Acronym troct (alt.: octoct octtric) Name triangle-octahedron duoprism,vertex figure of brag,(oct,oct,oct)-trigonic Circumradius sqrt(5/6) = 0.912871 Lace cityin approx. ASCII-art ```O O O = o3x3o (oct) O ``` Volume sqrt(6)/12 = 0.204124 Confer more general: n,oct-dip   related segmentotera: tisquippy   general polytopal classes: segmentotera   lace simplices Externallinks

Incidence matrix according to Dynkin symbol

```x3o x3o4o

. . . . . | 18 |  2  4 | 1  8  4 |  4  8 1 | 4 2
----------+----+-------+---------+---------+----
x . . . . |  2 | 18  * | 1  4  0 |  4  4 0 | 4 1
. . x . . |  2 |  * 36 | 0  2  2 |  1  4 1 | 2 2
----------+----+-------+---------+---------+----
x3o . . . |  3 |  3  0 | 6  *  * |  4  0 0 | 4 0
x . x . . |  4 |  2  2 | * 36  * |  1  2 0 | 2 1
. . x3o . |  3 |  0  3 | *  * 24 |  0  2 1 | 1 2
----------+----+-------+---------+---------+----
x3o x . . ♦  6 |  6  3 | 2  3  0 | 12  * * | 2 0
x . x3o . ♦  6 |  3  6 | 0  3  2 |  * 24 * | 1 1
. . x3o4o ♦  6 |  0 12 | 0  0  8 |  *  * 3 | 0 2
----------+----+-------+---------+---------+----
x3o x3o . ♦  9 |  9  9 | 3  9  3 |  3  3 0 | 8 *
x . x3o4o ♦ 12 |  6 24 | 0 12 16 |  0  8 2 | * 3
```

```x3o o3x3o

. . . . . | 18 |  2  4 | 1  8  2  2 |  4  4  4 1 | 2 2 2
----------+----+-------+------------+------------+------
x . . . . |  2 | 18  * | 1  4  0  0 |  4  2  2 0 | 2 2 1
. . . x . |  2 |  * 36 | 0  2  1  1 |  1  2  2 1 | 1 1 2
----------+----+-------+------------+------------+------
x3o . . . |  3 |  3  0 | 6  *  *  * |  4  0  0 0 | 2 2 0
x . . x . |  4 |  2  2 | * 36  *  * |  1  1  1 0 | 1 1 1
. . o3x . |  3 |  0  3 | *  * 12  * |  0  2  0 1 | 1 0 2
. . . x3o |  3 |  0  3 | *  *  * 12 |  0  0  2 1 | 0 1 2
----------+----+-------+------------+------------+------
x3o . x . ♦  6 |  6  3 | 2  3  0  0 | 12  *  * * | 1 1 0
x . o3x . ♦  6 |  3  6 | 0  3  2  0 |  * 12  * * | 1 0 1
x . . x3o ♦  6 |  3  6 | 0  3  0  2 |  *  * 12 * | 0 1 1
. . o3x3o ♦  6 |  0 12 | 0  0  4  4 |  *  *  * 3 | 0 0 2
----------+----+-------+------------+------------+------
x3o o3x . ♦  9 |  9  9 | 3  9  3  0 |  3  3  0 0 | 4 * *
x3o . x3o ♦  9 |  9  9 | 3  9  0  3 |  3  0  3 0 | * 4 *
x . o3x3o ♦ 12 |  6 24 | 0 12  8  8 |  0  4  4 2 | * * 3
```

```ox xx3oo4oo&#x   → height = sqrt(3)/2 = 0.866025
(oct || ope)

...
