{"id":"openai-math/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three","pdf":"https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026","tags":["tilings","discrete-geometry","periodic-tiling"],"score":0,"title":"A translational tile with no fully periodic tiling in dimension three","domain":"Combinatorics","family":"155","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We construct a finite translational tile in ℤ 3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ 3 , even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":2,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://www.quantamagazine.org/update/168049/#:~:text=Goodman%2DStrauss%20of%20the%20Museum%20of%20Mathematics%20noted","kind":"web","site":"Quanta Magazine","text":"Goodman-Strauss of the Museum of Mathematics noted in an email that a paper describing a three-dimensional aperiodic tile has some novel mathematical ideas but is “unnecessarily difficult to read… I figure this is pretty representative across the corpus.” It is, he notes, not remotely close to…","added":"2026-11-03","title":"Transformation | Quanta Magazine"},{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=that%20admits%20no%20fully%20periodic%20tiling%2C%20reaching%20the%20smallest%20possible%20dimension%20for%20the%20lattice%20problem","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"that admits no fully periodic tiling, reaching the smallest possible dimension for the lattice problem. Covering density of convex bodies. I described the breakthrough of Ordentlich, Regev, and Weiss on efficient lattice coverings in this post . OpenAI announces a universal lattice-covering bound…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans","author":"Gil Kalai"}],"formalized":true,"family_title":"A counterexample to periodic tiling in dimension three"}