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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}