```

```ox oo3xx3oo&#x   → height = sqrt(3)/2 = 0.866025
(oct || ope)

o. o.3o.3o.    | 6  * |  4  2 0  0 | 2 2 1  8  0 0 0 | 1  4 4 4 0 0 0 | 2 2 2 0
.o .o3.o3.o    | * 12 |  0  1 1  4 | 0 0 1  4  4 2 2 | 0  4 2 2 2 2 1 | 2 2 1 1
---------------+------+------------+-----------------+----------------+--------
.. .. x. ..    | 2  0 | 12  * *  * | 1 1 0  2  0 0 0 | 1  1 2 2 0 0 0 | 1 1 2 0
oo oo3oo3oo&#x | 1  1 |  * 12 *  * | 0 0 1  4  0 0 0 | 0  4 2 2 0 0 0 | 2 2 1 0
.x .. .. ..    | 0  2 |  *  * 6  * | 0 0 1  0  4 0 0 | 0  4 0 0 2 2 0 | 2 2 0 1
.. .. .x ..    | 0  2 |  *  * * 24 | 0 0 0  1  1 1 1 | 0  1 1 1 1 1 1 | 1 1 1 1
---------------+------+------------+-----------------+----------------+--------
.. o.3x. ..    | 3  0 |  3  0 0  0 | 4 * *  *  * * * | 1  0 2 0 0 0 0 | 1 0 2 0
.. .. x.3o.    | 3  0 |  3  0 0  0 | * 4 *  *  * * * | 1  0 0 2 0 0 0 | 0 1 2 0
ox .. .. ..&#x | 1  2 |  0  2 1  0 | * * 6  *  * * * | 0  4 0 0 0 0 0 | 2 2 0 0
.. .. xx ..&#x | 2  2 |  1  2 0  1 | * * * 24  * * * | 0  1 1 1 0 0 0 | 1 1 1 0
.x .. .x ..    | 0  4 |  0  0 2  2 | * * *  * 12 * * | 0  1 0 0 1 1 0 | 1 1 0 1
.. .o3.x ..    | 0  3 |  0  0 0  3 | * * *  *  * 8 * | 0  0 1 0 1 0 1 | 1 0 1 1
.. .. .x3.o    | 0  3 |  0  0 0  3 | * * *  *  * * 8 | 0  0 0 1 0 1 1 | 0 1 1 1
---------------+------+------------+-----------------+----------------+--------
.. o.3x.3o.    ♦ 6  0 | 12  0 0  0 | 4 4 0  0  0 0 0 | 1  * * * * * * | 0 0 2 0
ox .. xx ..&#x ♦ 2  4 |  1  4 2  2 | 0 0 2  2  1 0 0 | * 12 * * * * * | 1 1 0 0
.. oo3xx ..&#x ♦ 3  3 |  3  3 0  3 | 1 0 0  3  0 1 0 | *  * 8 * * * * | 1 0 1 0
.. .. xx3oo&#x ♦ 3  3 |  3  3 0  3 | 0 1 0  3  0 0 1 | *  * * 8 * * * | 0 1 1 0
.x .o3.x ..    ♦ 0  6 |  0  0 3  6 | 0 0 0  0  3 2 0 | *  * * * 4 * * | 1 0 0 1
.x .. .x3.o    ♦ 0  6 |  0  0 3  6 | 0 0 0  0  3 0 2 | *  * * * * 4 * | 0 1 0 1
.. .o3.x3.o    ♦ 0  6 |  0  0 0 12 | 0 0 0  0  0 4 4 | *  * * * * * 2 | 0 0 1 1
---------------+------+------------+-----------------+----------------+--------
ox oo3xx ..&#x ♦ 3  6 |  3  6 3  6 | 1 0 3  6  3 2 0 | 0  3 2 0 1 0 0 | 4 * * *
ox .. xx3oo&#x ♦ 3  6 |  3  6 3  6 | 0 1 3  6  3 0 2 | 0  3 0 2 0 1 0 | * 4 * *
.. oo3xx3oo&#x ♦ 6  6 | 12  6 0 12 | 4 4 0 12  0 4 4 | 1  0 4 4 0 0 1 | * * 2 *
