  In the Escherization problem, given a closed figure in a plane, the objective
is to find a closed figure that is as close as possible to the input figure and
tiles the plane. Koizumi and Sugihara's formulation reduces this problem to an
eigenvalue problem in which the tile and input figures are represented as
$n$-point polygons. In their formulation, the same number of points are
assigned to every tiling edge, which forms a tiling template, to parameterize
the tile shape. By considering all possible configurations for the assignment
of the $n$ points to the tiling edges, we can achieve much flexibility in terms
of the possible tile shapes and the quality of the optimal tile shape improves
drastically, at the cost of enormous computational effort. In this paper, we
propose an efficient algorithm to find the optimal tile shape for this extended
formulation of the Escherization problem.
