**Beiträge zur Algebra und Geometrie
**

Contributions to Algebra and Geometry

Vol. 39, No. 2, pp. 481-495 (1998)

#
Unilateral Tilings of the Plane with Squares of Three Sizes

##
Horst Martini, Endre Makai, Valeriu Soltan

Fakultät für Mathematik, TU Chemnitz, D-09009 Chemnitz, Germany, e-mail: Martini@mathematik.tu-chemnitz.de
Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest, Pb. 127, Hungary, e-mail: makai@math-inst.hu

Mathematical Institute of the Academy of Sciences of Moldova, MD-2028 Chisinau, Moldova, e-mail: 17soltan@mathem.moldova.su

**Abstract:** \font\calf=cmsy10 \def\cal#1{\hbox{\calf #1}} A tiling $\cal H$ of the Euclidean plane by squares of three distinct sizes is said to be a unilateral 3-tiling (or U3-tiling) if no two congruent tiles have a common side, and it is called equitransitive provided for any pair $S$, $S'$ of congruent tiles of $\cal H$ there is a congruence transform of the plane mapping $S$ onto $S'$ and also mapping $\cal H$ onto itself. Contrary to the assertion by D. Schattschneider that there are only five equitransitive U3-tilings, we present three new such tilings (two of them are due to B. Grünbaum). We describe the local environments of each tile in a U3-tiling with their global extensions, as well as further global properties of U3-tilings.

**Keywords:** tiling of the plane, squares, unilaterality, equitransitivity

**Classification (MSC91):** 52C20

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