Tilings and patterns
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Let A and B be convex polygons. We say that A and B are D-equivalent if there are convex polygons A = A 1 ,A 2 ,...,A n = B and Dudeney dissections of A i to A i + 1 (1 ≤ i ≤ n *** 1). A polygon is called a tile if the 2-dimensional Euclidean plane can be tiled by congruent copies of the polygon. A polygon is called a normal tile if the plane can be tiled by congruent copies of the polygon which are obtained without turning over the polygon. The numbers of types of convex tiles and convex normal tiles are still uncertain. In this paper, we prove that all convex normal tiles with the same area that we know so far are D-equivalent .