.x .o3.x3.o    ♦ 0 12 |  0  0 6 24 | 0 0 0  0 12 8 8 | 0  0 0 0 4 4 2 | * * * 1
```

```xo3ox xx3oo&#x   → height = sqrt(2/3) = 0.816497
(triddip || gyro triddip)

o.3o. o.3o.    | 9 * | 2 2  2 0 0 | 1 4 1 2 1  4 0 0 0 | 2 2 1 4 2 2 0 0 | 1 2 2 1 0
.o3.o .o3.o    | * 9 | 0 0  2 2 2 | 0 0 0 1 2  4 1 4 1 | 0 0 1 2 4 2 2 2 | 0 2 1 2 1
---------------+-----+------------+--------------------+-----------------+----------
x. .. .. ..    | 2 0 | 9 *  * * * | 1 2 0 1 0  0 0 0 0 | 2 1 1 2 0 0 0 0 | 1 2 1 0 0
.. .. x. ..    | 2 0 | * 9  * * * | 0 2 1 0 0  2 0 0 0 | 1 2 0 2 1 2 0 0 | 1 2 2 1 0
oo3oo oo3oo&#x | 1 1 | * * 18 * * | 0 0 0 1 1  2 0 0 0 | 0 0 1 2 2 1 0 0 | 0 1 1 1 0
.. .x .. ..    | 0 2 | * *  * 9 * | 0 0 0 0 1  0 1 2 0 | 0 0 1 0 2 0 2 1 | 0 2 0 1 1
.. .. .x ..    | 0 2 | * *  * * 9 | 0 0 0 0 0  2 0 2 1 | 0 0 0 1 2 2 1 2 | 0 2 1 2 1
---------------+-----+------------+--------------------+-----------------+----------
x.3o. .. ..    | 3 0 | 3 0  0 0 0 | 3 * * * *  * * * * | 2 0 1 0 0 0 0 0 | 1 2 0 0 0
x. .. x. ..    | 4 0 | 2 2  0 0 0 | * 9 * * *  * * * * | 1 1 0 1 0 0 0 0 | 1 1 1 0 0
.. .. x.3o.    | 3 0 | 0 3  0 0 0 | * * 3 * *  * * * * | 0 2 0 0 0 2 0 0 | 1 0 2 1 0
xo .. .. ..&#x | 2 1 | 1 0  2 0 0 | * * * 9 *  * * * * | 0 0 1 2 0 0 0 0 | 0 2 1 0 0
.. ox .. ..&#x | 1 2 | 0 0  2 1 0 | * * * * 9  * * * * | 0 0 1 0 2 0 0 0 | 0 2 0 1 0
.. .. xx ..&#x | 2 2 | 0 1  2 0 1 | * * * * * 18 * * * | 0 0 0 1 1 1 0 0 | 0 1 1 1 0
.o3.x .. ..    | 0 3 | 0 0  0 3 0 | * * * * *  * 3 * * | 0 0 1 0 0 0 2 0 | 0 2 0 0 1
.. .x .x ..    | 0 4 | 0 0  0 2 2 | * * * * *  * * 9 * | 0 0 0 0 1 0 1 1 | 0 1 0 1 1
.. .. .x3.o    | 0 3 | 0 0  0 0 3 | * * * * *  * * * 3 | 0 0 0 0 0 2 0 2 | 0 0 1 2 1
---------------+-----+------------+--------------------+-----------------+----------
x.3o. x. ..    ♦ 6 0 | 6 3  0 0 0 | 2 3 0 0 0  0 0 0 0 | 3 * * * * * * * | 1 1 0 0 0
x. .. x.3o.    ♦ 6 0 | 3 6  0 0 0 | 0 3 2 0 0  0 0 0 0 | * 3 * * * * * * | 1 0 1 0 0
xo3ox .. ..&#x ♦ 3 3 | 3 0  6 3 0 | 1 0 0 3 3  0 1 0 0 | * * 3 * * * * * | 0 2 0 0 0
xo .. xx ..&#x ♦ 4 2 | 2 2  4 0 1 | 0 1 0 2 0  2 0 0 0 | * * * 9 * * * * | 0 1 1 0 0
.. ox xx ..&#x ♦ 2 4 | 0 1  4 2 2 | 0 0 0 0 2  2 0 1 0 | * * * * 9 * * * | 0 1 0 1 0
.. .. xx3oo&#x ♦ 3 3 | 0 3  3 0 3 | 0 0 1 0 0  3 0 0 1 | * * * * * 6 * * | 0 0 1 1 0
