo3o .o . o. o4o
1o3x4oo3x .
{3} first
o . o
vertex first
. x4o
{4} first
2x3x .x . q
vertex figure
. o4q
3x3o .
opposite {3}
u . o. x4o
opposite {4}
4 x . q 
5o . o
opposite vertex
 o3o3oo3o .o . o. o3o
1x3o3xx3o .
{3} first
x . x
{4} first
. o3x
{3} first
2ax3x .q . o. x3x
2bo . q
3o3x .
opposite {3}
x . x
opposite {4}
. x3o
opposite {3}
Lace city
in approx. ASCII-art
  x  o 
x  u  x
 o  x  
o q o
q   q
o q o
Coordinates (1/sqrt(2), 1/sqrt(2), 0)   & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral members: cho   oho – other edge facetings)
Dual rad
Dihedral angles
  • between {3} and {4}:   arccos(-1/sqrt(3)) = 125.264390°
Confer
Grünbaumian relatives:
2co   2co+16{3}  
related Johnson solids:
tricu   tobcu   etigybcu  
variations:
a3b3c   q3o3x   f3o3x   v3o3x   u3o3x  
other axial symmetries:
oqo3coc&#xt   oqo5coc&#xt  
decompositions:
copy  
compounds:
arie  
unit-edged relatives:
pexco  
general polytopal classes:
partial Stott expansions   bistratic lace towers  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   Polyedergarten   quickfur

Note that co can be thought of as the external blend of 8 tets + 6 squippies. This decomposition is described as the degenerate segmentochoron oo3ox4oo&#xt.


Incidence matrix according to Dynkin symbol

o3x4o

. . . | 12 |  4 | 2 2
------+----+----+----
. x . |  2 | 24 | 1 1
------+----+----+----
o3x . |  3 |  3 | 8 *
. x4o |  4 |  4 | * 6

snubbed forms: o3β4o

o3/2x4o

.   . . | 12 |  4 | 2 2
--------+----+----+----
.   x . |  2 | 24 | 1 1
--------+----+----+----
o3/2x . |  3 |  3 | 8 *
.   x4o |  4 |  4 | * 6

o4/3x3o

.   . . | 12 |  4 | 2 2
--------+----+----+----
.   x . |  2 | 24 | 1 1
--------+----+----+----
o4/3x . |  4 |  4 | 6 *
.   x3o |  3 |  3 | * 8

o4/3x3/2o

.   .   . | 12 |  4 | 2 2
----------+----+----+----
.   x   . |  2 | 24 | 1 1
----------+----+----+----
o4/3x   . |  4 |  4 | 6 *
.   x3/2o |  3 |  3 | * 8

x3o3x

. . . | 12 |  2  2 | 1 2 1
------+----+-------+------
x . . |  2 | 12  * | 1 1 0
. . x |  2 |  * 12 | 0 1 1
------+----+-------+------
x3o . |  3 |  3  0 | 4 * *
x . x |  4 |  2  2 | * 6 *
. o3x |  3 |  0  3 | * * 4

snubbed forms: β3o3x, β3o3β

x3/2o3/2x

.   .   . | 12 |  2  2 | 1 2 1
----------+----+-------+------
x   .   . |  2 | 12  * | 1 1 0
.   .   x |  2 |  * 12 | 0 1 1
----------+----+-------+------
x3/2o   . |  3 |  3  0 | 4 * *
x   .   x |  4 |  2  2 | * 6 *
.   o3/2x |  3 |  0  3 | * * 4

s4x3o

demi( . . . ) | 12 |  2  2 | 2 1 1
--------------+----+-------+------
demi( . x . ) |  2 | 12  * | 1 1 0
sefa( s4x . ) |  2 |  * 12 | 1 0 1
--------------+----+-------+------
      s4x .     4 |  2  2 | 6 * *
demi( . x3o ) |  3 |  3  0 | * 4 *
sefa( s4x3o ) |  3 |  0  3 | * * 4

starting figure: x4x3o

xxo3oxx&#xt   → both heights = sqrt(2/3) = 0.816497
({3} || pseudo {6} || dual {3})

