An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023

An 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. The drawing above shows a patch of hats produced using a few rounds of substitution.

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Sample images

Here are some sample images you can use in publications, media, etc. Feel free to modify these images to suit your tastes.

Creative Commons License
All images, and the MP4 animation above, are licensed under a Creative Commons Attribution 4.0 International License.

A zoomed-in patch of hat tiles, each one decorated with fine lines showing the underlying kites.
[1200x882 PNG] [Scalable PDF]


An alternative patch, more zoomed out and using a warmer palette.
[1200x755 PNG] [Scalable PDF]


A still more zoomed out patch with the same colouring as the first example, with a local centre of threefold rotation in the centre of the drawing.
[1200x1200 PNG] [Scalable PDF]


A looping animated GIF similar to the animation mentioned above. [500x500 GIF]


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