I read that among the 24 heptiamonds there is one piece that does not tile the Euclidean plane. My question is the following, given a particular polyiamond how do you prove that the piece does tile the Euclidean plane? ( I thought about arranging the pieces and possibly stretching the piece to a form of which it is known that it tiles the plane, i.e. triangle, square, parallelogram or hexagon. )
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The V heptiamond will not tile the plane. All others will.
The proof of tiling is usually to find a tiling. That's often very simple for the smaller iamonds, polyhexes, and polyominoes.
For larger polyforms, such as the wheelbarrow polyhex, finding the tiling can be quite tricky. See the tiling database for more.