Acronym ... Name pentagonal rhombic barrel © Vertex figure [3,5,3,R], [3,r,R,r] Dihedral angles (at margins) between {3} and {5}:   arccos(-sqrt[(5+2 sqrt(5))/15]) = 142.622632° between {3} and {(r,R)2}:   arccos(-sqrt[(5+2 sqrt(5))/15]) = 142.622632° between {(r,R)2} and {(r,R)2}:   arccos(-1/sqrt(5)) = 116.565051° Confer uniform relatives: doe   id

The rhombs {(r,R)2} have vertex angles r = 72° resp. R = 108°. Esp. rr : rR = (1+sqrt(5))/2.

This polyhedron can be sliced into 3 monostratic segments. The polar ones then are identic to the pentagonal caps of id, while the equatorial one is identic to that of doe. The rhombs thereby get sliced along their long diagonales rr, thus showing up their relationship to the corresponding sections of regular pentagons.

Incidence matrix according to Dynkin symbol

xofo5ofox&#xt   → all heights = sqrt[(5-sqrt(5))/10] = 0.525731
({5} || pseudo dual f-{5} || pseudo f-{5} || dual {5})

o...5o...     & | 10  * |  2  2  0 | 1  2  1  [3,5,3,R]
.o..5.o..     & |  * 10 |  0  2  2 | 0  1  3  [3,r,R,r]
----------------+-------+----------+--------
x... ....     & |  2  0 | 10  *  * | 1  1  0
oo..5oo..&#x  & |  1  1 |  * 20  * | 0  1  1
.oo.5.oo.&#x    |  0  2 |  *  * 10 | 0  0  2
----------------+-------+----------+--------
x...5o...     & |  5  0 |  5  0  0 | 2  *  *  {5}
xo.. ....&#x  & |  2  1 |  1  2  0 | * 10  *  {3}
.... ofo.&#xt & |  1  3 |  0  2  2 | *  * 10  {(r,R)2}