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Suppose I have a regular grid of identical hexagons that tile the plane. How can I determine if a connected subset of these hexagons can tile the plane by translations that form a (Bravais) lattice? For reference, here is a picture:

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The four shapes on the right can tile the plane, while the left-most shape cannot.

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You mean just translations, no rotations allowed? That may be decidable. –  Will Jagy Mar 12 '13 at 4:14
    
That is correct, only translations, and no rotations. Not only that, but the overall tiling should be periodic. –  Victor Liu Mar 12 '13 at 5:01
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The polyhexes that tile by translation, up to order 9, can be found by links from polyomino.org.uk/mathematics/polyform-tiling --- e.g., there are 6572 polyhexes of order 9, of which only 387 tile by translation. There are also references there to the literature. See also mysite.verizon.net/vze16nctz/Tilings/Article/planar.ps –  Gerry Myerson Mar 12 '13 at 5:04

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