
TL;DR
This paper introduces a convex tile set that can tile the plane in infinitely many noncongruent ways, addressing a longstanding open problem in tiling theory.
Contribution
It provides the first known construction of a convex $\sigma$-morphic tile set, expanding the understanding of tiling possibilities with convex shapes.
Findings
Constructed a convex $\sigma$-morphic tile set
Demonstrated infinite noncongruent tilings with convex tiles
Extended tiling theory beyond non-convex constructions
Abstract
We say that a tile is -morphic if it tiles the plane in exactly many noncongruent ways (up to an isometry). It is an unsolved problem of whether a -morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is -morphic. The result is interesting since all the constructions of -morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cleverly dividing the tiles of the set of tiles discovered by Schmitt into convex tiles so that they behave in the same manner.
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