Papers · Preprint
A Strongly Aperiodic Monotile in Three Dimensions
A single solid — the seven-cube “chair” Chair44, with tiny features on its faces — that fills space but never repeats, answering the Socolar–Taylor question for a simply connected three-dimensional tile.
In plain words
Take a 2×2×2 block of eight unit cubes and remove one corner. What is left is a seven-cube “chair.” Cover its twenty-four exposed faces with a fixed pattern of tiny square bumps and dents. That single shape is Chair44.
Copies of Chair44 — turned any way, and mirror images allowed — fill all of three-dimensional space with no gaps. But the bumps and dents act as matching rules: they let copies meet in only one way, and that forces every tiling to be non-periodic. Slide a filled space by any whole vector and it never lands back on itself.
Socolar and Taylor asked for exactly this: one simply connected three-dimensional shape that forbids periodicity by its geometry alone. Earlier candidates leaked — the Schmitt–Conway–Danzer biprism allows a screw motion, and the 3D Socolar–Taylor tile allows a periodic stacking direction. Chair44 is offered as a shape with no such escape.
What it shows
- Chair44 admits tilings of ℝ³ by congruent copies, reflections allowed.
- 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 whole construction reduces to one finite test: the tile’s own contact rule survives coarsening, so the decoded parent tiling obeys that rule and no other.
How it is checked
The proof pairs 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 geometric lemmas and the logical assembly are Lean theorems as well. The design itself came from reading the aperiodic-monotile phenomenon through the Six Birds emergence calculus.
What it does not claim
This is a proof submission — a review version. The finite gates are machine-checked, but the written geometric lemmas have not been externally reviewed, and the Lean result holds modulo the named compiler hooks. It is not peer reviewed.
See it and run it. An interactive 3D viewer lets you rotate the tile and inspect the feature pattern: the Chair44 toy. A plain-language walkthrough is in the post What the 3D Einstein Cannot Erase. The construction, census and Lean checks live in six-birds-tiles.
Cite
Tsiokos, I. (2026). A Strongly Aperiodic Monotile in Three Dimensions. Zenodo. https://doi.org/10.5281/zenodo.22792358