Acronym octasc
Name octahedral scalene,
octahedron-pyramid pyramid,
line atop fully orthogonal octahedron,
1/4-lune of triacontiditeron
Circumradius 1/sqrt(2) = 0.707107
Lace city
in approx. ASCII-art
o    		where:
     		o = o3o4o (point)
O   o		O = x3o4o (oct)
Coordinates
  1. (0, 0, 0, 0, 1/sqrt(2))
    top-tip
  2. (0, 0, 0, 1/sqrt(2), 0)
    right-tip
  3. (1/sqrt(2), 0, 0, 0, 0)   & permutations and changes of sign in all but last 2 coordinates
    octahedron
Dihedral angles
  • at tet between pen and pen:   arccos(-3/5) = 126.869898°
  • at oct between octpy and octpy:   90°
  • at tet between octpy and pen:   arccos(1/sqrt(5)) = 63.434949°
Confer
uniform relative:
tac  
related segmentotera:
hexpy  
general polytopal classes:
segmentotera  

Incidence matrix according to Dynkin symbol

xo ox3oo4oo&#x   → height = 1/2
(line || perp oct)

o. o.3o.4o.    | 2 *  1  6  0 | 6 12 0 | 12  8 0 | 8 1
.o .o3.o4.o    | * 6  0  2  4 | 1  8 4 |  4  8 1 | 4 2
---------------+-----+---------+--------+---------+----
x. .. .. ..    | 2 0 | 1  *  *  6  0 0 | 12  0 0 | 8 0
oo oo3oo4oo&#x | 1 1 | * 12  *  1  4 0 |  4  4 0 | 4 1
.. .x .. ..    | 0 2 | *  * 12 | 0  2 2 |  1  4 1 | 2 2
---------------+-----+---------+--------+---------+----
xo .. .. ..&#x | 2 1 | 1  2  0 | 6  * * |  4  0 0 | 4 0
.. ox .. ..&#x | 1 2 | 0  2  1 | * 24 * |  1  2 0 | 2 1
.. .x3.o ..    | 0 3 | 0  0  3 | *  * 8 |  0  2 1 | 1 2
---------------+-----+---------+--------+---------+----
xo ox .. ..&#x  2 2 | 1  4  1 | 2  2 0 | 12  * * | 2 0
.. ox3oo ..&#x  1 3 | 0  3  3 | 0  3 1 |  * 16 * | 1 1
.. .x3.o4.o     0 6 | 0  0 12 | 0  0 8 |  *  * 1 | 0 2
---------------+-----+---------+--------+---------+----
xo ox3oo ..&#x  2 3 | 1  6  3 | 3  6 1 |  3  2 0 | 8 *
.. ox3oo4oo&#x  1 6 | 0  6 12 | 0 12 8 |  0  8 1 | * 2

xo oo3ox3oo&#x   → height = 1/2
(line || perp oct)

o. o.3o.3o.    | 2 *  1  6  0 | 6 12 0 0 | 12 4 4 0 | 4 4 1
.o .o3.o3.o    | * 6  0  2  4 | 1  8 2 2 |  4 4 4 1 | 2 2 2
---------------+-----+---------+----------+----------+------
x. .. .. ..    | 2 0 | 1  *  *  6  0 0 0 | 12 0 0 0 | 4 4 0
oo oo3oo3oo&#x | 1 1 | * 12  *  1  4 0 0 |  4 2 2 0 | 2 2 1
.. .. .x ..    | 0 2 | *  * 12 | 0  2 1 1 |  1 2 2 1 | 1 1 2
---------------+-----+---------+----------+----------+------
xo .. .. ..&#x | 2 1 | 1  2  0 | 6  * * * |  4 0 0 0 | 2 2 0
.. .. ox ..&#x | 1 2 | 0  2  1 | * 24 * * |  1 1 1 0 | 1 1 1
.. .o3.x ..    | 0 3 | 0  0  3 | *  * 4 * |  0 2 0 1 | 1 0 2
.. .. .x3.o    | 0 3 | 0  0  3 | *  * * 4 |  0 0 2 1 | 0 1 2
---------------+-----+---------+----------+----------+------
xo .. ox ..&#x  2 2 | 1  4  1 | 2  2 0 0 | 12 * * * | 1 1 0
.. oo3ox ..&#x  1 3 | 0  3  3 | 0  3 1 0 |  * 8 * * | 1 0 1
.. .. ox3oo&#x  1 3 | 0  3  3 | 0  3 0 1 |  * * 8 * | 0 1 1
.. .o3.x3.o     0 6 | 0  0 12 | 0  0 4 4 |  * * * 1 | 0 0 2
---------------+-----+---------+----------+----------+------
xo oo3ox ..&#x  2 3 | 1  6  3 | 3  6 1 0 |  3 2 0 0 | 4 * *
xo .. ox3oo&#x  2 3 | 1  6  3 | 3  6 0 1 |  3 0 2 0 | * 4 *
.. oo3ox3oo&#x  1 6 | 0  6 12 | 0 12 4 4 |  0 4 4 1 | * * 2

