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The hull of each sheet of this polyhedron allows also for an all unit edge variant. that one then is the expanded rhombic triacontahedron. Thereby then the squares {4_o} become rhombs, while the trapezia {(T_o²,t_o²)} become squares.

This toroidal polyhedron can be viewed as finite modwrap of the infinite hyperbolic tiling {4,5} (pesquat).

Incidence matrix

60  *  *  * |  2  1   2  0   0  0  0  0 |  2  2  1  0  0  0  [(To,4o,To,t32)]
 * 60  *  * |  0  1   0  2   2  0  0  0 |  2  0  0  2  1  0  [(Ti,4i,Ti,T32)]
 *  * 60  * |  0  0   2  0   0  2  1  0 |  0  2  1  0  0  2  [(to,4o,to,t52)]
 *  *  * 60 |  0  0   0  0   2  0  1  2 |  0  0  0  2  1  2  [(ti,4i,ti,T52)]
------------+---------------------------+------------------
 2  0  0  0 | 60  *   *  *   *  *  *  * |  1  1  0  0  0  0
 1  1  0  0 |  * 60   *  *   *  *  *  * |  2  0  0  0  0  0
 1  0  1  0 |  *  * 120  *   *  *  *  * |  0  1  1  0  0  0
 0  2  0  0 |  *  *   * 60   *  *  *  * |  1  0  0  1  0  0
 0  1  0  1 |  *  *   *  * 120  *  *  * |  0  0  0  1  1  0
 0  0  2  0 |  *  *   *  *   * 60  *  * |  0  1  0  0  0  1
 0  0  1  1 |  *  *   *  *   *  * 60  * |  0  0  0  0  0  2
 0  0  0  2 |  *  *   *  *   *  *  * 60 |  0  0  0  1  0  1
------------+---------------------------+------------------
 2  2  0  0 |  1  2   0  1   0  0  0  0 | 60  *  *  *  *  *  {(To2,Ti2)}
 2  0  2  0 |  1  0   2  0   0  1  0  0 |  * 60  *  *  *  *  {(To2,to2)}
 2  0  2  0 |  0  0   4  0   0  0  0  0 |  *  * 30  *  *  *  {(To,to)2}
 0  2  0  2 |  0  0   0  1   2  0  0  1 |  *  *  * 60  *  *  {(Ti2,ti2)}
 0  2  0  2 |  0  0   0  0   4  0  0  0 |  *  *  *  * 30  *  {(Ti,ti)2}
 0  0  2  2 |  0  0   0  0   0  1  2  1 |  *  *  *  *  * 60  {(to2,ti2)}

(combinatorically)

240 |   5 |   5
----+-----+----
  2 | 600 |   2
----+-----+----
  4 |   4 | 300