Last year we reported on “The hat”: A single shape that can tile the plane aperiodically but not periodically. A commenter pointed out that the aperiodic “hat” tiling included mirror reflections, which led to the question of whether there’s a single tile that can do the job without flipping.
Bob points us to the answer, pictured above from this source: It’s called “the spectre” and it’s an aperiodic tiling for which “reflected copies of the tile are not needed to form a tiling and no tiling with unreflected copies has a repeating pattern.”
Here’s the research paper, A chiral aperiodic monotile, by David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss.
So cool!
As before, I haven’t checked this result myself, but I have no reason to doubt it.
There is quite a cool youtube video about it (if you ignore the advertising) with some of the discoverers/authors – https://www.youtube.com/watch?v=A1BhOVW8qZU
That was a very interesting video, though it left me wanting to learn more about David Smith and his seemingly genius talent for shapes. The late Seth Roberts would have loved the fact that David Smith is an amateur with no academic affiliation, and yet he has made valuable contributions to mathematics.