Acronym cubasc
Name (degenerate) cubical scalene,
line atop fully orthogonal cube
|,>,O device line prism prism pyramid pyramid = |||>>
Circumradius ∞   i.e. flat in euclidean space
Confer
general polytopal classes:
decomposition  

The hull of this degenerate shape is cute. In fact, the latter provides 2 different decompositions, either into 6 squascs (kind in the sense of a 4D orange) or equatorially into 2 cubpies. The blend of these 2 decompositions, blending out right those cute cells each, then would result in cubasc again.

In fact, it is only this degeneracy, which lets become here the hull of a scalene to be a mere bipyramid.


Incidence matrix according to Dynkin symbol

xo oo3oo4ox&#x   → height = 0
(line || perp cube)

o. o.3o.4o.    | 2 *  1  8  0 | 8 12 0 | 12  6 0 | 6 1
.o .o3.o4.o    | * 8 | 0  2  3 | 1  6 3 |  3  6 1 | 3 2
---------------+-----+---------+--------+---------+----
x. .. .. ..    | 2 0 | 1  *  *  8  0 0 | 12  0 0 | 6 0
oo oo3oo4oo&#x | 1 1 | * 16  * | 1  3 0 |  3  3 0 | 3 1
.. .. .. .x    | 0 2 | *  * 12 | 0  2 2 |  1  4 1 | 2 2
---------------+-----+---------+--------+---------+----
xo .. .. ..&#x | 2 1 | 1  2  0 | 8  * * |  3  0 0 | 3 0
.. .. .. ox&#x | 1 2 | 0  2  1 | * 24 * |  1  2 0 | 2 1
.. .. .o4.x    | 0 4 | 0  0  4 | *  * 6 |  0  2 1 | 1 2
---------------+-----+---------+--------+---------+----
xo .. .. ox&#x  2 2 | 1  4  1 | 2  2 0 | 12  * * | 2 0
.. .. oo4ox&#x  1 4 | 0  4  4 | 0  4 1 |  * 12 * | 1 1
.. .o3.o4.x     0 8 | 0  0 12 | 0  0 6 |  *  * 1 | 0 2
---------------+-----+---------+--------+---------+----
xo .. oo4ox&#x  2 4 | 1  8  4 | 4  8 1 |  4  2 0 | 6 *
.. oo3oo4ox&#x  1 8 | 0  8 12 | 0 12 6 |  0  6 1 | * 2

ooo3ooo4oox&#x   → height(1,2) = 1
                   height(1,3) = height(2,3) = 1/2

