'Nasty' Geometry Breaks Decades-Old Tiling Conjecture | Quanta Magazine

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Abstraction: Greenfeld and Tao disprove periodic tiling conjecture with high-dimensional aperiodic tile

Key points:

Rachel Greenfeld (IAS) and Terence Tao (UCLA) disproved the periodic tiling conjecture by constructing a single tile that can fill a high-dimensional space using only translations, but only aperiodically — never periodically
The conjecture was known true in 1D and 2D (Bhattacharya 2016); the disproof required working in an astronomically high-dimensional space, possibly as large as $2^{100^{100}}$
Their construction reframed the tiling problem as a system of equations (like logic circuits built from AND/OR gates), encoding constraints that force any valid tiling to be aperiodic
The resulting tile is described as "nasty" — full of holes and twists, far from optimal; the authors believe aperiodic tiles likely exist in much lower dimensions (possibly 4D)
The result connects to Gödel incompleteness and computability: tiling problems can be undecidable; Greenfeld and Tao believe their techniques may yield a tile for which it is undecidable whether it tiles space at all
Roger Penrose previously showed two tiles suffice for aperiodic plane tiling (kites and darts); this work pushes toward a single tile in higher dimensions

Connections: Terence Tao · Institute For Advanced Study · Aperiodic Tiling · Mathematical Proof · Computational Undecidability