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Acronym
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co (alt.: ratet – old: ratit, rotet)
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TOCID symbol
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CO, rTT
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Name
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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),
equatorial cross-section of tet-first spid,
equatorial cross-section of oct-first ico
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 © ©
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Circumradius
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1
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Edge radius
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sqrt(3)/2 = 0.866025
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Inradius
wrt. {3}
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sqrt(2/3) = 0.816497
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Inradius
wrt. {4}
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1/sqrt(2) = 0.707107
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Vertex figure
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[(3,4)2] = x q
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Snub derivation
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Vertex layers
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| o3x4o | o3x .
{3} first | o . o
vertex first | . x4o
{4} first |
| x3x . | x . q
vertex figure | . o4q |
x3o .
opposite {3} | u . o | . x4o
opposite {4} |
| | x . q | |
o . o
opposite vertex |
| x3o3x | x3o .
{3} first | x . x
{4} first | . o3x
{3} first |
| x3x . | q . o | . x3x |
| o . q |
o3x .
opposite {3} | x . x
opposite {4} | . x3o
opposite {3} |
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Lace city
in approx. ASCII-art
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x o
x u x
o x
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o q o
q q
o q o
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Coordinates
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(1/sqrt(2), 1/sqrt(2), 0) & all permutations, all changes of sign
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Volume
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5 sqrt(2)/3 = 2.357023
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Surface
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6+2 sqrt(3) = 9.464102
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General of army
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(is itself convex)
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Colonel of regiment
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(is itself locally convex
– other uniform polyhedral members:
cho
oho
– other edge facetings)
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Dual
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rad
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Dihedral angles
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- between {3} and {4}: arccos[-1/sqrt(3)] = 125.264390°
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Face vector
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12, 24, 14
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Confer
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- more general:
-
xPo3o...o3oPxQ*a
oqo-n/d-coc&#xt
- Grünbaumian relatives:
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2co
2co+16{3}
- related Johnson solids:
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tricu
tobcu
etigybcu
- further diminishings:
-
xo3ox&#h
- variations:
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a3b3c
q3o3x
f3o3x
v3o3x
u3o3x
d3o3x
- other axial symmetries:
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retrip
oqo5coc&#xt
- decompositions:
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copy
- compounds:
-
arie
- unit-edged relatives:
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pexco
- ambification:
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reco
- ambification pre-image:
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cube
oct
- general polytopal classes:
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Wythoffian polyhedra
partial Stott expansions
bistratic lace towers
lace simplices
- analogs:
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maximal epanded simplex eSn
rectified orthoplex rOn
rectified hypercube rCn
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External
links
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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