Acronym rapesc Name rectified-pentachoric scalene,rectified-pentachoron-pyramid pyramid,line atop fully orthogonal rectified pentachoron Circumradius sqrt(2/3) = 0.816497 Confer general polytopal classes: segmentopeta   scalene

It might be worth noting that 12 copies of this polypeton can be aligned around their special edge with A5 around symmetry, then forming a dotet.

Incidence matrix according to Dynkin symbol

```xo oo3ox3oo3oo&#x   → height = sqrt(3/20) = 0.387298
(line || perp rap)

o. o.3o.3o.3o.    | 2  * ♦ 1 10  0 | 10 30  0  0 | 30 10 20 0 0 | 10 20  5  5 0 | 5 5 1
.o .o3.o3.o3.o    | * 10 ♦ 0  2  6 |  1 12  3  6 |  6  6 12 3 2 |  3  6  6  4 1 | 3 2 2
------------------+------+---------+-------------+--------------+---------------+------
x. .. .. .. ..    | 2  0 | 1  *  * ♦ 10  0  0  0 | 30  0  0 0 0 | 10 20  0  0 0 | 5 5 0
oo oo3oo3oo3oo&#x | 1  1 | * 20  * ♦  1  6  0  0 |  6  3  6 0 0 |  3  6  3  2 0 | 3 2 1
.. .. .x .. ..    | 0  2 | *  * 30 |  0  2  1  2 |  1  2  4 2 1 |  1  2  4  2 1 | 2 1 2
------------------+------+---------+-------------+--------------+---------------+------
xo .. .. .. ..&#x | 2  1 | 1  2  0 | 10  *  *  * ♦  6  0  0 0 0 |  3  6  0  0 0 | 3 2 0
.. .. ox .. ..&#x | 1  2 | 0  2  1 |  * 60  *  * |  1  1  2 0 0 |  1  2  2  1 0 | 2 1 1
.. .o3.x .. ..    | 0  3 | 0  0  3 |  *  * 10  * |  0  2  0 2 0 |  1  0  4  0 1 | 2 0 2
.. .. .x3.o ..    | 0  3 | 0  0  3 |  *  *  * 20 |  0  0  2 1 1 |  0  1  2  2 1 | 1 1 2
------------------+------+---------+-------------+--------------+---------------+------
xo .. ox .. ..&#x ♦ 2  2 | 1  4  1 |  2  2  0  0 | 30  *  * * * |  1  2  0  0 0 | 2 1 0
.. oo3ox .. ..&#x ♦ 1  3 | 0  3  3 |  0  3  1  0 |  * 20  * * * |  1  0  2  0 0 | 2 0 1
.. .. ox3oo ..&#x ♦ 1  3 | 0  3  3 |  0  3  0  1 |  *  * 40 * * |  0  1  1  1 0 | 1 1 1
.. .o3.x3.o ..    ♦ 0  6 | 0  0 12 |  0  0  4  4 |  *  *  * 5 * |  0  0  2  0 1 | 1 0 2
.. .. .x3.o3.o    ♦ 0  4 | 0  0  6 |  0  0  0  4 |  *  *  * * 5 |  0  0  0  2 1 | 0 1 2
------------------+------+---------+-------------+--------------+---------------+------
xo oo3ox .. ..&#x ♦ 2  3 | 1  6  3 |  3  6  1  0 |  3  2  0 0 0 | 10  *  *  * * | 2 0 0
xo .. ox3oo ..&#x ♦ 2  3 | 1  6  3 |  3  6  0  1 |  3  0  2 0 0 |  * 20  *  * * | 1 1 0
