The tile. A 2×2×2 cube with one of its eight cells taken out — a chair, the three-dimensional L-tile. Each of its 24 unit facets carries exactly one of three marks, drawn along that facet's diagonal and pointing at one of its corners. Three facets meet at each of the eight marked corners, and each corner gets one of each mark. (The sketch draws 21 of the 24; the three it leaves blank are forced by the two facts below, and are reconstructed here.)

• filled purple
• hollow purple
• blue

The rule. Two chairs may share a facet only when their two marks point at the same corner of it and the marks are complementary:

filled purple meets hollow purple · blue meets blue

What Chaim Goodman-Strauss pointed out. First: eight chairs are forced together into a chair twice as big, seven around the outside and one in the middle, plugging the notch. Second: the marks on the corners of that big chair are exactly the marks of the middle one — so the big chairs obey the same rule, are forced into chairs twice as big again, and the argument repeats at every scale.

What has been checked by computer (details in tiling-spec.md, scripts in verify/): the substitution is consistent at levels 1–3 (12 000 facet contacts, no violation); the supertile read at double scale carries the same marking, exactly, at levels 1 and 2; a chair and the copy shifted by (1, 1, 1) into its notch match; a chair has exactly 44 legal neighbours, the same 44 poses as the certified atlas of I. Tsiokos's independent bumps-and-dents realization, with the three marks matching his three motif classes panel for panel; and of the 1 679 616 candidate markings this is the only one agreeing with every symbol readable in the sketch.

What has not been proved. Aperiodicity. Forcing is proved only as a local theorem — from the shell of neighbours around a chair you can tell whether it is a middle one — and the induction to aperiodicity is not formalised. The marks are also blind to reflection, and registration (facets meeting square on) is assumed rather than derived.

Construction explained by Chaim Goodman-Strauss in conversation, 17 September 2026 (unpublished); reconstructed from a hand sketch — errors are ours. Context: I. Tsiokos, A Strongly Aperiodic Monotile in Three Dimensions (Zenodo, 2026); C. Goodman-Strauss, An aperiodic pair of tiles in Eⁿ for all n ≥ 3 (1999).
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