A Strongly Aperiodic Monotile in Three Dimensions

We exhibit a rational polyhedral 3-ball, Chair44 (R44), that tiles Euclidean three-space by congruent copies while every such tiling has no nonzero translation period and has a symmetry group of order at most 24. Every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles.

The tile is a seven-cube chair whose exposed panels carry small rational square-pyramid features. The proof combines written geometry with exhaustive finite enumerations, independent replay implementations, and a Lean 4 formalization. The decisive finite result is that the tile's admitted parent contact language, after rescaling, is exactly its original 44-contact language.

The accompanying repository provides the exact solid, certificates, verification code, formal development, paper source, viewer, simulations, and reader utilities:
https://github.com/ioannist/six-birds-tiles

Cryptographic release checkpoint (15 September 2026). The repository source is sealed by the parentless Git commit 137e46b15d36266c37879478cfc62af6e4469147, with root tree b7882f5054ba9f0e56a1a26078d651e75b347728 and signed paper-v1 tag object 9134e6a07fc1d6cb1f3ed5bf0aacbbe6d8cb7da3. The signed external release manifest has SHA-256 9f9b73d115cf34991e8100e255cf3c6cc0485d1a3004b188e0657c036715613e and is recorded in the Sigstore transparency log at Rekor index 2841963304. An OpenTimestamps commitment has also been submitted. The verification files accompany the release artifacts.