{ "_about": "Reconstruction of the marked 3-D chair that Chaim Goodman-Strauss drew on 2026-09-17, with the marking M24 determined by the drawing and verified by computation. See tiling-spec.md.", "_generated": "2026-09-17T13:44:44", "conventions": { "cell": "an integer triple c denotes the closed unit cube [c, c+1].", "chair": "C = [0,2]^3 minus [1,2]^3 = the 7 cells {0,1}^3 minus (1,1,1).", "far_corner": [ 0, 0, 0 ], "notch_cell": [ 1, 1, 1 ], "concave_corner": [ 1, 1, 1 ], "facet": "a unit square of the boundary, written (cell, outward normal); there are 24, of which 21 outer and 3 in the notch.", "site": "the marked corner of a facet: the concave corner (1,1,1) for the three notch facets, otherwise 2*cell. There are 8 sites, each shared by exactly 3 facets.", "pose": "a signed permutation (perm, signs) with (L v)_i = signs_i * v_{perm_i}, plus an integer translation.", "orientations": "the 24 orientation-preserving cubic frames. Reflected copies are NOT allowed (see the caveats in tiling-spec.md)." }, "symbols": { "F": { "name": "purple-filled", "glyph": "solid triangular arrowhead", "matches": "H" }, "H": { "name": "purple-hollow", "glyph": "open hook / V arrowhead", "matches": "F" }, "B": { "name": "blue", "glyph": "pale-blue arrow", "matches": "B" } }, "matching_rule": { "statement": "Two chairs may share a unit facet only if the two marks on it point at the SAME corner of that square and their types are complementary: purple-filled with purple-hollow, blue with blue.", "complement": { "F": "H", "H": "F", "B": "B" } }, "stars": { "000": { "site": [ 0, 0, 0 ], "x": "H", "y": "F", "z": "B" }, "001": { "site": [ 0, 0, 2 ], "x": "F", "y": "H", "z": "B" }, "010": { "site": [ 0, 2, 0 ], "x": "H", "y": "B", "z": "F" }, "011": { "site": [ 0, 2, 2 ], "x": "B", "y": "H", "z": "F" }, "100": { "site": [ 2, 0, 0 ], "x": "B", "y": "F", "z": "H" }, "101": { "site": [ 2, 0, 2 ], "x": "F", "y": "B", "z": "H" }, "110": { "site": [ 2, 2, 0 ], "x": "H", "y": "F", "z": "B" }, "dent": { "site": [ 1, 1, 1 ], "x": "F", "y": "H", "z": "B" } }, "facets": [ { "id": 0, "cell": [ 0, 0, 0 ], "outward_normal": [ -1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ], [ 0, 1, 1 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 0, 0, 0 ], "arrow_tail": [ 0, 1, 1 ], "arrow_head": [ 0, 0, 0 ] } }, { "id": 1, "cell": [ 0, 0, 0 ], "outward_normal": [ 0, -1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ], [ 1, 0, 1 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 0, 0, 0 ], "arrow_tail": [ 1, 0, 1 ], "arrow_head": [ 0, 0, 0 ] } }, { "id": 2, "cell": [ 0, 0, 0 ], "outward_normal": [ 0, 0, -1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 0, 0, 0 ], [ 0, 1, 0 ], [ 1, 0, 0 ], [ 1, 1, 0 ] ], "mark": { "type": "B", "name": "blue", "site": [ 0, 0, 0 ], "arrow_tail": [ 1, 1, 0 ], "arrow_head": [ 0, 0, 0 ] } }, { "id": 3, "cell": [ 0, 0, 1 ], "outward_normal": [ -1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 0, 0, 1 ], [ 0, 0, 2 ], [ 0, 1, 1 ], [ 0, 1, 2 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 0, 0, 2 ], "arrow_tail": [ 0, 1, 1 ], "arrow_head": [ 0, 0, 2 ] } }, { "id": 4, "cell": [ 0, 0, 1 ], "outward_normal": [ 0, -1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 0, 0, 1 ], [ 0, 0, 2 ], [ 1, 0, 1 ], [ 1, 0, 2 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 0, 0, 2 ], "arrow_tail": [ 1, 0, 1 ], "arrow_head": [ 0, 0, 2 ] } }, { "id": 5, "cell": [ 0, 0, 1 ], "outward_normal": [ 0, 0, 1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 0, 0, 2 ], [ 0, 1, 2 ], [ 1, 0, 2 ], [ 1, 1, 2 ] ], "mark": { "type": "B", "name": "blue", "site": [ 0, 0, 2 ], "arrow_tail": [ 1, 1, 2 ], "arrow_head": [ 0, 0, 2 ] } }, { "id": 6, "cell": [ 0, 1, 0 ], "outward_normal": [ -1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 0, 1, 0 ], [ 0, 1, 1 ], [ 0, 2, 0 ], [ 0, 2, 1 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 0, 2, 0 ], "arrow_tail": [ 0, 1, 1 ], "arrow_head": [ 0, 2, 0 ] } }, { "id": 7, "cell": [ 0, 1, 0 ], "outward_normal": [ 0, 0, -1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 0, 1, 0 ], [ 0, 2, 0 ], [ 1, 1, 0 ], [ 1, 2, 0 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 0, 2, 0 ], "arrow_tail": [ 1, 1, 0 ], "arrow_head": [ 0, 2, 0 ] } }, { "id": 8, "cell": [ 0, 1, 0 ], "outward_normal": [ 0, 1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 0, 2, 0 ], [ 0, 2, 1 ], [ 1, 2, 0 ], [ 1, 