.. oo3ox3oo ..&#x ♦ 1  6 | 0  6 12 |  0 12  4  4 |  0  4  4 1 0 |  *  * 10  * * | 1 0 1
.. .. ox3oo3oo&#x ♦ 1  4 | 0  4  6 |  0  6  0  4 |  0  0  4 0 1 |  *  *  * 10 * | 0 1 1
.. .o3.x3.o3.o    ♦ 0 10 | 0  0 30 |  0  0 10 20 |  0  0  0 5 5 |  *  *  *  * 1 | 0 0 2
------------------+------+---------+-------------+--------------+---------------+------
xo oo3ox3oo ..&#x ♦ 2  6 | 1 12 12 |  6 24  4  4 | 12  8  8 1 0 |  4  4  2  0 0 | 5 * *
xo .. ox3oo3oo&#x ♦ 2  4 | 1  8  6 |  4 12  0  4 |  6  0  8 0 1 |  0  4  0  2 0 | * 5 *
.. oo3ox3oo3oo&#x ♦ 1 10 | 0 10 30 |  0 30 10 20 |  0 10 20 5 5 |  0  0  5  5 1 | * * 2
```

```ooo3oxo3ooo3ooo&#x   → height(1,2) = height(2,3) = sqrt(2/5) = 0.632456
height(1,3) = 1

o..3o..3o..3o..    | 1  * * ♦ 10 1  0  0 | 30 10  0  0  0 | 10 20 30 0 0  0  0 | 5 5 10 20 0 0 0 | 1 5 5 0
.o.3.o.3.o.3.o.    | * 10 * ♦  1 0  6  1 |  6  1  3  6  6 |  3  6  6 3 2  3  6 | 3 2  3  6 1 3 2 | 1 3 2 1
..o3..o3..o3..o    | *  * 1 ♦  0 1  0 10 |  0 10  0  0 30 |  0  0 30 0 0 10 20 | 0 0 10 20 0 5 5 | 0 5 5 1
-------------------+--------+------------+----------------+--------------------+-----------------+--------
oo.3oo.3oo.3oo.&#x | 1  1 0 | 10 *  *  * ♦  6  1  0  0  0 |  3  6  6 0 0  0  0 | 3 2  3  6 0 0 0 | 1 3 2 0
o.o3o.o3o.o3o.o&#x | 1  0 1 |  * 1  *  * ♦  0 10  0  0  0 |  0  0 30 0 0  0  0 | 0 0 10 20 0 0 0 | 0 5 5 0
... .x. ... ...    | 0  2 0 |  * * 30  * |  1  0  1  2  1 |  1  2  1 2 1  1  2 | 2 1  1  2 1 2 1 | 1 2 1 1
.oo3.oo3.oo3.oo&#x | 0  1 1 |  * *  * 10 ♦  0  1  0  0  6 |  0  0  6 0 0  3  6 | 0 0  3  6 0 3 2 | 0 3 2 1
-------------------+--------+------------+----------------+--------------------+-----------------+--------
... ox. ... ...&#x | 1  2 0 |  2 0  1  0 | 30  *  *  *  * |  1  2  1 0 0  0  0 | 2 1  1  2 0 0 0 | 1 2 1 0
ooo3ooo3ooo3ooo&#x | 1  1 1 |  1 1  0  1 |  * 10  *  *  * ♦  0  0  6 0 0  0  0 | 0 0  3  6 0 0 0 | 0 3 2 0
.o.3.x. ... ...    | 0  3 0 |  0 0  3  0 |  *  * 10  *  * |  1  0  0 2 0  1  0 | 2 0  1  0 1 2 0 | 1 2 0 1
... .x.3.o. ...    | 0  3 0 |  0 0  3  0 |  *  *  * 20  * |  0  1  0 1 1  0  1 | 1 1  0  1 1 1 1 | 1 1 1 1
... .xo ... ...&#x | 0  2 1 |  0 0  1  2 |  *  *  *  * 30 |  0  0  1 0 0  1  2 | 0 0  1  2 0 2 1 | 0 2 1 1
-------------------+--------+------------+----------------+--------------------+-----------------+--------
