A Strongly Aperiodic Monotile in Three Dimensions

View PDF HTML (experimental) Abstract:Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction. We exhibit a rational polyhedral $3$-ball $Q$, which we call Chair44 (R44): a seven-cube chair whose $24$ exposed unit panels carry tiny square-pyramid features, and prove, as a proof submission, that $Q$ admits tilings of $\mathbb{R}^3$ by congruent copies, reflections allowed, and that every such tiling has no translational period and a symmetry group of order at most $24$; every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus (Section 3.3), and the construction turns on a single finite test, checked by machine: the tile's own contact rule survives coarsening, so that the decoded parent tiling obeys the tile's rule and no other. The proof combines a written geometric argument, that the features force every tiling onto a registered lattice, with exhaustive finite enumerations; the companion census is replayed by two independent implementations, every finite gate is kernel-checked in Lean 4 (modulo a named compiler hook per native_decide theorem), and the written geometric lemmas and the logical assembly are Lean theorems as well, so that the theorem is kernel-checked modulo the named compiler hooks; the written proofs remain as exposition. Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Metric Geometry (math.MG) Cite as: arXiv:2609.19214 [math.CO] (or arXiv:2609.19214v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2609.19214 arXiv-issued DOI via DataCite (pending registration) Submission history From: Ioannis Tsiokos [view email] [v1] Wed, 16 Sep 2026 12:15:33 UTC (182 KB)