Discovery history of the Hat and Spectre
A dated account of physical search, computation, proof, and the official records behind the 2023 discoveries.
The Hat: November 2022 to March 2023
David Smith combined paper cutouts, card experiments, and polyform-search software rather than relying on one lucky picture. In November 2022 he sent Craig Kaplan a candidate assembled from kites. Kaplan’s computation grew larger patches; Joseph Myers and Chaim Goodman-Strauss joined the effort to identify forced metatiles and turn the candidate into a proof. The Hat preprint appeared in March 2023.[1][3]
Official project pages, Kaplan’s dated retrospective, Smith’s Hedraweb posts, and the PolyForm Puzzle Solver document discovery practice; the 2024 open-access journal paper is the authority for the theorem. A large finite patch was evidence of compatibility, while forced hierarchy and incommensurability supplied the global aperiodicity argument.
The Spectre: weak to strict chirality
Firm public dates: Smith contacted Kaplan on 17 November 2022; he suggested an einstein candidate on 24 November; the Hat preprint appeared on 20 March 2023; the Spectre preprint on 28 May 2023; and the peer-reviewed papers followed on 30 June and 30 September 2024.[1][2][3]
The straight equilateral Tile(1,1) initially looked like a limiting family member with a periodic escape: equal numbers of both handednesses tile periodically. Smith’s experiments showed that when reflections were forbidden, recurring eight- and nine-tile clusters and Mystic pairs exposed a homochiral hierarchy. Edge modifications then made the no-reflection rule geometric, producing strict Spectres. Anecdotes about a “six-day turn” appear in informal histories; this wiki keeps only the dated public record unless a first-person source is cited for that specific claim.[2][3]
Discovery narrative and theorem should remain distinct. First-person posts explain who noticed what and which tools were used; the peer-reviewed paper defines weak versus strict chirality and proves aperiodicity. Community models and press articles are useful orientation sources, not replacements for the proof.
Evidence levels and later work
- Primary theorem: the final Hat and Spectre papers.[1][2]
- Independent mathematics: alternative proof, direct construction, group and rhombic formulations.[4][5][8][9]
- Discoverer history: dated project pages, repositories, and first-person accounts.
- Community and journalism: museum events, Hatfest, videos, and reporting that explain reception but do not independently certify theorem details.
- Unverified claim: ref. 28’s golden-ratio argument remains quarantined because an irrational count limit for one construction does not rule out all periodic tilings.[28]
See also
Hat tile, Spectre tile, Computational generation and navigation, Resources and tools
Categories: History · Mathematics