oxo xoo3oox&#x   → all heights = sqrt(2/3) = 0.816497

o.. o..3o..    | 3 * *  2 2 2 0 0 0 | 1 1 4 2 1  4 0 0 0 | 2 2 1 2 4 2 0 0 | 1 2 1 2 0
.o. .o.3.o.    | * 2 *  0 3 0 1 3 0 | 0 3 3 0 0  6 3 3 0 | 3 1 0 6 3 3 3 1 | 1 3 3 1 1
..o ..o3..o    | * * 3  0 0 2 0 2 2 | 0 0 0 1 2  4 1 4 1 | 0 0 1 2 2 4 2 2 | 0 1 2 2 1
---------------+-------+-------------+--------------------+-----------------+----------
... x.. ...    | 2 0 0 | 3 * * * * * | 1 0 2 1 0  0 0 0 0 | 1 2 1 0 2 0 0 0 | 1 1 0 2 0
oo. oo.3oo.&#x | 1 1 0 | * 6 * * * *  0 1 2 0 0  2 0 0 0 | 2 1 0 2 2 1 0 0 | 1 2 1 1 0
o.o o.o3o.o&#x | 1 0 1 | * * 6 * * * | 0 0 0 1 1  2 0 0 0 | 0 0 1 1 2 2 0 0 | 0 1 1 2 0
.x. ... ...    | 0 2 0 | * * * 1 * *  0 3 0 0 0  0 3 0 0 | 3 0 0 6 0 0 3 0 | 1 3 3 0 1
.oo .oo3.oo&#x | 0 1 1 | * * * * 6 *  0 0 0 0 0  2 1 2 0 | 0 0 0 2 1 2 2 1 | 0 1 2 1 1
... ... ..x    | 0 0 2 | * * * * * 3 | 0 0 0 0 1  0 0 2 1 | 0 0 1 0 0 2 1 2 | 0 0 1 2 1
---------------+-------+-------------+--------------------+-----------------+----------
... x..3o..    | 3 0 0 | 3 0 0 0 0 0 | 1 * * * *  * * * * | 0 2 1 0 0 0 0 0 | 1 0 0 2 0
ox. ... ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 3 * * *  * * * * | 2 0 0 2 0 0 0 0 | 1 2 1 0 0
... xo. ...&#x | 2 1 0 | 1 2 0 0 0 0 | * * 6 * *  * * * * | 1 1 0 0 1 0 0 0 | 1 1 0 1 0
... x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 3 *  * * * * | 0 0 1 0 2 0 0 0 | 0 1 0 2 0
... ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 3  * * * * | 0 0 1 0 0 2 0 0 | 0 0 1 2 0
ooo ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 12 * * * | 0 0 0 1 1 1 0 0 | 0 1 1 1 0
.xo ... ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * *  * 3 * * | 0 0 0 2 0 0 2 0 | 0 1 2 0 1
... ... .ox&#x | 0 1 2 | 0 0 0 0 2 1 | * * * * *  * * 6 * | 0 0 0 0 0 1 1 1 | 0 0 1 1 1
... ..o3..x    | 0 0 3 | 0 0 0 0 0 3 | * * * * *  * * * 1 | 0 0 1 0 0 0 0 2 | 0 0 0 2 1
---------------+-------+-------------+--------------------+-----------------+----------
ox. xo. ...&#x  2 2 0 | 1 4 0 1 0 0 | 0 2 2 0 0  0 0 0 0 | 3 * * * * * * * | 1 1 0 0 0
... xo.3oo.&#x  3 1 0 | 3 3 0 0 0 0 | 1 0 3 0 0  0 0 0 0 | * 2 * * * * * * | 1 0 0 1 0
... x.o3o.x&#x  3 0 3 | 3 0 6 0 0 3 | 1 0 0 3 3  0 0 0 1 | * * 1 * * * * * | 0 0 0 2 0
oxo ... ...&#x  1 2 1 | 0 2 1 1 2 0 | 0 1 0 0 0  2 1 0 0 | * * * 6 * * * * | 0 1 1 0 0
... xoo ...&#x  2 1 1 | 1 2 2 0 1 0 | 0 0 1 1 0  2 0 0 0 | * * * * 6 * * * | 0 1 0 1 0
... ... oox&#x  1 1 2 | 0 1 2 0 2 1 | 0 0 0 0 1  2 0 1 0 | * * * * * 6 * * | 0 0 1 1 0
.xo ... .ox&#x  0 2 2 | 0 0 0 1 4 1 | 0 0 0 0 0  0 2 2 0 | * * * * * * 3 * | 0 0 1 0 1
... .oo3.ox&#x  0 1 3 | 0 0 0 0 3 3 | 0 0 0 0 0  0 0 3 1 | * * * * * * * 2 | 0 0 0 1 1
---------------+-------+-------------+--------------------+-----------------+----------
ox. xo.3oo.&#x  3 2 0 | 3 6 0 1 0 0 | 1 3 6 0 0  0 0 0 0 | 3 2 0 0 0 0 0 0 | 1 * * * *
oxo xoo ...&#x  2 2 1 | 1 4 2 1 2 0 | 0 2 2 1 0  4 1 0 0 | 1 0 0 2 2 0 0 0 | * 3 * * *
oxo ... oox&#x  1 2 2 | 0 2 2 1 4 1 | 0 1 0 0 1  4 2 2 0 | 0 0 0 2 0 2 1 0 | * * 3 * *
... xoo3oox&#x  3 1 3 | 3 3 6 0 3 3 | 1 0 3 3 3  6 0 3 1 | 0 1 1 0 3 3 0 1 | * * * 2 *
.xo .oo3.ox&#x  0 2 3 | 0 0 0 1 6 3 | 0 0 0 0 0  0 3 6 1 | 0 0 0 0 0 0 3 2 | * * * * 1