oo.3ox. ... ...&#x ♦ 1  3 0 |  3 0  3  0 |  3  0  1  0  0 | 10  *  * * *  *  * | 2 0  1  0 0 0 0 | 1 2 0 0
... ox.3oo. ...&#x ♦ 1  3 0 |  3 0  3  0 |  3  0  0  1  0 |  * 20  * * *  *  * | 1 1  0  1 0 0 0 | 1 1 1 0
... oxo ... ...&#x ♦ 1  2 1 |  2 1  1  2 |  1  2  0  0  1 |  *  * 30 * *  *  * | 0 0  1  2 0 0 0 | 0 2 1 0
.o.3.x.3.o. ...    ♦ 0  6 0 |  0 0 12  0 |  0  0  4  4  0 |  *  *  * 5 *  *  * | 1 0  0  0 1 1 0 | 1 1 0 1
... .x.3.o.3.o.    ♦ 0  4 0 |  0 0  6  0 |  0  0  0  4  0 |  *  *  * * 5  *  * | 0 1  0  0 1 0 1 | 1 0 1 1
.oo3.xo ... ...&#x ♦ 0  3 1 |  0 0  3  3 |  0  0  1  0  3 |  *  *  * * * 10  * | 0 0  1  0 0 2 0 | 0 2 0 1
... .xo3.oo ...&#x ♦ 0  3 1 |  0 0  3  3 |  0  0  0  1  3 |  *  *  * * *  * 20 | 0 0  0  1 0 1 1 | 0 1 1 1
-------------------+--------+------------+----------------+--------------------+-----------------+--------
oo.3ox.3oo. ...&#x ♦ 1  6 0 |  6 0 12  0 | 12  0  4  4  0 |  4  4  0 1 0  0  0 | 5 *  *  * * * * | 1 1 0 0
... ox.3oo.3oo.&#x ♦ 1  4 0 |  4 0  6  0 |  6  0  0  4  0 |  0  4  0 0 1  0  0 | * 5  *  * * * * | 1 0 1 0
ooo3oxo ... ...&#x ♦ 1  3 1 |  3 1  3  3 |  3  3  1  0  3 |  1  0  3 0 0  1  0 | * * 10  * * * * | 0 2 0 0
... oxo3ooo ...&#x ♦ 1  3 1 |  3 1  3  3 |  3  3  0  1  3 |  0  1  3 0 0  0  1 | * *  * 20 * * * | 0 1 1 0
.o.3.x.3.o.3.o.    ♦ 0 10 0 |  0 0 30  0 |  0  0 10 20  0 |  0  0  0 5 5  0  0 | * *  *  * 1 * * | 1 0 0 1
.oo3.xo3.oo ...&#x ♦ 0  6 1 |  0 0 12  6 |  0  0  4  4 12 |  0  0  0 1 0  4  4 | * *  *  * * 5 * | 0 1 0 1
... .xo3.oo3.oo&#x ♦ 0  4 1 |  0 0  6  4 |  0  0  0  4  6 |  0  0  0 0 1  0  4 | * *  *  * * * 5 | 0 0 1 1
-------------------+--------+------------+----------------+--------------------+-----------------+--------
oo.3ox.3oo.3oo.&#x ♦ 1 10 0 | 10 0 30  0 | 30  0 10 20  0 | 10 20  0 5 5  0  0 | 5 5  0  0 1 0 0 | 1 * * *
ooo3oxo3ooo ...&#x ♦ 1  6 1 |  6 1 12  6 | 12  6  4  4 12 |  4  4 12 1 0  4  4 | 1 0  4  4 0 1 0 | * 5 * *
... oxo3ooo3ooo&#x ♦ 1  4 1 |  4 1  6  4 |  6  4  0  4  6 |  0  4  6 0 1  0  4 | 0 1  0  4 0 0 1 | * * 5 *
.oo3.xo3.oo3.oo&#x ♦ 0 10 1 |  0 0 30 10 |  0  0 10 20 30 |  0  0  0 5 5 10 20 | 0 0  0  0 1 5 5 | * * * 1
```

```xoo oxo3oox3ooo&#x   → height(1,2) = sqrt(3/8) = 0.612372
height(1,3) = 1/2
height(2,3) = sqrt(5/8) = 0.790569

o.. o..3o..3o..    | 2 * * ♦ 1 4  6 0  0  0 | 4  6 6 12 12 0 0  0 0 0 | 6 4 12 4 4 12  6 12 0 0 0 0 | 4 1 4 4 1 6 12 4 4 0 | 1 1 4 4 1
.o. .o.3.o.3.o.    | * 4 * ♦ 0 2  0 3  3  0 | 1  6 0  0  6 3 3  3 0 0 | 3 6  0 0 0  3  6  6 1 3 1 0 | 3 2 0 0 0 3  3 6 2 1 | 1 0 3 1 2
..o ..o3..o3..o    | * * 6 ♦ 0 0  2 0  2  4 | 0  0 1  8  4 0 1  4 2 2 | 0 0  4 4 4  2  2  8 0 2 2 1 | 0 0 2 2 2 1  4 4 4 1 | 0 1 2 2 2
-------------------+-------+----------------+-------------------------+-----------------------------+----------------------+----------
x.. ... ... ...    | 2 0 0 | 1 *  * *  *  * ♦ 4  0 6  0  0 0 0  0 0 0 | 6 0 12 0 0 12  0  0 0 0 0 0 | 4 0 4 4 0 6 12 0 0 0 | 1 1 4 4 0
oo. oo.3oo.3oo.&#x | 1 1 0 | * 8  * *  *  * ♦ 1  3 0  0  3 0 0  0 0 0 | 3 3  0 0 0  3  3  3 0 0 0 0 | 3 1 0 0 0 3  3 3 1 0 | 1 0 3 1 1
o.o o.o3o.o3o.o&#x | 1 0 1 | * * 12 *  *  * ♦ 0  0 1  4  2 0 0  0 0 0 | 0 0  4 2 2  2  1  4 0 0 0 0 | 0 0 2 2 1 1  4 2 2 0 | 0 1 2 2 1
... .x. ... ...    | 0 2 0 | * *  * 6  *  * | 0  2 0  0  0 2 1  0 0 0 | 1 4  0 0 0  0  2  0 1 2 0 0 | 2 2 0 0 0 1  0 4 0 1 | 1 0 2 0 2
.oo .oo3.oo3.oo&#x | 0 1 1 | * *  * * 12  * | 0  0 0  0  2 0 1  2 0 0 | 0 0  0 0 0  1  2  4 0 2 1 0 | 0 0 0 0 0 1  2 4 2 1 | 0 0 2 1 2
... ... .x. ...    | 0 0 2 | * *  * *  * 12 | 0  0 0  2  0 0 0  1 1 1 | 0 0  1 2 2  0  0  2 0 1 1 1 | 0 0 1 1 2 0  1 2 2 1 | 0 1 1 1 2
-------------------+-------+----------------+-------------------------+-----------------------------+----------------------+----------
xo. ... ... ...&#x | 2 1 0 | 1 2  0 0  0  0 | 4  * *  *  * * *  * * * ♦ 3 0  0 0 0  3  0  0 0 0 0 0 | 3 0 0 0 0 3  3 0 0 0 | 1 0 3 1 0
... ox. ... ...&#x | 1 2 0 | 0 2  0 1  0  0 | * 12 *  *  * * *  * * * | 1 2  0 0 0  0  1  0 0 0 0 0 | 2 1 0 0 0 1  0 2 0 0 | 1 0 2 0 1
x.o ... ... ...&#x | 2 0 1 | 1 0  2 0  0  0 | *  * 6  *  * * *  * * * ♦ 0 0  4 0 0  2  0  0 0 0 0 0 | 0 0 2 2 0 1  4 0 0 0 | 0 1 2 2 0
... ... o.x ...&#x | 1 0 2 | 0 0  2 0  0  1 | *  * * 24  * * *  * * * | 0 0  1 1 1  0  0  1 0 0 0 0 | 0 0 1 1 1 0  1 1 1 0 | 0 1 1 1 1
ooo ooo3ooo3ooo&#x | 1 1 1 | 0 1  1 0  1  0 | *  * *  * 24 * *  * * * | 0 0  0 0 0  1  1  2 0 0 0 0 | 0 0 0 0 0 1  2 2 1 0 | 0 0 2 1 1
... .x.3.o. ...    | 0 3 0 | 0 0  0 3  0  0 | *  * *  *  * 4 *  * * * | 0 2  0 0 0  0  0  0 1 1 0 0 | 1 2 0 0 0 0  0 2 0 1 | 1 0 1 0 2
