Maximum Eigenvalue Problem for Escherization

A tiling of the plane is a collection of figures, called tiles, that cover the plane without gaps or overlaps except at their boundaries. Tilings have been used for various purposes, such as ornamentation for construction and design of clothes. The patterns made by tilings range from those made by simple geometrical figures to those made by intricate figures, and they attract a great deal of interest from both artists and mathematicians. From a mathematical point of view, the placement rules, incidence types, and other properties of tilings have been considered deeply [1]. The Dutch woodblock artist M. C. Escher is one of the greatest artists of artistic tilings. His works include many tilings made of animal forms based on his own trial-and-error approaches. Kaplan and Salesin [3] first introduced the problem of finding tilings automatically in which the tiles are similar to a given shape. This problem is called the Escherization problem, named after Escher and his elegant work. More precisely, the Escherization problem is as follows.