.o3.x .x ..    ♦ 0 6 | 0 0  0 6 3 | 0 0 0 0 0  0 2 3 0 | * * * * * * 3 * | 0 1 0 0 1
.. .x .x3.o    ♦ 0 6 | 0 0  0 3 6 | 0 0 0 0 0  0 0 3 2 | * * * * * * * 3 | 0 0 0 1 1
---------------+-----+------------+--------------------+-----------------+----------
x.3o. x.3o.    ♦ 9 0 | 9 9  0 0 0 | 3 9 3 0 0  0 0 0 0 | 3 3 0 0 0 0 0 0 | 1 * * * *
xo3ox xx ..&#x ♦ 6 6 | 6 6  6 6 6 | 2 3 0 6 6  6 2 3 0 | 1 0 2 3 3 0 1 0 | * 3 * * *
xo .. xx3oo&#x ♦ 6 3 | 3 6  6 0 3 | 0 3 2 3 0  6 0 0 1 | 0 1 0 3 0 2 0 0 | * * 3 * *
.. ox xx3oo&#x ♦ 3 6 | 0 3  6 3 6 | 0 0 1 0 3  6 0 3 2 | 0 0 0 0 3 2 0 1 | * * * 3 *
.o3.x .x3.o    ♦ 0 9 | 0 0  0 9 9 | 0 0 0 0 0  0 3 9 3 | 0 0 0 0 0 0 3 3 | * * * * 1
```

```ooo3xxx3ooo&#x   → all pairwise heights = 1

o..3o..3o..    | 6 * * |  4 1 1  0 0  0 | 2 2  4 1  4 0 0  0 0 0 | 1 2 2  4 2 2 0 0 0 0 | 1 2 2 1 0
.o.3.o.3.o.    | * 6 * |  0 1 0  4 1  0 | 0 0  4 1  0 2 2  4 0 0 | 0 2 2  4 0 0 1 2 2 0 | 1 2 2 0 1
..o3..o3..o    | * * 6 |  0 0 1  0 1  4 | 0 0  0 1  4 0 0  4 2 2 | 0 0 0  4 2 2 0 2 2 1 | 0 2 2 1 1
---------------+-------+----------------+------------------------+----------------------+----------
... x.. ...    | 2 0 0 | 12 * *  * *  * | 1 1  1 0  1 0 0  0 0 0 | 1 1 1  1 1 1 0 0 0 0 | 1 1 1 1 0
oo.3oo.3oo.&#x | 1 1 0 |  * 6 *  * *  * | 0 0  4 1  0 0 0  0 0 0 | 0 2 2  4 0 0 0 0 0 0 | 1 2 2 0 0
o.o3o.o3o.o&#x | 1 0 1 |  * * 6  * *  * | 0 0  0 1  4 0 0  0 0 0 | 0 0 0  4 2 2 0 0 0 0 | 0 2 2 1 0
... .x. ...    | 0 2 0 |  * * * 12 *  * | 0 0  1 0  0 1 1  1 0 0 | 0 1 1  1 0 0 1 1 1 0 | 1 1 1 0 1
.oo3.oo3.oo&#x | 0 1 1 |  * * *  * 6  * | 0 0  0 1  0 0 0  4 0 0 | 0 0 0  4 0 0 0 2 2 0 | 0 2 2 0 1
... ..x ...    | 0 0 2 |  * * *  * * 12 | 0 0  0 0  1 0 0  1 1 1 | 0 0 0  1 1 1 0 1 1 1 | 0 1 1 1 1
---------------+-------+----------------+------------------------+----------------------+----------
o..3x.. ...    | 3 0 0 |  3 0 0  0 0  0 | 4 *  * *  * * *  * * * | 1 1 0  0 1 0 0 0 0 0 | 1 1 0 1 0
... x..3o..    | 3 0 0 |  3 0 0  0 0  0 | * 4  * *  * * *  * * * | 1 0 1  0 0 1 0 0 0 0 | 1 0 1 1 0
... xx. ...&#x | 2 2 0 |  1 2 0  1 0  0 | * * 12 *  * * *  * * * | 0 1 1  1 0 0 0 0 0 0 | 1 1 1 0 0
ooo3ooo3ooo&#x | 1 1 1 |  0 1 1  0 1  0 | * *  * 6  * * *  * * * | 0 0 0  4 0 0 0 0 0 0 | 0 2 2 0 0
