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 up to a relabelling, which the notch marks in the sketch settle. At the level the page opens on, the reconstructed ones are the marks at the six corners of the hexagon.)

• 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.

Which way round. The marked tile is chiral: its mirror image satisfies everything on this page equally well, so only the drawing says which of the two it is. CGS wrote the axes beside his single-tile figure — +x to the lower left, +y to the right — and read with those, the pit shows the hollow purple V on its left wall and the solid head on its right. That is what is drawn here.

What has been checked by computer (scripts and log in verify/, write-up in tiling-spec.md): 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 up to rotation — the same 44 poses as the contact atlas reported for I. Tsiokos's independent bumps-and-dents realization, with the three marks matching his three motif classes panel for panel — and 16 more if reflected copies are allowed, which the arrows cannot rule out; and of the 1 679 616 candidate markings only 18 are consistent with the substitution, of which exactly one agrees 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, so the 16 reflected neighbours are excluded by fiat rather than by the rule, and registration (facets meeting square on) is assumed rather than derived.

Construction explained by Chaim Goodman-Strauss in conversation, 17 September 2026 (unpublished); the marking is reconstructed from a hand sketch — errors are ours. The tile and its substitution are the n = 3 case of C. Goodman-Strauss, An aperiodic pair of tiles in Eⁿ for all n ≥ 3 (1999); the 44 poses above are compared against the atlas reported for I. Tsiokos, A Strongly Aperiodic Monotile in Three Dimensions (Zenodo, 2026).
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