A translational tile with no periodic tiling in dimension three

A finite set T in Z^3 tiles Z^3 by translations, but no tiling of Z^3 by T is invariant under a subgroup of finite index, and the union of the unit cubes at the points of T has the same property in R^3, even with arbitrary real translations. An arithmetic rule for arrays along lines of integer slope has a solution with no constant column but none with a vertical period; tiling equations in Z^2 x V with V cyclic encode the rule through the dependence of outputs on input digits; one explicit common solution, a stacking with a fresh prime and a rigid passage to three dimensions complete the proof. The course is an exposition of OpenAI's proof (2026) that the periodic tiling conjecture fails in dimension three.

Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Tilings of abelian groups and periodicity
2. An arithmetic word rule with no vertical period
3. Encoding symbols by tiling equations
4. Forcing activity and excluding periodicity
5. A common solution of the tiling equations
6. One tile with a cyclic finite coordinate
7. Passing to three-dimensional space

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Groups, cosets, quotient groups, products of cyclic groups and the Chinese remainder theorem (core course Abstract Algebra I).
• Bases and dimension of Rⁿ (core course Linear Algebra); Lebesgue measure on R³ (core course Measure and Integration).
• Finite probability spaces: independence and the union bound.