2, 1 ] ], "mark": { "type": "B", "name": "blue", "site": [ 0, 2, 0 ], "arrow_tail": [ 1, 2, 1 ], "arrow_head": [ 0, 2, 0 ] } }, { "id": 9, "cell": [ 0, 1, 1 ], "outward_normal": [ -1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 0, 1, 1 ], [ 0, 1, 2 ], [ 0, 2, 1 ], [ 0, 2, 2 ] ], "mark": { "type": "B", "name": "blue", "site": [ 0, 2, 2 ], "arrow_tail": [ 0, 1, 1 ], "arrow_head": [ 0, 2, 2 ] } }, { "id": 10, "cell": [ 0, 1, 1 ], "outward_normal": [ 0, 0, 1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 0, 1, 2 ], [ 0, 2, 2 ], [ 1, 1, 2 ], [ 1, 2, 2 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 0, 2, 2 ], "arrow_tail": [ 1, 1, 2 ], "arrow_head": [ 0, 2, 2 ] } }, { "id": 11, "cell": [ 0, 1, 1 ], "outward_normal": [ 0, 1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 0, 2, 1 ], [ 0, 2, 2 ], [ 1, 2, 1 ], [ 1, 2, 2 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 0, 2, 2 ], "arrow_tail": [ 1, 2, 1 ], "arrow_head": [ 0, 2, 2 ] } }, { "id": 12, "cell": [ 0, 1, 1 ], "outward_normal": [ 1, 0, 0 ], "axis": "x", "is_notch_facet": true, "corners": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 2, 1 ], [ 1, 2, 2 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 1, 1, 1 ], "arrow_tail": [ 1, 2, 2 ], "arrow_head": [ 1, 1, 1 ] } }, { "id": 13, "cell": [ 1, 0, 0 ], "outward_normal": [ 0, -1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 1, 0, 0 ], [ 1, 0, 1 ], [ 2, 0, 0 ], [ 2, 0, 1 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 2, 0, 0 ], "arrow_tail": [ 1, 0, 1 ], "arrow_head": [ 2, 0, 0 ] } }, { "id": 14, "cell": [ 1, 0, 0 ], "outward_normal": [ 0, 0, -1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 1, 0, 0 ], [ 1, 1, 0 ], [ 2, 0, 0 ], [ 2, 1, 0 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 2, 0, 0 ], "arrow_tail": [ 1, 1, 0 ], "arrow_head": [ 2, 0, 0 ] } }, { "id": 15, "cell": [ 1, 0, 0 ], "outward_normal": [ 1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 2, 0, 0 ], [ 2, 0, 1 ], [ 2, 1, 0 ], [ 2, 1, 1 ] ], "mark": { "type": "B", "name": "blue", "site": [ 2, 0, 0 ], "arrow_tail": [ 2, 1, 1 ], "arrow_head": [ 2, 0, 0 ] } }, { "id": 16, "cell": [ 1, 0, 1 ], "outward_normal": [ 0, -1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 1, 0, 1 ], [ 1, 0, 2 ], [ 2, 0, 1 ], [ 2, 0, 2 ] ], "mark": { "type": "B", "name": "blue", "site": [ 2, 0, 2 ], "arrow_tail": [ 1, 0, 1 ], "arrow_head": [ 2, 0, 2 ] } }, { "id": 17, "cell": [ 1, 0, 1 ], "outward_normal": [ 0, 0, 1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 1, 0, 2 ], [ 1, 1, 2 ], [ 2, 0, 2 ], [ 2, 1, 2 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 2, 0, 2 ], "arrow_tail": [ 1, 1, 2 ], "arrow_head": [ 2, 0, 2 ] } }, { "id": 18, "cell": [ 1, 0, 1 ], "outward_normal": [ 0, 1, 0 ], "axis": "y", "is_notch_facet": true, "corners": [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 2, 1, 1 ], [ 2, 1, 2 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 1, 1, 1 ], "arrow_tail": [ 2, 1, 2 ], "arrow_head": [ 1, 1, 1 ] } }, { "id": 19, "cell": [ 1, 0, 1 ], "outward_normal": [ 1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 2, 0, 1 ], [ 2, 0, 2 ], [ 2, 1, 1 ], [ 2, 1, 2 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 2, 0, 2 ], "arrow_tail": [ 2, 1, 1 ], "arrow_head": [ 2, 0, 2 ] } }, { "id": 20, "cell": [ 1, 1, 0 ], "outward_normal": [ 0, 0, -1 ], "axis": "z", "is_notch_facet": false, "corners": [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 2, 1, 0 ], [ 2, 2, 0 ] ], "mark": { "type": "B", "name": "blue", "site": [ 2, 2, 0 ], "arrow_tail": [ 1, 1, 0 ], "arrow_head": [ 2, 2, 0 ] } }, { "id": 21, "cell": [ 1, 1, 0 ], "outward_normal": [ 0, 0, 1 ], "axis": "z", "is_notch_facet": true, "corners": [ [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ] ], "mark": { "type": "B", "name": "blue", "site": [ 1, 1, 1 ], "arrow_tail": [ 2, 2, 1 ], "arrow_head": [ 1, 1, 1 ] } }, { "id": 22, "cell": [ 1, 1, 0 ], "outward_normal": [ 0, 1, 0 ], "axis": "y", "is_notch_facet": false, "corners": [ [ 1, 2, 0 ], [ 1, 2, 1 ], [ 2, 2, 0 ], [ 2, 2, 1 ] ], "mark": { "type": "F", "name": "purple-filled", "site": [ 2, 2, 0 ], "arrow_tail": [ 1, 2, 1 ], "arrow_head": [ 2, 2, 0 ] } }, { "id": 23, "cell": [ 1, 1, 0 ], "outward_normal": [ 1, 0, 0 ], "axis": "x", "is_notch_facet": false, "corners": [ [ 2, 1, 0 ], [ 2, 1, 1 ], [ 2, 2, 0 ], [ 2, 2, 1 ] ], "mark": { "type": "H", "name": "purple-hollow", "site": [ 