The c edge, provided in the below description, only qualifies as pseudo edge wrt. the full polyhedron.

Incidence matrix according to Dynkin symbol

co oo3ox&#zy   → height = 0
                 c = (sqrt(5)-1)/3 = 0.412023 (tip-to-tip distance)
                 y = sqrt[(9-sqrt(5))/18] = 0.613004 (lacing edge size)
(tegum product of c-line with {3})

o. o.3o.    | 2 * | 3 0 | 3
.o .o3.o    | * 3 | 2 2 | 4
------------+-----+-----+--
oo oo3oo&#y | 1 1 | 6 * | 2
.. .o3.x    | 0 2 | * 3 | 2
------------+-----+-----+--
.. .. ox&#y | 1 2 | 2 1 | 6

oxo3ooo&#yt   → both heights = c/2 = (sqrt(5)-1)/6 = 0.206011
                y = sqrt[(9-sqrt(5))/18] = 0.613004 (lacing edge size)
(pt || pseudo {3} || pt)

o..3o..    | 1 * * | 3 0 0 | 3 0
.o.3.o.    | * 3 * | 1 2 1 | 2 2
..o3..o    | * * 1 | 0 0 3 | 0 3
-----------+-------+-------+---
oo.3oo.&#y | 1 1 0 | 3 * * | 2 0
.x.3...    | 0 2 0 | * 3 * | 1 1
.oo3.oo&#y | 0 1 1 | * * 3 | 0 2
-----------+-------+-------+----
ox.3...&#y | 1 2 0 | 2 1 0 | 3 *
.xo3...&#y | 0 2 1 | 0 1 2 | * 3