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,
lattice A3 contact polytope (span of its roots),
lattice B3 contact polytope (span of its small roots)
 
 © ©
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   lace simplices  
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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