o..3o..4o..    | 1 * *  1 8 0  0 | 8 12  0 0 | 12 6 0 0 | 6 1 0
.o.3.o.4.o.    | * 1 *  1 0 8  0 | 8  0 12 0 | 12 0 6 0 | 6 0 1
..o3..o4..o    | * * 8 | 0 1 1  3 | 1  3  3 3 |  3 3 3 1 | 3 1 1
---------------+-------+----------+-----------+----------+------
oo.3oo.4oo.&#x | 1 1 0 | 1 * *  *  8  0  0 0 | 12 0 0 0 | 6 0 0
o.o3o.o4o.o&#x | 1 0 1 | * 8 *  * | 1  3  0 0 |  3 3 0 0 | 3 1 0
.oo3.oo4.oo&#x | 0 1 1 | * * 8  * | 1  0  3 0 |  3 0 3 0 | 3 0 1
... ... ..x    | 0 0 2 | * * * 12 | 0  1  1 2 |  1 2 2 1 | 2 1 1
---------------+-------+----------+-----------+----------+------
ooo3ooo4ooo&#x | 1 1 1 | 1 1 1  0 | 8  *  * * |  3 0 0 0 | 3 0 0
... ... o.x&#x | 1 0 2 | 0 2 0  1 | * 12  * * |  1 2 0 0 | 2 1 0
... ... .ox&#x | 0 1 2 | 0 0 2  1 | *  * 12 * |  1 0 2 0 | 2 0 1
... ..o4..x    | 0 0 4 | 0 0 0  4 | *  *  * 6 |  0 1 1 1 | 1 1 1
---------------+-------+----------+-----------+----------+------
... ... oox&#x  1 1 2 | 1 2 2  1 | 2  1  1 0 | 12 * * * | 2 0 0
... o.o4o.x&#x  1 0 4 | 0 4 0  4 | 0  4  0 1 |  * 6 * * | 1 1 0
... .oo4.ox&#x  0 1 4 | 0 0 4  4 | 0  0  4 1 |  * * 6 * | 1 0 1
..o3..o4..x     0 0 8 | 0 0 0 12 | 0  0  0 6 |  * * * 1 | 0 1 1
---------------+-------+----------+-----------+----------+------
... ooo4oox&#x  1 1 4 | 1 4 4  4 | 4  4  4 1 |  4 1 1 0 | 6 * *
o.o3o.o4o.x&#x  1 0 8 | 0 8 0 12 | 0 12  0 6 |  0 6 0 1 | * 1 *
.oo3.oo4.ox&#x  0 1 8 | 0 0 8 12 | 0  0 12 6 |  0 0 6 1 | * * 1
or
o..3o..4o..    & | 2 *  1  8  0 | 8 12 0 | 12  6 0 | 6 1
..o3..o4..o      | * 8 | 0  2  3 | 1  6 3 |  3  6 1 | 3 2
-----------------+-----+---------+--------+---------+----
oo.3oo.4oo.&#x   | 2 0 | 1  *  *  8  0 0 | 12  0 0 | 6 0
o.o3o.o4o.o&#x & | 1 1 | * 16  * | 1  3 0 |  3  3 0 | 3 1
... ... ..x      | 0 2 | *  * 12 | 0  2 2 |  1  4 1 | 2 2
-----------------+-----+---------+--------+---------+----
ooo3ooo4ooo&#x   | 2 1 | 1  2  0 | 8  * * |  3  0 0 | 3 0
... ... o.x&#x & | 1 2 | 0  2  1 | * 24 * |  1  2 0 | 2 1
... ..o4..x      | 0 4 | 0  0  4 | *  * 6 |  0  2 1 | 1 2
-----------------+-----+---------+--------+---------+----
... ... oox&#x    2 2 | 1  4  1 | 2  2 0 | 12  * * | 2 0
... o.o4o.x&#x &  1 4 | 0  4  4 | 0  4 1 |  * 12 * | 1 1
..o3..o4..x       0 8 | 0  0 12 | 0  0 6 |  *  * 1 | 0 2
-----------------+-----+---------+--------+---------+----
... ooo4oox&#x    2 4 | 1  8  4 | 4  8 1 |  4  2 0 | 6 *
o.o3o.o4o.x&#x &  1 8 | 0  8 12 | 0 12 6 |  0  6 1 | * 2

pt || cubpy   → height = 0

o..3o..4o..    | 1 * *  1 8 0  0 | 8 12  0 0 | 12 6 0 0 | 6 1 0
.o.3.o.4.o.    | * 1 *  1 0 8  0 | 8  0 12 0 | 12 0 6 0 | 6 0 1
..o3..o4..o    | * * 8 | 0 1 1  3 | 1  3  3 3 |  3 3 3 1 | 3 1 1
---------------+-------+----------+-----------+----------+------
oo.3oo.4oo.&#x | 1 1 0 | 1 * *  *  8  0  0 0 | 12 0 0 0 | 6 0 0
o.o3o.o4o.o&#x | 1 0 1 | * 8 *  * | 1  3  0 0 |  3 3 0 0 | 3 1 0
.oo3.oo4.oo&#x | 0 1 1 | * * 8  * | 1  0  3 0 |  3 0 3 0 | 3 0 1
... ... ..x    | 0 0 2 | * * * 12 | 0  1  1 2 |  1 2 2 1 | 2 1 1
---------------+-------+----------+-----------+----------+------
ooo3ooo4ooo&#x | 1 1 1 | 1 1 1  0 | 8  *  * * |  3 0 0 0 | 3 0 0
... ... o.x&#x | 1 0 2 | 0 2 0  1 | * 12  * * |  1 2 0 0 | 2 1 0
... ... .ox&#x | 0 1 2 | 0 0 2  1 | *  * 12 * |  1 0 2 0 | 2 0 1
... ..o4..x    | 0 0 4 | 0 0 0  4 | *  *  * 6 |  0 1 1 1 | 1 1 1
---------------+-------+----------+-----------+----------+------
... ... oox&#x  1 1 2 | 1 2 2  1 | 2  1  1 0 | 12 * * * | 2 0 0
... o.o4o.x&#x  1 0 4 | 0 4 0  4 | 0  4  0 1 |  * 6 * * | 1 1 0
... .oo4.ox&#x  0 1 4 | 0 0 4  4 | 0  0  4 1 |  * * 6 * | 1 0 1
..o3..o4..x     0 0 8 | 0 0 0 12 | 0  0  0 6 |  * * * 1 | 0 1 1
---------------+-------+----------+-----------+----------+------
... ooo4oox&#x  1 1 4 | 1 4 4  4 | 4  4  4 1 |  4 1 1 0 | 6 * *
o.o3o.o4o.x&#x  1 0 8 | 0 8 0 12 | 0 12  0 6 |  0 6 0 1 | * 1 *
.oo3.oo4.ox&#x  0 1 8 | 0 0 8 12 | 0  0 12 6 |  0 0 6 1 | * * 1

