This tile can tile the plane, but only in a non-periodic manner. This was proven by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss on March 20, 2023.

This is a solution to the famous "einstein problem" (no relation to Albert Einstein: https://en.wikipedia.org/wiki/Einstein_problem).

The paper can be found here: https://arxiv.org/pdf/2303.10798.pdf