oxo3ooo4ooo&#x   → height(1,2) = height(2,3) = 1/sqrt(2) = 0.707107
                   height(1,3) = 1

o..3o..4o..    | 1 * *  6 1  0 0 | 12 6 0  0 | 8 12 0 0 | 1 8 0
.o.3.o.4.o.    | * 6 *  1 0  4 1 |  4 1 4  4 | 4  4 1 4 | 1 4 1
..o3..o4..o    | * * 1  0 1  0 6 |  0 6 0 12 | 0 12 0 8 | 0 8 1
---------------+-------+----------+-----------+----------+------
oo.3oo.4oo.&#x | 1 1 0 | 6 *  * *   4 1 0  0 | 4  4 0 0 | 1 4 0
o.o3o.o4o.o&#x | 1 0 1 | * 1  * *   0 6 0  0 | 0 12 0 0 | 0 8 0
.x. ... ...    | 0 2 0 | * * 12 * |  1 0 2  1 | 2  1 1 2 | 1 2 1
.oo3.oo4.oo&#x | 0 1 1 | * *  * 6   0 1 0  4 | 0  4 0 4 | 0 4 1
---------------+-------+----------+-----------+----------+------
ox. ... ...&#x | 1 2 0 | 2 0  1 0 | 12 * *  * | 2  1 0 0 | 1 2 0
ooo3ooo4ooo&#x | 1 1 1 | 1 1  0 1 |  * 6 *  * | 0  4 0 0 | 0 4 0
.x.3.o. ...    | 0 3 0 | 0 0  3 0 |  * * 8  * | 1  0 1 1 | 1 1 1
.xo ... ...&#x | 0 2 1 | 0 0  1 2 |  * * * 12 | 0  1 0 2 | 0 2 1
---------------+-------+----------+-----------+----------+------
ox.3oo. ...&#x  1 3 0 | 3 0  3 0 |  3 0 1  0 | 8  * * * | 1 1 0
oxo ... ...&#x  1 2 1 | 2 1  1 2 |  1 2 0  1 | * 12 * * | 0 2 0
.x.3.o.4.o.     0 6 0 | 0 0 12 0 |  0 0 8  0 | *  * 1 * | 1 0 1
.xo3.oo ...&#x  0 3 1 | 0 0  3 3 |  0 0 1  3 | *  * * 8 | 0 1 1
---------------+-------+----------+-----------+----------+------
ox.3oo.4oo.&#x  1 6 0 | 6 0 12 0 | 12 0 8  0 | 8  0 1 0 | 1 * *
oxo3ooo ...&#x  1 3 1 | 3 1  3 3 |  3 3 1  3 | 1  3 0 1 | * 8 *
.xo3.oo4.oo&#x  0 6 1 | 0 0 12 6 |  0 0 8 12 | 0  0 1 8 | * * 1