o..3o..    | 3 * * | 2 2 0 0 0 0 | 1 2 1 0 0 0
.o.3.o.    | * 6 * | 0 1 1 1 1 0 | 0 1 1 1 1 0
..o3..o    | * * 3 | 0 0 0 0 2 2 | 0 0 0 1 2 1
-----------+-------+-------------+------------
x.. ...    | 2 0 0 | 3 * * * * * | 1 1 0 0 0 0
oo.3oo.&#x | 1 1 0 | * 6 * * * * | 0 1 1 0 0 0
.x. ...    | 0 2 0 | * * 3 * * * | 0 1 0 1 0 0
... .x.    | 0 2 0 | * * * 3 * * | 0 0 1 0 1 0
.oo3.oo&#x | 0 1 1 | * * * * 6 * | 0 0 0 1 1 0
... ..x    | 0 0 2 | * * * * * 3 | 0 0 0 0 1 1
-----------+-------+-------------+------------
x..3o..    | 3 0 0 | 3 0 0 0 0 0 | 1 * * * * *
xx. ...&#x | 2 2 0 | 1 2 1 0 0 0 | * 3 * * * *
... ox.&#x | 1 2 0 | 0 2 0 1 0 0 | * * 3 * * *
.xo ...&#x | 0 2 1 | 0 0 1 0 2 0 | * * * 3 * *
... .xx&#x | 0 2 2 | 0 0 0 1 2 1 | * * * * 3 *
..o3..x    | 0 0 3 | 0 0 0 0 0 3 | * * * * * 1
or
o..3o..    & | 6 * | 2  2 0 | 1 2 1
.o.3.o.      | * 6 | 0  2 2 | 0 2 2
-------------+-----+--------+------
x.. ...    & | 2 0 | 6  * * | 1 1 0
oo.3oo.&#x & | 1 1 | * 12 * | 0 1 1
.x. ...    & | 0 2 | *  * 6 | 0 1 1
-------------+-----+--------+------
x..3o..    & | 3 0 | 3  0 0 | 2 * *
xx. ...&#x & | 2 2 | 1  2 1 | * 6 *
.xo ...&#x & | 1 2 | 0  2 1 | * * 6

xox4oqo&#xt   → both heights = 1/sqrt(2) = 0.707107
({4} || dual pseudo q-{4} || {4})

o..4o..     | 4 * * | 2 2 0 0 | 1 2 1 0 0
.o.4.o.     | * 4 * | 0 2 2 0 | 0 1 2 1 0
..o4..o     | * * 4 | 0 0 2 2 | 0 0 1 2 1
------------+-------+---------+----------
x.. ...     | 2 0 0 | 4 * * * | 1 1 0 0 0
oo.4oo.&#x  | 1 1 0 | * 8 * * | 0 1 1 0 0
.oo4.oo&#x  | 0 1 1 | * * 8 * | 0 0 1 1 0
..x ...     | 0 0 2 | * * * 4 | 0 0 0 1 1
------------+-------+---------+----------
x..4o..     | 4 0 0 | 4 0 0 0 | 1 * * * *
xo. ...&#x  | 2 1 0 | 1 2 0 0 | * 4 * * *
... oqo&#xt | 1 2 1 | 0 2 2 0 | * * 4 * *
.ox ...&#x  | 0 1 2 | 0 0 2 1 | * * * 4 *
..x4..o     | 0 0 4 | 0 0 0 4 | * * * * 1
or
o..4o..     & | 8 * | 2  2 | 1 2 1
.o.4.o.       | * 4 | 0  4 | 0 2 2
--------------+-----+------+------
x.. ...     & | 2 0 | 8  * | 1 1 0
oo.4oo.&#x  & | 1 1 | * 16 | 0 1 1
--------------+-----+------+------
x..4o..     & | 4 0 | 4  0 | 2 * *
xo. ...&#x  & | 2 1 | 1  2 | * 8 *
... oqo&#xt   | 2 2 | 0  4 | * * 4

oxuxo oqoqo&#xt   → all heights = 1/2
(pt || pseudo (x,q)-{4} || pseudo line || pseudo (x,q)-{4} || pt)

o.... o....     | 1 * * * * | 4 0 0 0 0 0 0 | 2 2 0 0 0 0
.o... .o...     | * 4 * * * | 1 1 1 1 0 0 0 | 1 1 1 1 0 0
..o.. ..o..     | * * 2 * * | 0 0 2 0 2 0 0 | 0 1 2 0 1 0
...o. ...o.     | * * * 4 * | 0 0 0 1 1 1 1 | 0 0 1 1 1 1
....o ....o     | * * * * 1 | 0 0 0 0 0 0 4 | 0 0 0 0 2 2
----------------+-----------+---------------+------------
oo... oo...&#x  | 1 1 0 0 0 | 4 * * * * * * | 1 1 0 0 0 0
.x... .....     | 0 2 0 0 0 | * 2 * * * * * | 1 0 0 1 0 0
.oo.. .oo..&#x  | 0 1 1 0 0 | * * 4 * * * * | 0 1 1 0 0 0
.o.o. .o.o.&#x  | 0 1 0 1 0 | * * * 4 * * * | 0 0 1 1 0 0
..oo. ..oo.&#x  | 0 0 1 1 0 | * * * * 4 * * | 0 0 1 0 1 0
...x. .....     | 0 0 0 2 0 | * * * * * 2 * | 0 0 0 1 0 1
...oo ...oo&#x  | 0 0 0 1 1 | * * * * * * 4 | 0 0 0 0 1 1
----------------+-----------+---------------+------------
ox... .....&#x  | 1 2 0 0 0 | 2 1 0 0 0 0 0 | 2 * * * * *
..... oqo..&#xt | 1 2 1 0 0 | 2 0 2 0 0 0 0 | * 2 * * * *
.ooo. .ooo.&#xt | 0 1 1 1 0 | 0 0 1 1 1 0 0 | * * 4 * * *
.x.x. .....&#x  | 0 2 0 2 0 | 0 1 0 2 0 1 0 | * * * 2 * *
..... ..oqo&#xt | 0 0 1 2 1 | 0 0 0 0 2 0 2 | * * * * 2 *
...xo .....&#x  | 0 0 0 2 1 | 0 0 0 0 0 1 2 | * * * * * 2
or
o.... o....      & | 2 * * | 4 0 0 0 | 2 2 0 0
.o... .o...      & | * 8 * | 1 1 1 1 | 1 1 1 1
..o.. ..o..        | * * 2 | 0 0 4 0 | 0 2 2 0
-------------------+-------+---------+--------
oo... oo...&#x   & | 1 1 0 | 8 * * * | 1 1 0 0
.x... .....      & | 0 2 0 | * 4 * * | 1 0 0 1
.oo.. .oo..&#x   & | 0 1 1 | * * 8 * | 0 1 1 0
.o.o. .o.o.&#x     | 0 2 0 | * * * 4 | 0 0 1 1
-------------------+-------+---------+--------
ox... .....&#x   & | 1 2 0 | 2 1 0 0 | 4 * * *
..... oqo..&#xt  & | 1 2 1 | 2 0 2 0 | * 4 * *
.ooo. .ooo.&#xt    | 0 2 1 | 0 0 2 1 | * * 4 *
.x.x. .....&#x     | 0 4 0 | 0 2 0 2 | * * * 2

qo xo4oq&#zx   → height = 0
(tegum sum of (q,x,x)-cube and gyro q-{4})

o. o.4o.     | 8 * | 2  2 | 1 2 1
.o .o4.o     | * 4 | 0  4 | 0 2 2
-------------+-----+------+------
.. x. ..     | 2 0 | 8  * | 1 1 0
oo oo4oo&#x  | 1 1 | * 16 | 0 1 1
-------------+-----+------+------
.. x.4o.     | 4 0 | 4  0 | 2 * *
.. xo ..&#x  | 2 1 | 1  2 | * 8 *
qo .. oq&#zx | 2 2 | 0  4 | * * 4

x(uo)x x(ou)x&#xt   → both heights = 1/sqrt(2) = 0.707107
({4} || compound of 2 mutual perp pseudo u-lines || {4})

o(..). o(..).     | 4 * * * | 1 1 1 1 0 0 0 0 | 1 1 1 1 0 0 0
.(o.). .(o.).     | * 2 * * | 0 0 2 0 2 0 0 0 | 0 1 0 2 1 0 0
.(.o). .(.o).     | * * 2 * | 0 0 0 2 0 2 0 0 | 0 0 1 2 0 1 0
.(..)o .(..)o     | * * * 4 | 0 0 0 0 1 1 1 1 | 0 0 0 1 1 1 1
------------------+---------+-----------------+--------------
x(..). .(..).     | 2 0 0 0 | 2 * * * * * * * | 1 0 1 0 0 0 0
.(..). x(..).     | 2 0 0 0 | * 2 * * * * * * | 1 1 0 0 0 0 0
o(o.). o(o.).&#x  | 1 1 0 0 | * * 4 * * * * * | 0 1 0 1 0 0 0
o(.o). o(.o).&#x  | 1 0 1 0 | * * * 4 * * * * | 0 0 1 1 0 0 0
.(o.)o .(o.)o&#x  | 0 1 0 1 | * * * * 4 * * * | 0 0 0 1 1 0 0
.(.o)o .(.o)o&#x  | 0 0 1 1 | * * * * * 4 * * | 0 0 0 1 0 1 0
.(..)x .(..).     | 0 0 0 2 | * * * * * * 2 * | 0 0 0 0 0 1 1
.(..). .(..)x     | 0 0 0 2 | * * * * * * * 2 | 0 0 0 0 1 0 1
------------------+---------+-----------------+--------------
x(..). x(..).     | 4 0 0 0 | 2 2 0 0 0 0 0 0 | 1 * * * * * *
.(..). x(o.).&#x  | 2 1 0 0 | 0 1 2 0 0 0 0 0 | * 2 * * * * *
x(.o). .(..).&#x  | 2 0 1 0 | 1 0 0 2 0 0 0 0 | * * 2 * * * *
o(oo)o o(oo)o&#xr | 1 1 1 1 | 0 0 1 1 1 1 0 0 | * * * 4 * * *
.(..). .(o.)x&#x  | 0 1 0 2 | 0 0 0 0 2 0 0 1 | * * * * 2 * *
.(.o)x .(..).&#x  | 0 0 1 2 | 0 0 0 0 0 2 1 0 | * * * * * 2 *
.(..)x .(..)x     | 0 0 0 4 | 0 0 0 0 0 0 2 2 | * * * * * * 1

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