Edited by Jane Rosemarin
Sy Chen’s origami Einstein Hat. The lower photo outlines the eight kite shapes that form the tile’s face. An assembly of these tiles forms a non-repeating pattern. Folded and photographed by the author. See PDF diagrams.

What Is an Einstein Hat?

This 13-sided, hat-shaped polygon is the first known example of an “einstein,” a single shape that can be tiled to cover a plane with a pattern that never repeats. The word means one stone in German.

David Smith published his discovery of the polygon on March 20, 2023, in an article entitled “An aperiodic monotile.” The article appeared on Cornell University’s arXiv.org website, an archive of non-peer-reviewed research papers. In addition to Smith, who calls himself a shape hobbyist, there are three academic co-authors.

The Origami Version

After several failed attempts with a modular approach using either a four-unit diamond shape or an eight-unit kite shape, I started to work on a single-piece version. Finally, I came up with the one-piece solution published here.

Editor’s note: Smith later found another einstein shape, and Sy recently discovered an origami version of this second einstein hat. Both versions can be found in Sy’s diagrams.

Tiling Challenge

Putting “the hat” pieces together turns out to be quite a challenge without any guidelines or templates. The following tiling arrangement is one example. A second color was used for the mirrored shapes.

A 41-hat assembly. The black tiles are mirror images of the red tiles. Folded and photographed by the author.


A downloadable copy of “An aperiodic monotile” by Smith et al. is available on the arXiv website.

An article about the einstein hat is on CNN’s website: “Newly discovered ‘einstein’ shape can do something no other tile can do” by Jacopo Prisco.

A more expansive article, “Elusive ‘Einstein’ Solves a Longstanding Math Problem” by Siobhan Roberts, was published in the New York Times on March 28, 2023. Access requires a subscription.