Acronym poct
Name pentagon - octahedron duoprism
Circumradius sqrt[(10+sqrt(5))/10] = 1.106168
Confer
more general:
n,oct-dip  
general polytopal classes:
segmentotera  

Incidence matrix according to Dynkin symbol

x5o x3o4o

. . . . . | 30 |  2  4 | 1  8  4 |  4  8 1 | 4 2
----------+----+-------+---------+---------+----
x . . . . |  2 | 30  * | 1  4  0 |  4  4 0 | 4 1
. . x . . |  2 |  * 60 | 0  2  2 |  1  4 1 | 2 2
----------+----+-------+---------+---------+----
x5o . . . |  5 |  5  0 | 6  *  * |  4  0 0 | 4 0
x . x . . |  4 |  2  2 | * 60  * |  1  2 0 | 2 1
. . x3o . |  3 |  0  3 | *  * 40 |  0  2 1 | 1 2
----------+----+-------+---------+---------+----
x5o x . .  10 | 10  5 | 2  5  0 | 12  * * | 2 0
x . x3o .   6 |  3  6 | 0  3  2 |  * 40 * | 1 1
. . x3o4o   6 |  0 12 | 0  0  8 |  *  * 5 | 0 2
----------+----+-------+---------+---------+----
x5o x3o .  15 | 15 15 | 3 15  5 |  3  5 0 | 8 *
x . x3o4o  12 |  6 24 | 0 12 16 |  0  8 2 | * 5

x5o o3x3o

. . . . . | 30 |  2  4 | 1  8  2  2 |  4  4  4 1 | 2 2 2
----------+----+-------+------------+------------+------
x . . . . |  2 | 30  * | 1  4  0  0 |  4  2  2 0 | 2 2 1
. . . x . |  2 |  * 60 | 0  2  1  1 |  1  2  2 1 | 1 1 2
----------+----+-------+------------+------------+------
x5o . . . |  5 |  5  0 | 6  *  *  * |  4  0  0 0 | 2 2 0
x . . x . |  4 |  2  2 | * 60  *  * |  1  1  1 0 | 1 1 1
. . o3x . |  3 |  0  3 | *  * 20  * |  0  2  0 1 | 1 0 2
. . . x3o |  3 |  0  3 | *  *  * 20 |  0  0  2 1 | 0 1 2
----------+----+-------+------------+------------+------
x5o . x .  10 | 10  5 | 2  5  0  0 | 12  *  * * | 1 1 0
x . o3x .   6 |  3  6 | 0  3  2  0 |  * 20  * * | 1 0 1
x . . x3o   6 |  3  6 | 0  3  0  2 |  *  * 20 * | 0 1 1
. . o3x3o   6 |  0 12 | 0  0  4  4 |  *  *  * 5 | 0 0 2
----------+----+-------+------------+------------+------
x5o o3x .  15 | 15 15 | 3 15  5  0 |  3  5  0 0 | 4 * *
x5o . x3o  15 | 15 15 | 3 15  0  5 |  3  0  5 0 | * 4 *
x . o3x3o  12 |  6 24 | 0 12  8  8 |  0  4  4 2 | * * 5

xo3ox xx5oo&#x     → height = sqrt(2/3) = 0.816497
(trapedip || {3}-gyro trapedip)

o.3o. o.5o.    | 15  * |  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 .o5.o    |  * 15 |  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 | 15  *  *  *  * | 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 |  * 15  *  *  * | 0  2 1  0  0  2 0  0 0 | 1 2 0  2  1 2 0 0 | 1 2 2 1 0
oo3oo oo5oo&#x |  1  1 |  *  * 30  *  * | 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 |  *  *  * 15  * | 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 |  *  *  *  * 15 | 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 | 5  * *  *  *  * *  * * | 2 0 1  0  0 0 0 0 | 1 2 0 0 0
x. .. x. ..    |  4  0 |  2  2  0  0  0 | * 15 *  *  *  * *  * * | 1 1 0  1  0 0 0 0 | 1 1 1 0 0
.. .. x.5o.    |  5  0 |  0  5  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 | *  * * 15  *  * *  * * | 0 0 1  2  0 0 0 0 | 0 2 1 0 0
.. ox .. ..&#x |  1  2 |  0  0  2  1  0 | *  * *  * 15  * *  * * | 0 0 1  0  2 0 0 0 | 0 2 0 1 0
.. .. xx ..&#x |  2  2 |  0  1  2  0  1 | *  * *  *  * 30 *  * * | 0 0 0  1  1 1 0 0 | 0 1 1 1 0
.o3.x .. ..    |  0  3 |  0  0  0  3  0 | *  * *  *  *  * 5  * * | 0 0 1  0  0 0 2 0 | 0 2 0 0 1
.. .x .x ..    |  0  4 |  0  0  0  2  2 | *  * *  *  *  * * 15 * | 0 0 0  0  1 0 1 1 | 0 1 0 1 1
.. .. .x5.o    |  0  5 |  0  0  0  0  5 | *  * *  *  *  * *  * 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 | 5 * *  *  * * * * | 1 1 0 0 0
x. .. x.5o.     10  0 |  5 10  0  0  0 | 0  5 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 | * * 5  *  * * * * | 0 2 0 0 0
xo .. xx ..&#x   4  2 |  2  2  4  0  1 | 0  1 0  2  0  2 0  0 0 | * * * 15  * * * * | 0 1 1 0 0
.. ox xx ..&#x   2  4 |  0  1  4  2  2 | 0  0 0  0  2  2 0  1 0 | * * *  * 15 * * * | 0 1 0 1 0
.. .. xx5oo&#x   5  5 |  0  5  5  0  5 | 0  0 1  0  0  5 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 | * * *  *  * * 5 * | 0 1 0 0 1
.. .x .x5.o      0 10 |  0  0  0  5 10 | 0  0 0  0  0  0 0  5 2 | * * *  *  * * * 3 | 0 0 0 1 1
---------------+-------+----------------+------------------------+-------------------+----------
x.3o. x.5o.     15  0 | 15 15  0  0  0 | 5 15 3  0  0  0 0  0 0 | 5 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 | * 5 * * *
xo .. xx5oo&#x  10  5 |  5 10 10  0  5 | 0  5 2  5  0 10 0  0 1 | 0 1 0  5  0 2 0 0 | * * 3 * *
.. ox xx5oo&#x   5 10 |  0  5 10  5 10 | 0  0 1  0  5 10 0  5 2 | 0 0 0  0  5 2 0 1 | * * * 3 *
.o3.x .x5.o      0 15 |  0  0  0 15 15 | 0  0 0  0  0  0 5 15 3 | 0 0 0  0  0 0 5 3 | * * * * 1

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