Acronym	cute (old: cubdipy)
Name	cubical (line-)tegum, cubical bipyramid
Circumradius	...
Volume	1/4 = 0.25
Dual	ope variant
Dihedral angles	at {3} between squippy and squippy:   120° at {4} between squippy and squippy:   90°
Face vector	10, 28, 30, 12
Confer	segmentochora: cubpy   squasc   uniform relative: ico   general polytopal classes: isohedral polytope   n-prismatic bipyramids   bistratic lace towers   Hanner polytopes
External links

Just as ico can be decomposed into 8 cutes according to ooq3ooo3ooo4oxo&#zx (where the vertices of the inscribed q-hex each together with the center point just span their edge-line factor, while each facet-cube of its dual and also ico-inscribed tes outlines the other factor), so also cute itself can either be decomposed equatorially into 2 cubpies or alternatively, kind like a 4D orange, into 6 squascs. (The blend of these 2 decompositions of cute, blending out out the boundary cells of cute, then would be the degenerate 5D segmentoteron cubasc.)

The dual of the uniform version of ope rather would become qo oo3oo4ox&#zy instead, where then y = sqrt(5)/2 = 1.118034 would again result into the required zero height. Within this variant, as necessary for uniform duals, all dihedral angles would become alike, in fact arccos(-1/3) = 109.471221°. The still single cell type then would become oo4ox&#h instead.

Incidence matrix according to Dynkin symbol

ooo3ooo4oxo&#xt   → both heights = 1/2
(pt || pseudo cube || pt)

o..3o..4o..    | 1 * * ♦ 8  0 0 | 12 0  0 | 6 0
.o.3.o.4.o.    | * 8 * | 1  3 1 |  3 3  3 | 3 3
..o3..o4..o    | * * 1 ♦ 0  0 8 |  0 0 12 | 0 6
---------------+-------+--------+---------+----
oo.3oo.4oo.&#x | 1 1 0 | 8  * * |  3 0  0 | 3 0
... ... .x.    | 0 2 0 | * 12 * |  1 2  1 | 2 2
.oo3.oo4.oo&#x | 0 1 1 | *  * 8 |  0 0  3 | 0 3
---------------+-------+--------+---------+----
... ... ox.&#x | 1 2 0 | 2  1 0 | 12 *  * | 2 0
... .o.4.x.    | 0 4 0 | 0  4 0 |  * 6  * | 1 1
... ... .xo&#x | 0 2 1 | 0  1 2 |  * * 12 | 0 2
---------------+-------+--------+---------+----
... oo.4ox.&#x ♦ 1 4 0 | 4  4 0 |  4 1  0 | 6 *
... .oo4.xo&#x ♦ 0 4 1 | 0  4 4 |  0 1  4 | * 6

((xo oo3oo4ox))&#zx   → height = 0
(tegum product of pseudo line with cube)

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

oxo oxo4ooo&#xt   → both heights = 1/2
(pt || pseudo cube || pt)

o.. o..4o..    & | 2 * ♦  8 0 0 | 4  8 0 0 | 4 2
.o. .o. .o.      | * 8 |  2 1 2 | 2  4 2 1 | 4 2
-----------------+-----+--------+----------+----
oo. oo.4oo.&#x & | 1 1 | 16 * * | 1  2 0 0 | 2 1
.x. ... ...      | 0 2 |  * 4 * | 2  0 2 0 | 4 0
... .x. ...      | 0 2 |  * * 8 | 0  2 1 1 | 2 2
-----------------+-----+--------+----------+----
ox. ... ...&#x & | 1 2 |  2 1 0 | 8  * * * | 2 0
... ox. ...&#x & | 1 2 |  2 0 1 | * 16 * * | 1 1
.x. .x. ...      | 0 4 |  0 2 2 | *  * 4 * | 2 0
... .x.4.o.      | 0 4 |  0 0 4 | *  * * 2 | 0 2
-----------------+-----+--------+----------+----
ox. ox. ...&#x & ♦ 1 4 |  4 2 2 | 2  2 1 0 | 8 *
... ox.4oo.&#x & ♦ 1 4 |  4 0 4 | 0  4 0 1 | * 4

oxo oxo oxo&#xt   → both heights = 1/2
(pt || pseudo cube || pt)

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

oxo xox4ooo&#xt   → both heights = 1/2
({4} || pseudo ortho line || {4})

o.. o.. o..    & | 8 * | 2  2 1 | 1  4 2 2 | 2 4
.o. .o. .o.      | * 2 ♦ 0  8 0 | 0  8 0 4 | 2 4
-----------------+-----+--------+----------+----
... x.. ...    & | 2 0 | 8  * * | 1  2 1 0 | 2 2
oo. oo.4oo.&#x & | 1 1 | * 16 * | 0  2 0 1 | 1 2
o.o o.o4o.o&#x   | 2 0 | *  * 4 | 0  0 2 2 | 0 4
-----------------+-----+--------+----------+----
... x..4o..    & | 4 0 | 4  0 0 | 2  * * * | 2 0
... xo. ...&#x & | 2 1 | 1  2 0 | * 16 * * | 1 1
... x.x ...&#x   | 4 0 | 2  0 2 | *  * 4 * | 0 2
ooo ooo4ooo&#x   | 2 1 | 0  2 1 | *  * * 8 | 0 2
-----------------+-----+--------+----------+----
... xo.4oo.&#x & ♦ 4 1 | 4  4 0 | 1  4 0 0 | 4 *
... xox ...&#x   ♦ 4 1 | 2  4 2 | 0  2 1 2 | * 8

oxo xox xox&#xt   → both heights = 1/2
({4} || pseudo ortho line || {4})

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