Acronym oct (alt.: trap, tatet)
TOCID symbol O, TT, (3)Q
Name octahedron,
rectified tetrahedron,
tricross3),
tetratetrahedron,
aerochor(id),
trigonal antiprism,
larger Delone cell of face-centered cubic (fcc) lattice,
equatorial cross-section of (vertex first) 1/q-tes,
vertex figure of hex,
Gosset polytope 01,1,
lattice C3 contact polytope (span of its small roots)

` © ©    ©`
Vertex figure [34] = x4o
Snub derivation
Vertex layers
 Layer Symmetry Subsymmetries o3o4o o3o . o . o . o4o 1 x3o4o x3o .{3} first x . oedge first . o4overtex first 2 o3x .opposite {3} o . q . x4overtex figure 3 x . oopposite edge . o4oopposite vertex o3o3o o3o . o . o . o3o 1 o3x3o o3x .{3} first o . overtex first . x3o{3} first 2 x3o .opposite {3} x . xvertex figure . o3xopposite {3} 3 o . oopposite vertex
Lace city
in approx. ASCII-art
 ``` © ``` ``` x o o x ```
 ``` © ``` ``` o o q o o ```
Lace hyper city
in approx. ASCII-art
``` ©
```
 ``` o o o ``` ``` o o o ```
Coordinates (1/sqrt(2), 0, 0)   & all permutations, all changes of sign
Volume sqrt(2)/3 = 0.471405
Surface 2 sqrt(3) = 3.464102
Rel. Roundness π sqrt(3)/9 = 60.459979 %
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral member: thah – other edge facetings)
Dual cube
Dihedral angles
• between {3} and {3}:   arccos(-1/3) = 109.471221°
Confer
more general:
xPo3o...o3o4o
general antiprisms:
n-ap   n/2-ap   n/d-ap
special bipyramids:
m mNo
variations:
qo3oq&#x   xo3ox&#q   xo3ox&#h   ho3oh&#q
Grünbaumian relatives:
oct+6{4}   2oct   2oct+6{4}   2oct+8{3}   2oct+12{4}   4oct
uniform relative:
thah
related Johnson solids:
squippy
related scaliform:
bobipyr
compounds:
se   sno   daso   dissit   si   gissi   addasi   dasi
general polytopal classes:
deltahedra   regular   noble polytopes   orthoplex   partial Stott expansions   segmentohedra   fundamental lace prisms   bistratic lace towers   lace simplices   Coxeter-Elte-Gosset polytopes   Hanner polytopes
analogs:
rectified simplex rSn   mid-rectified simplex mrSn   regular orthoplex On   birectified hypercube brCn
External

The number of ways to color the octahedron with different colors per face is 8!/24 = 1 680. – This is because the color group is the permutation group of 8 elements and has size 8!, while the order of the pure rotational octahedral group is 24. (The reflectional octahedral group would have twice as many, i.e. 48 elements.)

When considered as a trigonal antiprism s s3s (and scaled by h = sqrt(3)), then the prismatic compound with its mirrored copy will have for hull metrically exact the hip-variant q x3x. Stated the other way round, the mere alternating faceting of that variant results in this (scaled) regular shape.

The acronym oct is being used for the full symmetrical (octahedral) variant, trap refers to the trigonal antiprismatic subsymmetry, and tatet refers to the tetratetrahedron, i.e. its tetrahedral subsymmetry.

Incidence matrix according to Dynkin symbol

```x3o4o

. . . | 6 |  4 | 4
------+---+----+--
x . . | 2 | 12 | 2
------+---+----+--
x3o . | 3 |  3 | 8

snubbed forms: β3o4o
```

```x3/2o4o

.   . . | 6 |  4 | 4
--------+---+----+--
x   . . | 2 | 12 | 2
--------+---+----+--
x3/2o . | 3 |  3 | 8

snubbed forms: β3/2o4o
```

```o4/3o3x

.   . . | 6 |  4 | 4
--------+---+----+--
.   . x | 2 | 12 | 2
--------+---+----+--
.   o3x | 3 |  3 | 8

snubbed forms: o4/3o3β
```

```o4/3o3/2x

.   .   . | 6 |  4 | 4
----------+---+----+--
.   .   x | 2 | 12 | 2
----------+---+----+--
.   o3/2x | 3 |  3 | 8

snubbed forms: o4/3o3/2β
```

```o3x3o

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

snubbed forms: o3β3o
```

```o3/2x3o

.   . . | 6 |  4 | 2 2
--------+---+----+----
.   x . | 2 | 12 | 1 1
--------+---+----+----
o3/2x . | 3 |  3 | 4 *
.   x3o | 3 |  3 | * 4

snubbed forms: o3/2β3o
```

```o3/2x3/2o

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

snubbed forms: o3/2β3/2o
```

```s2s3s

demi( . . .  ) | 6 | 1 1 2 | 1 3
---------------+---+-------+----
s2s .    | 2 | 3 * * | 0 2
s . s2*a | 2 | * 3 * | 0 2
sefa( . s3s  ) | 2 | * * 6 | 1 1
---------------+---+-------+----
. s3s    ♦ 3 | 0 0 3 | 2 *
sefa( s2s3s  ) | 3 | 1 1 1 | * 6
```
```or
demi( . . . )            | 6 | 2 2 | 1 3
-------------------------+---+-----+----
s2s .  &  s . s2*a | 2 | 6 * | 0 2
sefa( . s3s )            | 2 | * 6 | 1 1
-------------------------+---+-----+----
. s3s              ♦ 3 | 0 3 | 2 *
sefa( s2s3s )            | 3 | 2 1 | * 6

starting figure: x x3x
```

```s2s6o

demi( . . . ) | 6 | 2 2 | 1 3
--------------+---+-----+----
s2s .   | 2 | 6 * | 0 2
sefa( . s6o ) | 2 | * 6 | 1 1
--------------+---+-----+----
. s6o   ♦ 3 | 0 3 | 2 *
sefa( s2s6o ) | 3 | 2 1 | * 6

starting figure: x x6o
```

```xo3ox&#x   → height = sqrt(2/3) = 0.816497
({3} || dual {3})

o.3o.    | 3 * | 2 2 0 | 1 2 1 0
.o3.o    | * 3 | 0 2 2 | 0 1 2 1
---------+-----+-------+--------
x. ..    | 2 0 | 3 * * | 1 1 0 0
oo3oo&#x | 1 1 | * 6 * | 0 1 1 0
.. .x    | 0 2 | * * 3 | 0 0 1 1
---------+-----+-------+--------
x.3o.    | 3 0 | 3 0 0 | 1 * * *
xo ..&#x | 2 1 | 1 2 0 | * 3 * *
.. ox&#x | 1 2 | 0 2 1 | * * 3 *
.o3.x    | 0 3 | 0 0 3 | * * * 1
```

```oxo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo {4} || pt)

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

```oxo oxo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo {4} || pt)

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

```xox oqo&#xt   → both heights = 1/2
(line || perp pseudo q-line || line)

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

```oxox&#xr   → all cyclical heights = sqrt(3)/2 = 0.866025
in fact this lace simplex degenerates into a rhomb with diagonals:
height(1,3) = sqrt(2) = 1.414214
height(2,4) = 1

o...    | 1 * * * | 2 2 0 0 0 0 0 | 1 2 1 0 0 0
.o..    | * 2 * * | 1 0 1 1 1 0 0 | 1 1 0 1 1 0
..o.    | * * 1 * | 0 0 0 2 0 2 0 | 0 0 0 1 2 1
...o    | * * * 2 | 0 1 0 0 1 1 1 | 0 1 1 0 1 1
--------+---------+---------------+------------
oo..&#x | 1 1 0 0 | 2 * * * * * * | 1 1 0 0 0 0
o..o&#x | 1 0 0 1 | * 2 * * * * * | 0 1 1 0 0 0
.x..    | 0 2 0 0 | * * 1 * * * * | 1 0 0 1 0 0
.oo.&#x | 0 1 1 0 | * * * 2 * * * | 0 0 0 1 1 0
.o.o&#x | 0 1 0 1 | * * * * 2 * * | 0 1 0 0 1 0
..oo&#x | 0 0 1 1 | * * * * * 2 * | 0 0 0 0 1 1
...x    | 0 0 0 2 | * * * * * * 1 | 0 0 1 0 0 1
--------+---------+---------------+------------
ox..&#x | 1 2 0 0 | 2 0 1 0 0 0 0 | 1 * * * * *
oo.o&#x | 1 1 0 1 | 1 1 0 0 1 0 0 | * 2 * * * *
o..x&#x | 1 0 0 2 | 0 2 0 0 0 0 1 | * * 1 * * *
.xo.&#x | 0 2 1 0 | 0 0 1 2 0 0 0 | * * * 1 * *
.ooo&#x | 0 1 1 1 | 0 0 0 1 1 1 0 | * * * * 2 *
..ox&#x | 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * * 1
```

```qo ox4oo&#zx   → height = 0
(tegum sum of q-line and perp {4})
(tegum product of q-line with {4})

o. o.4o.    | 2 * | 4 0 | 4
.o .o4.o    | * 4 | 2 2 | 4
------------+-----+-----+--
oo oo4oo&#x | 1 1 | 8 * | 2
.. .x ..    | 0 2 | * 4 | 2
------------+-----+-----+--
.. ox ..&#x | 1 2 | 2 1 | 8
```

```qo ox ox&#zx   → height = 0
(tegum sum of q-line and perp {4})
(tegum product of q-line with {4})

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

```qoo oqo ooq&#zx   → all heights = 0
(tegum sum of 3 perp q-lines)
(tegum product of 3 q-lines)

o.. o.. o..    | 2 * * | 2 2 0 | 4
.o. .o. .o.    | * 2 * | 2 0 2 | 4
..o ..o ..o    | * * 2 | 0 2 2 | 4
---------------+-------+-------+--
oo. oo. oo.&#x | 1 1 0 | 4 * * | 2
o.o o.o o.o&#x | 1 0 1 | * 4 * | 2
.oo .oo .oo&#x | 0 1 1 | * * 4 | 2
---------------+-------+-------+--
ooo ooo ooo&#x | 1 1 1 | 1 1 1 | 8
```

```oooooo&#xr   → all consecutive pairwise heights = all alternating pairwise heights = 1
Note: these lengths show that this cycle is not flat, rather it is wobbling up and down!

o.....     & | 6 | 2 2 | 3 1
-------------+---+-----+----
oo....&#x  & | 2 | 6 * | 2 0
o.o...&#x  & | 2 | * 6 | 1 1
-------------+---+-----+----
ooo...&#x  & | 3 | 2 1 | 6 *
o.o.o.&#x  & | 3 | 0 3 | * 2
```

```oxxo&#xt   → height(1,2) = height(3,4) = 1/sqrt(12) = 0.288675
height(2,3) = 1/sqrt(3) = 0.577350
Note: these lengths show that this tower is not flat, rather it has additional leporello folds!

o...     & | 2 * | 2 2 0 0 | 1 1 2
.o..     & | * 4 | 1 1 1 1 | 1 1 2
-----------+-----+---------+------
oo..&#x  & | 1 1 | 4 * * * | 1 0 1
o.o.&#x  & | 1 1 | * 4 * * | 0 1 1
.x..     & | 0 2 | * * 2 * | 1 1 0
.oo.&#x    | 0 2 | * * * 2 | 0 0 2
-----------+-----+---------+------
ox..&#x  & | 1 2 | 2 0 1 0 | 2 * *
o.x.&#x  & | 1 2 | 0 2 1 0 | * 2 *
ooo.&#x  & | 1 2 | 1 1 0 1 | * * 4
```