... .xo ... ...&#x | 0 2 1 | 0 0  0 1  2  0 | *  * *  *  * * 6  * * * | 0 0  0 0 0  0  2  0 0 2 0 0 | 0 0 0 0 0 1  0 4 0 1 | 0 0 2 0 2
... ... .ox ...&#x | 0 1 2 | 0 0  0 0  2  1 | *  * *  *  * * * 12 * * | 0 0  0 0 0  0  0  2 0 1 1 0 | 0 0 0 0 0 0  1 2 2 1 | 0 0 1 1 2
... ..o3..x ...    | 0 0 3 | 0 0  0 0  0  3 | *  * *  *  * * *  * 4 * | 0 0  0 2 0  0  0  0 0 1 0 1 | 0 0 1 0 2 0  0 2 0 1 | 0 1 1 0 2
... ... ..x3..o    | 0 0 3 | 0 0  0 0  0  3 | *  * *  *  * * *  * * 4 | 0 0  0 0 2  0  0  0 0 0 1 1 | 0 0 0 1 2 0  0 0 2 1 | 0 1 0 1 2
-------------------+-------+----------------+-------------------------+-----------------------------+----------------------+----------
xo. ox. ... ...&#x ♦ 2 2 0 | 1 4  0 1  0  0 | 2  2 0  0  0 0 0  0 0 0 | 6 *  * * *  *  *  * * * * * | 2 0 0 0 0 1  0 0 0 0 | 1 0 2 0 0
... ox.3oo. ...&#x ♦ 1 3 0 | 0 3  0 3  0  0 | 0  3 0  0  0 1 0  0 0 0 | * 8  * * *  *  *  * * * * * | 1 1 0 0 0 0  0 1 0 0 | 1 0 1 0 1
x.o ... o.x ...&#x ♦ 2 0 2 | 1 0  4 0  0  1 | 0  0 2  2  0 0 0  0 0 0 | * * 12 * *  *  *  * * * * * | 0 0 1 1 0 0  1 0 0 0 | 0 1 1 1 0
... o.o3o.x ...&#x ♦ 1 0 3 | 0 0  3 0  0  3 | 0  0 0  3  0 0 0  0 1 0 | * *  * 8 *  *  *  * * * * * | 0 0 1 0 1 0  0 1 0 0 | 0 1 1 0 1
... ... o.x3o.o&#x ♦ 1 0 3 | 0 0  3 0  0  3 | 0  0 0  3  0 0 0  0 0 1 | * *  * * 8  *  *  * * * * * | 0 0 0 1 1 0  0 0 1 0 | 0 1 0 1 1
xoo ... ... ...&#x ♦ 2 1 1 | 1 2  2 0  1  0 | 1  0 1  0  2 0 0  0 0 0 | * *  * * * 12  *  * * * * * | 0 0 0 0 0 1  2 0 0 0 | 0 0 2 1 0
... oxo ... ...&#x ♦ 1 2 1 | 0 2  1 1  2  0 | 0  1 0  0  2 0 1  0 0 0 | * *  * * *  * 12  * * * * * | 0 0 0 0 0 1  0 2 0 0 | 0 0 2 0 1
... ... oox ...&#x ♦ 1 1 2 | 0 1  2 0  2  1 | 0  0 0  1  2 0 0  1 0 0 | * *  * * *  *  * 24 * * * * | 0 0 0 0 0 0  1 1 1 0 | 0 0 1 1 1
... .x.3.o.3.o.    ♦ 0 4 0 | 0 0  0 6  0  0 | 0  0 0  0  0 4 0  0 0 0 | * *  * * *  *  *  * 1 * * * | 0 2 0 0 0 0  0 0 0 1 | 1 0 0 0 2
... .xo3.ox ...&#x ♦ 0 3 3 | 0 0  0 3  6  3 | 0  0 0  0  0 1 3  3 1 0 | * *  * * *  *  *  * * 4 * * | 0 0 0 0 0 0  0 2 0 1 | 0 0 1 0 2
... ... .ox3.oo&#x ♦ 0 1 3 | 0 0  0 0  3  3 | 0  0 0  0  0 0 0  3 0 1 | * *  * * *  *  *  * * * 4 * | 0 0 0 0 0 0  0 0 2 1 | 0 0 0 1 2
... ..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 0 2 0  0 0 0 1 | 0 1 0 0 2
