Acronym co   (alt.: ratet – old: ratit, rotet)
TOCID symbol CO, rTT
Name cuboctahedron,
(small) 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)

` © ©`
wrt. {3}
sqrt(2/3) = 0.816497
wrt. {4}
1/sqrt(2) = 0.707107
Vertex figure [(3,4)2] = x q
Snub derivation
Vertex layers
 Layer Symmetry Subsymmetries o3o4o o3o . o . o . o4o 1 o3x4o o3x .{3} first o . overtex first . x4o{4} first 2 x3x . x . qvertex figure . o4q 3 x3o .opposite {3} u . o . x4oopposite {4} 4 x . q 5 o . oopposite vertex o3o3o o3o . o . o . o3o 1 x3o3x x3o .{3} first x . x{4} first . o3x{3} first 2a x3x . q . o . x3x 2b o . q 3 o3x .opposite {3} x . xopposite {4} . x3oopposite {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
Volume 5 sqrt(2)/3 = 2.357023
Surface 6+2 sqrt(3) = 9.464102
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral members: cho   oho – other edge facetings)
Dihedral angles
• between {3} and {4}:   arccos[-1/sqrt(3)] = 125.264390°
Confer
more general:
xPo3o...o3oPxQ*a
Grünbaumian relatives:
2co   2co+16{3}
related Johnson solids:
tricu   tobcu   etigybcu
further diminishings:
xo3ox&#h
variations:
a3b3c   q3o3x   f3o3x   v3o3x   u3o3x   d3o3x
other axial symmetries:
retrip   oqo5coc&#xt
decompositions:
copy
compounds:
arie
unit-edged relatives:
pexco
general polytopal classes:
partial Stott expansions   bistratic lace towers   lace simplices
analogs:
maximal epanded simplex eSn   rectified orthoplex rOn   rectified hypercube rCn
External

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.

The acronym co is being used for the full octahedral symmetrical variant, while ratet refers to the rhombitetratetrahedron, i.e. its tetrahedral subsymmetry.

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
```

```uxo oxu oqo&#zx   → height = 0
(tegum sum of 2 perp u-lines and lacing-para (x,x,q)-cube)

o.. o.. o..     & | 4 * |  4 0 | 2 2 0
.o. .o. .o.       | * 8 |  2 2 | 2 1 1
------------------+-----+------+------
oo. oo. oo.&#x  & | 1 1 | 16 * | 1 1 0
.x. ... ...     & | 0 2 |  * 8 | 1 0 1
------------------+-----+------+------
... ox. ...&#x  & | 1 2 |  2 1 | 8 * *
... ... oqo&#xt   | 2 2 |  4 0 | * 4 *
.x. .x. ...       | 0 4 |  0 4 | * * 2
```

```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
```