Most shapes that tile the plane do it the boring way: find a patch, repeat it forever. For sixty years the question was whether a single shape could tile the plane while never repeating — and the record crept down from 20,426 tiles to 104, to 6, to 2, and stopped. In November 2022 David Smith, a retired print technician cutting shapes out of cardboard, found one. Across 41 interactive demonstrations, build tilings by substitution, force aperiodicity with edge rules, slice a quasicrystal out of a higher-dimensional lattice, and grow the hat and the spectre yourself.
See also: Symmetry for the wallpaper groups a periodic tiling must belong to, Fractals for other things built by substitution, and Sphere Packing for another question about filling space that stayed open for centuries.
Cover the plane with copies of one shape, and try to make it repeat
Cut every tile into smaller copies, inflate, and repeat forever
Decorate the edges so the only tilings left are the ones that never repeat
Two tiles, fivefold symmetry, and the golden ratio hiding in the counts
Take an irrational slice through a lattice and a quasicrystal falls out
Sixty years of shrinking the record: 20,426 tiles, then 104, then 6, then 2
One shape, thirteen sides, found by an amateur with scissors in 2022
Four clusters, one substitution, and an inflation factor that cannot be rational
The hat needs its mirror image. Months later, a tile that does not
The hat is one point on a continuous path, and almost every point works