... x.x ...&#x | 2 0 2 |  1 0 2  0 0  1 | * *  * * 12 * *  * * * | 0 0 0  1 1 1 0 0 0 0 | 0 1 1 1 0
.o.3.x. ...    | 0 3 0 |  0 0 0  3 0  0 | * *  * *  * 4 *  * * * | 0 1 0  0 0 0 1 1 0 0 | 1 1 0 0 1
... .x.3.o.    | 0 3 0 |  0 0 0  3 0  0 | * *  * *  * * 4  * * * | 0 0 1  0 0 0 1 0 1 0 | 1 0 1 0 1
... .xx ...&#x | 0 2 2 |  0 0 0  1 2  1 | * *  * *  * * * 12 * * | 0 0 0  1 0 0 0 1 1 0 | 0 1 1 0 1
..o3..x ...    | 0 0 3 |  0 0 0  0 0  3 | * *  * *  * * *  * 4 * | 0 0 0  0 1 0 0 1 0 1 | 0 1 0 1 1
... ..x3..o    | 0 0 3 |  0 0 0  0 0  3 | * *  * *  * * *  * * 4 | 0 0 0  0 0 1 0 0 1 1 | 0 0 1 1 1
---------------+-------+----------------+------------------------+----------------------+----------
o..3x..3o..    ♦ 6 0 0 | 12 0 0  0 0  0 | 4 4  0 0  0 0 0  0 0 0 | 1 * *  * * * * * * * | 1 0 0 1 0
oo.3xx. ...&#x ♦ 3 3 0 |  3 3 0  3 0  0 | 1 0  3 0  0 1 0  0 0 0 | * 4 *  * * * * * * * | 1 1 0 0 0
... xx.3oo.&#x ♦ 3 3 0 |  3 3 0  3 0  0 | 0 1  3 0  0 0 1  0 0 0 | * * 4  * * * * * * * | 1 0 1 0 0
... xxx ...&#x ♦ 2 2 2 |  1 2 2  1 2  1 | 0 0  1 2  1 0 0  1 0 0 | * * * 12 * * * * * * | 0 1 1 0 0
o.o3x.x ...&#x ♦ 3 0 3 |  3 0 3  0 0  3 | 1 0  0 0  3 0 0  0 1 0 | * * *  * 4 * * * * * | 0 1 0 1 0
... x.x3o.o&#x ♦ 3 0 3 |  3 0 3  0 0  3 | 0 1  0 0  3 0 0  0 0 1 | * * *  * * 4 * * * * | 0 0 1 1 0
.o.3.x.3.o.    ♦ 0 6 0 |  0 0 0 12 0  0 | 0 0  0 0  0 4 4  0 0 0 | * * *  * * * 1 * * * | 1 0 0 0 1
.oo3.xx ...&#x ♦ 0 3 3 |  0 0 0  3 3  3 | 0 0  0 0  0 1 0  3 1 0 | * * *  * * * * 4 * * | 0 1 0 0 1
... .xx3.oo&#x ♦ 0 3 3 |  0 0 0  3 3  3 | 0 0  0 0  0 0 1  3 0 1 | * * *  * * * * * 4 * | 0 0 1 0 1
..o3..x3..o    ♦ 0 0 6 |  0 0 0  0 0 12 | 0 0  0 0  0 0 0  0 4 4 | * * *  * * * * * * 1 | 0 0 0 1 1
---------------+-------+----------------+------------------------+----------------------+----------
oo.3xx.3oo.&#x ♦ 6 6 0 | 12 6 0 12 0  0 | 4 4 12 0  0 4 4  0 0 0 | 1 4 4  0 0 0 1 0 0 0 | 1 * * * *
ooo3xxx ...&#x ♦ 3 3 3 |  3 3 3  3 3  3 | 1 0  3 3  3 1 0  3 1 0 | 0 1 0  3 1 0 0 1 0 0 | * 4 * * *
... xxx3ooo&#x ♦ 3 3 3 |  3 3 3  3 3  3 | 0 1  3 3  3 0 1  3 0 1 | 0 0 1  3 0 1 0 0 1 0 | * * 4 * *
o.o3x.x3o.o&#x ♦ 6 0 6 | 12 0 6  0 0 12 | 4 4  0 0 12 0 0  0 4 4 | 1 0 0  0 4 4 0 0 0 1 | * * * 1 *
.oo3.xx3.oo&#x ♦ 0 6 6 |  0 0 0 12 6 12 | 0 0  0 0  0 4 4 12 4 4 | 0 0 0  0 0 0 1 4 4 1 | * * * * 1
```