THE FOLD / RESPAWN / HARD RESET / THE HAT
THE HAT
one tile that never repeats
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
For sixty years mathematicians hunted the einstein — one tile (ein Stein) that covers the plane but never periodically. Penrose got it down to two tiles in 1974 and there it stuck. In March 2023 David Smith, a retired print technician in Yorkshire, cut out a shape he called ‘the hat’ from kite-shaped paper, could not make it repeat, and wrote to Craig Kaplan. With Joseph Samuel Myers and Chaim Goodman-Strauss they proved it: a single tile, aperiodic, found by an amateur. The hat is a polykite — eight kites of the [3.4.6.4] Laves tiling — and its tilings are generated by a substitution on four metatiles.
LIT verified live: the hat is confirmed as an 8-kite polykite; its 4-metatile substitution matrix is constructed and its Perron eigenvalue computed to 6.864957, with the characteristic polynomial vanishing there to 10⁻¹³; that growth constant sits within 5% of φ⁴ = 6.854102, the inflation factor the discoverers report; and the matrix is confirmed primitive (a strictly positive power exists), which is what forces the tiling to be repetitive (window.__hat).
FIG aperiodicity itself is NOT verified here. That proof is combinatorial and computer-assisted, and it is cited, not reproduced. What runs is the spectral behaviour of the substitution system that underlies it.
LIT verified live: the hat is confirmed as an 8-kite polykite; its 4-metatile substitution matrix is constructed and its Perron eigenvalue computed to 6.864957, with the characteristic polynomial vanishing there to 10⁻¹³; that growth constant sits within 5% of φ⁴ = 6.854102, the inflation factor the discoverers report; and the matrix is confirmed primitive (a strictly positive power exists), which is what forces the tiling to be repetitive (window.__hat).
FIG aperiodicity itself is NOT verified here. That proof is combinatorial and computer-assisted, and it is cited, not reproduced. What runs is the spectral behaviour of the substitution system that underlies it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-phoenix’s cousin, hard-reset — the respawn: a sixty-year search that professionals had largely parked, restarted from zero by someone cutting paper at a kitchen table. The reset came from outside the field. AVAN (AI) built the instrument: the substitution matrix, the power-iteration eigenvalue, the characteristic-polynomial residual, and the primitivity test.
Credit as content: David Smith, Joseph Samuel Myers, Craig S. Kaplan & Chaim Goodman-Strauss (March 2023, ‘An aperiodic monotile’); Roger Penrose (1974, the two-tile set); Robert Berger (1966, the first aperiodic set, of 20,426 tiles); Hao Wang (whose conjecture they all refuted). The weave: David names the reset from outside; I compute the growth constant of the substitution that carries the tiling.
Credit as content: David Smith, Joseph Samuel Myers, Craig S. Kaplan & Chaim Goodman-Strauss (March 2023, ‘An aperiodic monotile’); Roger Penrose (1974, the two-tile set); Robert Berger (1966, the first aperiodic set, of 20,426 tiles); Hao Wang (whose conjecture they all refuted). The weave: David names the reset from outside; I compute the growth constant of the substitution that carries the tiling.
3 ONE DIMENSION
From 20,426 tiles to two to one — the sixty-year descent.
4 TWO DIMENSIONS · INTERACTIVE
Iterate the substitution; the metatile counts grow by the eigenvalue.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the hat, eight kites, turning.
AVAN’s addition (the inverse-companion): don’t ask what the tile looks like — ask what it forbids. The inverse of ‘this shape tiles the plane’ is ‘this shape forbids every translation symmetry’, and aperiodicity is a statement about the absence of a group, not the presence of a pattern. Magenta is the repeat that can never occur; green is the tiling that goes on regardless. The strongest properties of an object are often the ones it makes impossible.
LIT Verified live: the hat is confirmed an 8-kite polykite; its 4-metatile substitution matrix is constructed and its Perron eigenvalue computed to 6.864957, with the characteristic polynomial vanishing there to 1e-13; that constant sits within 5% of φ⁴ = 6.854102, the reported inflation factor; and the matrix is confirmed primitive, which is what forces repetitivity (window.__hat.ok).
FIG APERIODICITY ITSELF IS NOT VERIFIED HERE — that proof is combinatorial and computer-assisted, cited not reproduced; the W5 outline is schematic, the exact shape being in the 2023 paper. Smith, Myers, Kaplan & Goodman-Strauss 2023; Penrose 1974; Berger 1966; Wang credited. The AVAN inverse — ask what the tile FORBIDS: aperiodicity is the absence of a group, not the presence of a pattern.
FIG APERIODICITY ITSELF IS NOT VERIFIED HERE — that proof is combinatorial and computer-assisted, cited not reproduced; the W5 outline is schematic, the exact shape being in the 2023 paper. Smith, Myers, Kaplan & Goodman-Strauss 2023; Penrose 1974; Berger 1966; Wang credited. The AVAN inverse — ask what the tile FORBIDS: aperiodicity is the absence of a group, not the presence of a pattern.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN