Classification of Unilateral and Equitransitive Tilings by Squares of Three Sizes

Abstract: After a break of 20 years and with the help of the fundamental book [A], the study of unilateral, equitransitive tilings of the plane by squares of three sizes was revived. First D. Schattschneider had found five possible tilings [A, p. 76] and Martini, Makai and Soltan generally characterized the unilateral tilings and obtained a new equitransitive arrangement [B]. They also described two other tilings constructed by B. Grünbaum. The problem to describe all possibilities remained open. In this paper we shall derive all the unilateral and equitransitive tilings using the classification of the fundamental planigons. We prove, that only the eight known tilings are possible. We have learned that D. Schattschneider parallelly solved this problem, too. Our method is different from hers. \item{[A]} Grünbaum, B.; Shepard, G. C.: Tilings and Patterns. W. H. Freeman 1987, 72-81. \item{[B]} Martini, H.; Makai, E.; Soltan, V.: Unilateral Tilings of the Plane with Squares of Three Sizes. Beitr. Algebra Geom. 39 (1998), 481-495.

Keywords: tiling of the plane by squares, unilaterality, equitransitivity, planigon, plane group

Classification (MSC2000): 52C20

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