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 https://en.wikipedia.org/wiki/Tetrad_(geometry_puzzle) (new Wikipedia article, not by me).
@11011110 the wikipage links to a site that provides multiple pentagonal tetrads: https://www.trump.de/tetrads/question-14/question-14.htm
@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.
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!