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Rectification wrt. a non-regular polytope is meant to be the singular instance of truncations on all vertices at such a depth that the hyperplane intersections on the former edges will coincide (provided such a choice exists). Within the specific case of tid as a pre-image these intersection points might differ on its 2 edge types. Therefore tid cannot be rectified (within this stronger sense). Nonetheless the Conway operator of ambification (chosing the former edge centers generally) clearly is applicable. This would result in 2 different edge sizes in the outcome polyhedron. That one here is scaled such so that the shorter one becomes unity. Then the larger edge will have size a = x(10) = sqrt[(5+sqrt(5))/2] = 1.902113.

Incidence matrix according to Dynkin symbol

xo3oA5Ao&#za   → height = 0
                 A = aa = u+f = (5+sqrt(5))/2 = 3.618034 (pseudo)
                 a = x(10) = sqrt[(5+sqrt(5))/2] = 1.902113
(tegum sum of (x,A)-srid and A-id)

o.3o.5o.     | 60  * |  2   2 |  1  2  1
.o3.o5.o     |  * 30 |  0   4 |  0  2  2
-------------+-------+--------+---------
x. .. ..     |  2  0 | 60   * |  1  1  0
oo3oo5oo&#a  |  1  1 |  * 120 |  0  1  1
-------------+-------+--------+---------
x.3o. ..     |  3  0 |  3   0 | 20  *  *  x-{3}
xo .. ..&#a  |  2  1 |  1   2 |  * 60  *  {(xaa)}
.. oA5Ao&#za |  5  5 |  0  10 |  *  * 12  a-{10}