Families of irreptiles

The search for tilings of the plane by congruent images of some given polygon A leads in a natural way to the concept of a reptile. A is called a reptile if it can be dissected into finitely many pairwise congruent pieces which are similar images of A. We speak of a dissection if the covering pieces can only have boundary points in common. Many known examples of reptiles are polyominoes or polyiamonds. A polygon is called a polyomino (polyiamond) if it has a connected interior and possesses an edge-to-edge dissection into finitely many congruent squares (equilateral triangles). A family of reptiles that are polyominoes can be obtained as follows (see [2], [3, p. 97], [6, p. 54], and the first illustration in Fig. 1). Fix an integer k ≥ 1 and dissect a square S into (2k)2 congruent smaller squares S1, . . . , S(2k)2 . Let δc denote the rotation about the centre c of S by an angle of π2 . Now choose a simple polygonal arc contained in the union of the boundaries of the pieces Si that connects c with a point on the boundary of S such that ∩ δc( ) = {c}. Then , δc( ), and a quarter of the boundary of S bound a reptile A ⊆ S (shaded in Fig. 1). Indeed, since A splits into k2 congruent squares Si and since S as well as any other square admits a dissection into four congruent similar copies of A, A can be