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.