This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: Software developer Ioannis Tsiokos has reported the discovery of a three-dimensional shape, which roughly resembles a chair, that can fill space infinitely without repeating a pattern. The shape follows certain mathematical rules that make it the first of its kind in three dimensions.
The AI-assisted findings are described in an arXiv preprint paper. Mathematicians have long been on the lookout for aperiodic monotiles—shapes that can fill space through tiling without gaps or overlaps, and without any globally repeating arrangements. The usual rules allow for locally recurring patterns, but shifting the entire tiling by a nonzero distance and recovering exactly the same arrangement is forbidden.
A two-dimensional version of such a shape was reported in 2023. This "hat" tile was a 13-sided polykite and was proven to be the first true aperiodic monotile. Later the same year, another 2D version, called the "spectre" tile, was reported.
The hat tile required its mirror image to properly tile the plane without gaps, while the spectre did not. Still, researchers wanted to go a step further and find a 3D aperiodic monotile, particularly one that excluded endlessly repeating combinations of rotation and movement, called screw motions. Some 3D candidates existed, but they retained a repeating stacking direction or allowed screw-motion symmetry.
The goal was a single connected solid whose geometry alone prevents these forms of repetition, without printed markings or extra assembly rules. By entering the extensive rules and framework required for a 3D aperiodic monotile into OpenAI's reasoning model, ChatGPT Astra, Tsiokos found a shape that fit the correct definition. According to Tsiokos, the shape, called "Chair44," can fill all three-dimensional space without a nonzero translational period.
The shape looks like a block of eight cubes with one corner removed. Its surface has 192 tiny pyramid-shaped bumps and dents that constrain how copies fit. Every allowed tiling with Chair44 has at most 24 symmetries, excluding screw motions and any other symmetry that can generate infinitely many distinct transformations.
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