Apparently it is unknown whether one can find a pentagon so that four congruent copies can be placed in the plane with each pair sharing a nonzero length of boundary. It is possible with hexagons: see (new Wikipedia article, not by me).

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@pierrebai So it does, thanks. I wonder why the wikiarticle doesn't mention that. The question of whether it's possible without leaving a hole does still seem to be open.

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