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Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set.

Now, I wish to characterize all the periodic tilings of this set (better if they are characterized by the appearing frequency of each type of the tiles)... Is there some way that I could do any such kind of characterization?

Thank you.

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I gather that you are mainly asking about which lists of 16 numbers adding to 1 could be the relative proportions . restricting to the 4 tiles having 1 on the top and bottom edges might be a good start. it is impossible to have the proportions be $1/3$ each for having sides (left right) $00$ $01$ $10$ so a certain obvious condition is not sufficient. –  Aaron Meyerowitz May 3 at 2:06

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