oox oox oox&#x   → height(1,2) = 1
                   height(1,3) = height(2,3) = 1/2

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

oox xxo xxo&#x   → height(1,2) = 1
                   height(1,3) = height(2,3) = 1/2

o.. o.. o..    | 4 * * | 1 1 1 2 0 0 0 0 | 1 1 1 1 2 2 2 0 0 0 0 | 1 1 1 2 1 2 2 0 0 0 | 1 1 1 2 0
.o. .o. .o.    | * 4 * | 0 0 1 0 1 1 2 0 | 0 1 1 0 0 0 2 1 1 2 2 | 1 0 0 0 1 2 2 1 1 2 | 0 1 1 2 1
..o ..o ..o    | * * 2  0 0 0 4 0 0 4 1 | 0 0 0 4 2 2 4 0 4 2 2 | 0 2 2 1 4 2 2 2 2 1 | 1 2 2 1 1
---------------+-------+-----------------+-----------------------+---------------------+----------
... x.. ...    | 2 0 0 | 2 * * * * * * * | 1 1 0 0 2 0 0 0 0 0 0 | 1 1 0 2 0 2 0 0 0 0 | 1 1 0 2 0
... ... x..    | 2 0 0 | * 2 * * * * * * | 1 0 1 0 0 2 0 0 0 0 0 | 1 0 1 2 0 0 2 0 0 0 | 1 0 1 2 0
oo. oo. oo.&#x | 1 1 0 | * * 4 * * * * * | 0 1 1 0 0 0 2 0 0 0 0 | 1 0 0 0 1 2 2 0 0 0 | 0 1 1 2 0
o.o o.o o.o&#x | 1 0 1 | * * * 8 * * * * | 0 0 0 1 1 1 1 0 0 0 0 | 0 1 1 1 1 1 1 0 0 0 | 1 1 1 1 0
... .x. ...    | 0 2 0 | * * * * 2 * * * | 0 1 0 0 0 0 0 1 0 2 0 | 1 0 0 0 0 2 0 1 0 2 | 0 1 0 2 1
... ... .x.    | 0 2 0 | * * * * * 2 * * | 0 0 1 0 0 0 0 1 0 0 2 | 1 0 0 0 0 0 2 0 1 2 | 0 0 1 2 1
.oo .oo .oo&#x | 0 1 1 | * * * * * * 8 * | 0 0 0 0 0 0 1 0 1 1 1 | 0 0 0 0 1 1 1 1 1 1 | 0 1 1 1 1
..x ... ...    | 0 0 2 | * * * * * * * 1  0 0 0 4 0 0 0 0 4 0 0 | 0 2 2 0 4 0 0 2 2 0 | 1 2 2 0 1
---------------+-------+-----------------+-----------------------+---------------------+----------
... x.. x..    | 4 0 0 | 2 2 0 0 0 0 0 0 | 1 * * * * * * * * * * | 1 0 0 2 0 0 0 0 0 0 | 1 0 0 2 0
... xx. ...&#x | 2 2 0 | 1 0 2 0 1 0 0 0 | * 2 * * * * * * * * * | 1 0 0 0 0 2 0 0 0 0 | 0 1 0 2 0
... ... xx.&#x | 2 2 0 | 0 1 2 0 0 1 0 0 | * * 2 * * * * * * * * | 1 0 0 0 0 0 2 0 0 0 | 0 0 1 2 0
o.x ... ...&#x | 1 0 2 | 0 0 0 2 0 0 0 1 | * * * 4 * * * * * * * | 0 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
... x.o ...&#x | 2 0 1 | 1 0 0 2 0 0 0 0 | * * * * 4 * * * * * * | 0 1 0 1 0 1 0 0 0 0 | 1 1 0 1 0
... ... x.o&#x | 2 0 1 | 0 1 0 2 0 0 0 0 | * * * * * 4 * * * * * | 0 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0