ooo3oox3ooo&#x   → height(1,2) = 1
                 height(1,3) = height(2,3) = 1/sqrt(2) = 0.707107

o..3o..3o..    | 1 * *  1 6 0  0 | 6 12  0 0 0 | 12 4 4 0 0 0 | 4 4 1 0
.o.3.o.3.o.    | * 1 *  1 0 6  0 | 6  0 12 0 0 | 12 0 0 4 4 0 | 4 4 0 1
..o3..o3..o    | * * 6  0 1 1  4 | 1  4  4 2 2 |  4 2 2 2 2 1 | 2 2 1 1
---------------+-------+----------+-------------+--------------+--------
oo.3oo.3oo.&#x | 1 1 0 | 1 * *  *  6  0  0 0 0 | 12 0 0 0 0 0 | 4 4 0 0
o.o3o.o3o.o&#x | 1 0 1 | * 6 *  *  1  4  0 0 0 |  4 2 2 0 0 0 | 2 2 1 0
.oo3.oo3.oo&#x | 0 1 1 | * * 6  *  1  0  4 0 0 |  4 0 0 2 2 0 | 2 2 0 1
... ..x ...    | 0 0 2 | * * * 12 | 0  1  1 1 1 |  1 1 1 1 1 1 | 1 1 1 1
---------------+-------+----------+-------------+--------------+--------
ooo3ooo3ooo&#x | 1 1 1 | 1 1 1  0 | 6  *  * * * |  4 0 0 0 0 0 | 2 2 0 0
... o.x ...&#x | 1 0 2 | 0 2 0  1 | * 12  * * * |  1 1 1 0 0 0 | 1 1 1 0
... .ox ...&#x | 0 1 2 | 0 0 2  1 | *  * 12 * * |  1 0 0 1 1 0 | 1 1 0 1
..o3..x ...    | 0 0 3 | 0 0 0  3 | *  *  * 4 * |  0 1 0 1 0 1 | 1 0 1 1
... ..x3..o    | 0 0 3 | 0 0 0  3 | *  *  * * 4 |  0 0 1 0 1 1 | 0 1 1 1
---------------+-------+----------+-------------+--------------+--------
... oox ...&#x  1 1 2 | 1 2 2  1 | 2  1  1 0 0 | 12 * * * * * | 1 1 0 0
o.o3o.x ...&#x  1 0 3 | 0 3 0  3 | 0  3  0 1 0 |  * 4 * * * * | 1 0 1 0
... o.x3o.o&#x  1 0 3 | 0 3 0  3 | 0  3  0 0 1 |  * * 4 * * * | 0 1 1 0
.oo3.ox ...&#x  0 1 3 | 0 0 3  3 | 0  0  3 1 0 |  * * * 4 * * | 1 0 0 1
... .ox3.oo&#x  0 1 3 | 0 0 3  3 | 0  0  3 0 1 |  * * * * 4 * | 0 1 0 1
..o3..x3..o     0 0 6 | 0 0 0 12 | 0  0  0 4 4 |  * * * * * 1 | 0 0 1 1
---------------+-------+----------+-------------+--------------+--------
ooo3oox ...&#x  1 1 3 | 1 3 3  3 | 3  3  3 1 0 |  3 1 0 1 0 0 | 4 * * *
... oox3ooo&#x  1 1 3 | 1 3 3  3 | 3  3  3 0 1 |  3 0 1 0 1 0 | * 4 * *
o.o3o.x3o.o&#x  1 0 6 | 0 6 0 12 | 0 12  0 4 4 |  0 4 4 0 0 1 | * * 1 *
.oo3.ox3.oo&#x  0 1 6 | 0 0 6 12 | 0  0 12 4 4 |  0 0 0 4 4 1 | * * * 1

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