2, 2, 0 ], "arrow_tail": [ 2, 1, 1 ], "arrow_head": [ 2, 2, 0 ] } } ], "substitution": { "supertile": "S = [0,4]^3 minus [2,4]^3 = the chair at scale 2", "children": [ { "slot": "000", "role": "corner", "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, 0 ], "det": 1 }, { "slot": "001", "role": "corner", "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ], "det": 1 }, { "slot": "010", "role": "corner", "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ], "det": 1 }, { "slot": "011", "role": "corner", "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ 0, 4, 4 ], "det": 1 }, { "slot": "100", "role": "corner", "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ], "det": 1 }, { "slot": "101", "role": "corner", "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 4, 0, 4 ], "det": 1 }, { "slot": "110", "role": "corner", "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, 0 ], "det": 1 }, { "slot": "dent", "role": "central", "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 1, 1, 1 ], "det": 1 } ], "note": "These eight frames are FORCED by the requirement that the supertile, read at double scale, carries the same marking." }, "contact_atlas": { "proper": [ { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, -2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, -1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, 1, -1 ], "translation": [ 2, -2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, 1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, -1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ -1, -1, -1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 1, 1, 1 ] }, { "perm": [ 0, 2, 1 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ -1, 3, -1 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ -1, -1, 3 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 0 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3, 1 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, 1, -1 ], "translation": [ 2, -2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, 1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -1, 3, 3 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 3, -1, -1 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] } ], "proper_count": 44, "improper_count": 16, "improper": [ { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 2, 0 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 2, 4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 4, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, -1 ], "translation": [ 4, 2, 2 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, -2 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 2, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, -2 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 2, 4 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 4, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, -1 ], "translation": [ 4, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 2, 4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, -1 ], "translation": [ 2, 4, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, -1 ], "translation": [ 4, 2, 2 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, -2 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, 1 ], "translation": [ 2, 0, 0 ] } ], "note": "the 44 orientation-preserving entries are identical, as a set of poses, to the 44-pose contact atlas A44 reported for Tsiokos's Chair44 (2026), as transcribed in tiling-matching-system.json. That transcription, not the paper itself, is what the comparison is against." }, "notch_fillers": [ { "slot": "110", "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "slot": "000", "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 1, 1, 1 ] }, { "slot": "010", "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "slot": "001", "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "slot": "011", "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "slot": "101", "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "slot": "100", "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] } ], "verification": { "facet_count": 24, "type_census_M24": { "F": 8, "H": 8, "B": 8 }, "marking_symmetry_group_order": 1, "shape_symmetry_group_order": 6, "child_frames": { "(0, 0, 0)": { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, 0 ], 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