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An Aperiodic Monotile (2023)
aperiodic monotile, sometimes called an "einstein", is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions.
We prove that this shape, a polykite that we call "the hat", must assemble into tilings based on a substitution system. You can create your own patches of hats, and save them as PNG or SVG files, using an interactive application that runs in your web browser. Separately, we have created a second interactive application that lets you construct patches of hats based on the H7/H8 substitution rules shown in Figure 2.11 of the paper.
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