ooo ooo ooo&#x | 1 1 1 | 0 0 1 1 0 0 1 0 | * * * * * * 8 * * * * | 0 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0
... .x. .x.    | 0 4 0 | 0 0 0 0 2 2 0 0 | * * * * * * * 1 * * * | 1 0 0 0 0 0 0 0 0 2 | 0 0 0 2 1
.ox ... ...&#x | 0 1 2 | 0 0 0 0 0 0 2 1 | * * * * * * * * 4 * * | 0 0 0 0 1 0 0 1 1 0 | 0 1 1 0 1
... .xo ...&#x | 0 2 1 | 0 0 0 0 1 0 2 0 | * * * * * * * * * 4 * | 0 0 0 0 0 1 0 1 0 1 | 0 1 0 1 1
... ... .xo&#x | 0 2 1 | 0 0 0 0 0 1 2 0 | * * * * * * * * * * 4 | 0 0 0 0 0 0 1 0 1 1 | 0 0 1 1 1
---------------+-------+-----------------+-----------------------+---------------------+----------
... xx. xx.&#x  4 4 0 | 2 2 4 0 2 2 0 0 | 1 2 2 0 0 0 0 1 0 0 0 | 1 * * * * * * * * * | 0 0 0 2 0
o.x x.o ...&#x  2 0 2 | 1 0 0 4 0 0 0 1 | 0 0 0 2 2 0 0 0 0 0 0 | * 2 * * * * * * * * | 1 1 0 0 0
o.x ... x.o&#x  2 0 2 | 0 1 0 4 0 0 0 1 | 0 0 0 2 0 2 0 0 0 0 0 | * * 2 * * * * * * * | 1 0 1 0 0
... x.o x.o&#x  4 0 1 | 2 2 0 4 0 0 0 0 | 1 0 0 0 2 2 0 0 0 0 0 | * * * 2 * * * * * * | 1 0 0 1 0
oox ... ...&#x  1 1 2 | 0 0 1 2 0 0 2 1 | 0 0 0 1 0 0 2 0 1 0 0 | * * * * 4 * * * * * | 0 1 1 0 0
... xxo ...&#x  2 2 1 | 1 0 2 2 1 0 2 0 | 0 1 0 0 1 0 2 0 0 1 0 | * * * * * 4 * * * * | 0 1 0 1 0
... ... xxo&#x  2 2 1 | 0 1 2 2 0 1 2 0 | 0 0 1 0 0 1 2 0 0 0 1 | * * * * * * 4 * * * | 0 0 1 1 0
.ox .xo ...&#x  0 2 2 | 0 0 0 0 1 0 4 1 | 0 0 0 0 0 0 0 0 2 2 0 | * * * * * * * 2 * * | 0 1 0 0 1
.ox ... .xo&#x  0 2 2 | 0 0 0 0 0 1 4 1 | 0 0 0 0 0 0 0 0 2 0 2 | * * * * * * * * 2 * | 0 0 1 0 1
... .xo .xo&#x  0 4 1 | 0 0 0 0 2 2 4 0 | 0 0 0 0 0 0 0 1 0 2 2 | * * * * * * * * * 2 | 0 0 0 1 1
---------------+-------+-----------------+-----------------------+---------------------+----------
o.x x.o x.o&#x  4 0 2 | 2 2 0 8 0 0 0 1 | 1 0 0 4 4 4 0 0 0 0 0 | 0 2 2 2 0 0 0 0 0 0 | 1 * * * *
oox xxo ...&#x  2 2 2 | 1 0 2 4 1 0 4 1 | 0 1 0 2 2 0 4 0 2 2 0 | 0 1 0 0 2 2 0 1 0 0 | * 2 * * *
oox ... xxo&#x  2 2 2 | 0 1 2 4 0 1 4 1 | 0 0 1 2 0 2 4 0 2 0 2 | 0 0 1 0 2 0 2 0 1 0 | * * 2 * *
... xxo xxo&#x  4 4 1 | 2 2 4 4 2 2 4 0 | 1 2 2 0 2 2 4 1 0 2 2 | 1 0 0 1 0 2 2 0 0 1 | * * * 2 *
.ox .xo .xo&#x  0 4 2 | 0 0 0 0 2 2 8 1 | 0 0 0 0 0 0 0 1 4 4 4 | 0 0 0 0 0 0 0 2 2 2 | * * * * 1

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