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Given any set of tiles (jordan curves) that can tile the plane, how to prove that the number of possible tilings using tiles from this set is either in bijection with the real numbers or a (possibly infinite) subset of the integers?

Two tilings are equal if they can be made to coincide by translations or rotations.

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Crossposted from Math.SE: math.stackexchange.com/q/71110 –  Theo Buehler Oct 10 '11 at 0:53
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Closed sets in complete separable metric spaces are either countable or of cardinality continuum. Once we fix the position of the first tile, the set of possible positions of each other tile is closed by the usual compactness argument (assuming your tiles are reasonable, e.g., closed sets with boundaries of zero area, etc.). –  fedja Oct 10 '11 at 1:03

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