Spectre aperiodic monotile or Einstein

This is a Spectre. It is an aperiodic monotile, or "Einstein": A shape that tiles the plane, but only nonperiodically. More specifically, it tiles the plane nonperiodically without using a mix of normal and mirror image shapes. Spectres were discovered in May 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. In this model, there is an "S" embossed in one side to make it easy to identify pieces that have been flipped over.

In March 2023 Smith et al. discovered a family of shapes, including ones called the "hat" and the "turtle", which can tile the plane but only nonperiodically. However, these tilings do require using both the normal and mirror image versions of the shape; in that sense these might be regarded as tilings using two shapes rather than one.

It has now been proved that one shape in that family can tile the plane only nonperiodically if mirror image shapes are not used, though it can tile periodically if a mix of normal and mirror images shapes are used. But the edges of the shape can be varied (in an infinite variety of ways) to prevent any tiling using mixed normal and mirror image shapes; for such shapes, called Spectres, the only tilings are nonperiodic ones using no mirror image shapes. This makes them aperiodic monotiles even under the stricter interpretation that regards mirror image shapes as distinct.

For more information about Spectres see https://cs.uwaterloo.ca/~csk/spectre/.