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I am working on projects in solving ground state of generalized ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results: enter image description here

every row corresponds to one constructible structure with basic tiles being "Generating configurations of the structure".

The resulting structures could be shown as followed:

enter image description here

9 and 12 are more complicated (can be periodic or aperiodic) shown as followed: enter image description here enter image description here

I realize (with the help of community), this is actually related to Wang Tiles. So the first few steps I wish to approach this problem are looking into all established theorems on Wang Tiles. Could someone suggest some relevant reference?

(PS:From Engineering respective, I think we (in engineer department) commonly would agree that as long as we could keep tiling the plane to a sufficiently large extent( e.g use 100*100 tiles without conflict), then we blindly think these few basics tiles could tile the plane. So the undecidability is not too crucial...)

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The book "Tilings and Patterns" by Grünbaum and Shephard has a section on Wang tiles. This might be a good (but outdated) starting point. – Gregor Samsa Nov 21 '13 at 21:50
For some context, see author's earlier post, – Gerry Myerson Nov 21 '13 at 22:22

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