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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 This might be a good (but outdated) starting point.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

Original references (Wang's "Proving theorems by pattern recognition" and Berger's "The undecidability of the domino problem") are not very friendly. You may find some reviews around, though.

I'll suggest you to take a look at Jarkko Kari's website where you will find some introductory material and these slides for some basic notions and references.

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