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This hypercompact hyperbolic tesselation uses hihexat in the sense of an infinite bollohedron as cell.

This hemiation indeed is uniform again, because all edges by mere alternation already do have the same size: diagonals of either of the former squares. I.e. no afterwards edge resizements are required (except of a homogenuous global scaling).

Incidence matrix according to Dynkin symbol

o3o3o4s4*b   (N,M → ∞)

demi( . . . .    ) | NM ♦  12   6 |  12  24 |  4  4 12
-------------------+----+---------+---------+---------
      . o . s4*b   |  2 | 6NM   * |   2   2 |  1  1  2
      . . o4s      |  2 |   * 3NM |   0   4 |  0  2  2
-------------------+----+---------+---------+---------
sefa( o3o . s4*b ) |  3 |   3   0 | 4NM   * |  1  0  1
sefa( . o3o4s4*b ) |  6 |   3   3 |   * 4NM |  0  1  1
-------------------+----+---------+---------+---------
      o3o . s4*b   ♦  4 |   6   0 |   4   0 | NM  *  *
      . o3o4s4*b   ♦ 2M |  3M  3M |   0  2M |  * 2N  *
sefa( o3o3o4s4*b ) ♦ 12 |  12   6 |   4   4 |  *  * NM

starting figure: o3o3o4x4*b