A $\sigma$-morphic convex protoset

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 $\sigma$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $\sigma$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $\sigma$-morphic. The result is interesting since all the constructions of $\sigma$-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.