Acronym ...
Name u x5o,
variation of pentagonal prism
Circumradius sqrt[(15+sqrt(5))/10] = 1.312862
Lace city
in approx. ASCII-art
o    o
f    f
x    x
   x
x     x

 x   x
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Confer pip   q x5o   f x5o   v x5o   x v5o   f v5o  

Incidence matrix according to Dynkin symbol

u x5o

. . . | 10 | 1  2 | 2 1
------+----+------+----
u . . |  2 | 5  * | 2 0
. x . |  2 | * 10 | 1 1
------+----+------+----
u x . |  4 | 2  2 | 5 *
. x5o |  5 | 0  5 | * 2

xx5oo&#u   → height = 2
({5} || {5})

o.5o.    | 5 * | 2 1 0 | 1 2 0
.o5.o    | * 5 | 0 1 2 | 0 2 1
---------+-----+-------+------
x. ..    | 2 0 | 5 * * | 1 1 0
oo5oo&#u | 1 1 | * 5 * | 0 2 0
.x ..    | 0 2 | * * 5 | 0 1 1
---------+-----+-------+------
x.5o.    | 5 0 | 5 0 0 | 1 * *
xx ..&#u | 2 2 | 1 2 1 | * 5 *
.x5.o    | 0 5 | 0 0 5 | * * 1

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