Is there a strongly aperiodic monotile in three dimensions?

Claimed yes: Chair44 (R44), a rational polyhedral 3-ball built from seven cubes in a chair, whose 24 exposed unit panels carry small square-pyramid features. The claim is that it tiles $\mathbb R^3$ by congruent copies with reflections allowed, that every such tiling has no translational period and a symmetry group of order at most 24, and that every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The features force each tiling onto a registered lattice; the finite facts - dissection, the 44-contact atlas, 33 first shells, a 6,862 to 44 companion census with 299,975 collision boxes, mesh and angle audits - are enumerated exhaustively and checked in Lean, with a second independent Python implementation replaying them.

Claimed yes: Chair44 (R44), a rational polyhedral 3-ball built from seven cubes in a chair, whose 24 exposed unit panels carry small square-pyramid features. The claim is that it tiles $\mathbb R^3$ by congruent copies with reflections allowed, that every such tiling has no translational period and a symmetry group of order at most 24, and that every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The features force each tiling onto a registered lattice; the finite facts - dissection, the 44-contact atlas, 33 first shells, a 6,862 to 44 companion census with 299,975 collision boxes, mesh and angle audits - are enumerated exhaustively and checked in Lean, with a second independent Python implementation replaying them.