Mathematicians Unveil a 3D Monotile That Never Repeats
A Strongly Aperiodic Monotile in Three Dimensions
Socolar and Taylor asked for a single, simply connected 3D tile that forces nonperiodicity by shape alone. The new rational polyhedral 3-ball, Chair44 (R44), is a seven-cube chair whose exposed panels carry tiny square-pyramid features. It tiles space by congruent copies, yet every tiling lacks translational period, has symmetry group of order at most 24, is homochiral, and carries a unique infinite hierarchy of nested supertiles. The proof combines geometric argument with exhaustive finite enumerations, kernel-checked in Lean 4.
The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus, 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.