A single shape that tiles the plane but never repeats — discovered March 2023
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In March 2023, Smith, Myers, Kaplan & Goodman-Strauss announced the "Hat" — the first single shape (monotile) proven to tile the plane only aperiodically.
Unlike Penrose tiles (which require 2 shapes), the Hat needs just one. It's a polykite made of 8 kite pieces arranged in a specific pattern.
The tiling uses both the Hat and its mirror image — the ratio of reflected tiles follows the silver ratio. A purely chiral version (Spectre) was found shortly after.
Inflation rule: Each Hat can be replaced by a cluster of Hats at a larger scale, proving the tiling extends infinitely without repeating.