-------------------+-------+----------------+-------------------------+-----------------------------+----------------------+----------
xo. ox.3oo. ...&#x ♦ 2 3 0 | 1 6  0 3  0  0 | 3  6 0  0  0 1 0  0 0 0 | 3 2  0 0 0  0  0  0 0 0 0 0 | 4 * * * * *  * * * * | 1 0 1 0 0
... ox.3oo.3oo.&#x ♦ 1 4 0 | 0 4  0 6  0  0 | 0  6 0  0  0 4 0  0 0 0 | 0 4  0 0 0  0  0  0 1 0 0 0 | * 2 * * * *  * * * * | 1 0 0 0 1
x.o o.o3o.x ...&#x ♦ 2 0 3 | 1 0  6 0  0  3 | 0  0 3  6  0 0 0  0 1 0 | 0 0  3 2 0  0  0  0 0 0 0 0 | * * 4 * * *  * * * * | 0 1 1 0 0
x.o ... o.x3o.o&#x ♦ 2 0 3 | 1 0  6 0  0  3 | 0  0 3  6  0 0 0  0 0 1 | 0 0  3 0 2  0  0  0 0 0 0 0 | * * * 4 * *  * * * * | 0 1 0 1 0
... o.o3o.x3o.o&#x ♦ 1 0 6 | 0 0  6 0  0 12 | 0  0 0 12  0 0 0  0 4 4 | 0 0  0 4 4  0  0  0 0 0 0 1 | * * * * 2 *  * * * * | 0 1 0 0 1
xoo oxo ... ...&#x ♦ 2 2 1 | 1 4  2 1  2  0 | 2  2 1  0  4 0 1  0 0 0 | 1 0  0 0 0  2  2  0 0 0 0 0 | * * * * * 6  * * * * | 0 0 2 0 0
xoo ... oox ...&#x ♦ 2 1 2 | 1 2  4 0  2  1 | 1  0 2  2  4 0 0  1 0 0 | 0 0  1 0 0  2  0  2 0 0 0 0 | * * * * * * 12 * * * | 0 0 1 1 0
... oxo3oox ...&#x ♦ 1 3 3 | 0 3  3 3  6  3 | 0  3 0  3  6 1 3  3 1 0 | 0 1  0 1 0  0  3  3 0 1 0 0 | * * * * * *  * 8 * * | 0 0 1 0 1
... ... oox3ooo&#x ♦ 1 1 3 | 0 1  3 0  3  3 | 0  0 0  3  3 0 0  3 0 1 | 0 0  0 0 1  0  0  3 0 0 1 0 | * * * * * *  * * 8 * | 0 0 0 1 1
... .xo3.ox3.oo&#x ♦ 0 4 6 | 0 0  0 6 12 12 | 0  0 0  0  0 4 6 12 4 4 | 0 0  0 0 0  0  0  0 1 4 4 1 | * * * * * *  * * * 1 | 0 0 0 0 2
-------------------+-------+----------------+-------------------------+-----------------------------+----------------------+----------
xo. ox.3oo.3oo.&#x ♦ 2 4 0 | 1 8  0 6  0  0 | 4 12 0  0  0 4 0  0 0 0 | 6 8  0 0 0  0  0  0 1 0 0 0 | 4 2 0 0 0 0  0 0 0 0 | 1 * * * *
x.o o.o3o.x3o.o&#x ♦ 2 0 6 | 1 0 12 0  0 12 | 0  0 6 24  0 0 0  0 4 4 | 0 0 12 8 8  0  0  0 0 0 0 1 | 0 0 4 4 2 0  0 0 0 0 | * 1 * * *
xoo oxo3oox ...&#x ♦ 2 3 3 | 1 6  6 3  6  3 | 3  6 3  6 12 1 3  3 1 0 | 3 2  3 2 0  6  6  6 0 1 0 0 | 1 0 1 0 0 3  3 2 0 0 | * * 4 * *
xoo ... oox3ooo&#x ♦ 2 1 3 | 1 2  6 0  3  3 | 1  0 3  6  6 0 0  3 0 1 | 0 0  3 0 2  3  0  6 0 0 1 0 | 0 0 0 1 0 0  3 0 2 0 | * * * 4 *
... oxo3oox3ooo&#x ♦ 1 4 6 | 0 4  6 6 12 12 | 0  6 0 12 12 4 6 12 4 4 | 0 4  0 4 4  0  6 12 1 4 4 1 | 0 1 0 0 1 0  0 4 4 1 | * * * * 2
```