Worked solution: Smith–Myers–Kaplan–Goodman-Strauss's aperiodic monotiles: the hat and the Spectre (2023)
Take a standard tiling of the plane made from identical kite shapes — quadrilaterals that look like the paper kites children fly — arranged around every vertex in a repeating pattern. Now glue exactly eight of these kites together, in a specific pattern, into a single -sided outline. That single glued-together piece is the "hat".
The surprise is that this one modest-looking shape can tile the entire infinite plane with no gaps or overlaps, yet it can never do so in a way that repeats itself by sliding — no matter how you arrange copies of it, the pattern never has translational symmetry.
In March 2023, David Smith (an amateur tiling enthusiast) discovered the "hat", a -sided polykite formed from eight kite tiles of the Laves tessellation (the dual of the Archimedean tiling). Together with Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss, he conjectured it could tile the plane, but only aperiodically.
Smith's initial computer experiments (using Kaplan's Heesch-number software and Myers's polyform-tiling software) already showed something extraordinary before any proof existed: if the hat failed to tile the plane, its Heesch number would have to be at least — far beyond the previous record of — and if it tiled only periodically, its isohedral number would have to be at least . Either outcome alone would have been a major discovery, which is what convinced the team the shape deserved a full proof.
The rest of this proof walks through two independent arguments (Sections 3 and 4–5 of Smith, Myers, Kaplan & Goodman-Strauss 2023) that settle the question: the hat admits tilings of the plane, and none of them can be periodic.
- Polykite
- A shape formed by gluing together several copies of a kite (a quadrilateral with two pairs of adjacent equal sides) from a fixed kite tiling, edge-to-edge.
- Heesch number and isohedral number
- The Heesch number of a shape is the largest number of complete concentric rings of copies that can surround it without gaps (infinite if it tiles the plane); the isohedral number of a tiling is the smallest number of orbit classes its tiles fall into under the tiling's symmetries.