Resources and tools
Generators, datasets, museums, fabrication files, OEIS sequences, and built installations.
How to use this index
The numbered bibliography holds peer-reviewed and preprint research with stable citation anchors. This page catalogs everything else: official generators, open fabrication files, museum exhibits, workshop kits, games, OEIS sequences, journalism, and documented installations.
Tile(1,1) is not the same as Spectre. The straight-edged Tile(1,1) polygon admits a periodic tiling if reflected copies are mixed; Spectres are edge-modified, strictly chiral shapes. Many community repositories reuse the word “Spectre” for straight-edged outlines, verify the geometry before fabrication.[2]
An automated crawl maintains a machine-readable source registry for literature discovery. The lists below are human-curated and tiered: link freely, but check each site’s license before republishing assets.
Trust tiers and actual capabilities
- Official theorem sources: the Hat/Spectre project pages and open-access papers define canonical terminology and geometry. Use them before community summaries.
- Official reference code:
hatvizgenerates Hat patches under BSD-3-Clause;hatvalidatereplays encoded proof checks. Generation and proof verification are different jobs. - Independent generators: TileOneOne (MIT) emits straight Tile(1,1) patches; symbolic- spectre-tiles (MPL-2.0) emphasizes exact coordinates; Tatham’s tools provide hierarchical navigation and transducers. Validate each against official rules.
- Viewers and editors: WebGPU/deep-zoom tools optimize display. Manual playgrounds may permit mirrors, overlaps, or illegal frontiers and are not legality certificates.
- Fabrication assets: CC0 multi-format outlines, GPL OpenSCAD, community Printables models, and laser-sheet generators differ in license, units, geometry variant, and manufacturing intent. A downloadable STL is not automatically canonical.
- Education and installations: museum archives, workshop kits, terraces, grout papers, and public builds establish engagement or feasibility, not theorem, pedagogy, or structural performance.
The computational distinctions, substitution generation, recognition, finite-state navigation, GPU rendering, and independent geometry checks, are explained in Computational generation and navigation. Discovery accounts are evaluated separately in Discovery history.
The funbin package accompanying ref. 27 can bin data into arbitrary polygons, including
Hat-family cells, using point-in-polygon tests and spatial indexing. Its paper is intentionally satirical
and supplies no controlled bias, error, readability, anti-aliasing, or reconstruction benchmark. Treat it
as a creative visualization tool, not scientific evidence that aperiodic bins improve inference.[27]
Catalog
Official project pages and discoverer accounts
- Kaplan, Aperiodic Monotiles, discovery chronology and community links
- David Smith, Hedraweb, discoverer blog and physical experiments
- David Smith, The Special One, first-person Tile(1,1) / Spectre story
- Joseph Myers, publications and preprints
- Chaim Goodman-Strauss, papers and notes
- Combinatorial Theory, Hat paper (open access)
- Combinatorial Theory, Spectre paper (open access)
Generators, code, and datasets
- isohedral/hatviz, official Hat patch builder (BSD-3-Clause)
- isohedral/hatvalidate, computer-assisted aperiodicity verification
- henningle/TileOneOne, reference Tile(1,1) MATLAB generator (MIT)
- jsm28/AperiodicMonotilesLean, Lean formalization staging repo
- reversi-fun/symbolic-spectre-tiles, symbolic coordinates and CSV/SVG export (MPL-2.0)
- ctkrug/monotile, infinite Spectre pan/zoom studio
- Ricky Reusser, WebGPU aperiodic monotile rendering notebook
- Ricky Reusser, deep-zoom aperiodic monotile notebook
- James Smith, AperiodicCube 3D interactive demo
- Simon Tatham, coordinate algorithms for Hat tilings
- Simon Tatham, finite-state transducers for Hat and Spectre
- Simon Tatham, refinable frontier (H7/H8 and Spectre hex types)
- Jaap Scherphuis, PolyForm Puzzle Solver (discovery-era search tool)
- Spectre Tiling Playground, manual editor with TILE coordinate format
- Beach Spectre Practise, touch-friendly substitution trainer
Institutional references and OEIS
- Tilings Encyclopedia, Hat substitution record
- Tilings Encyclopedia, CAP representative
- Tilings Encyclopedia, aperiodic monotile glossary
- OEIS A363445, Hat perimeter fractal turn sequence
- OEIS A397115-A397123, Spectre hierarchical cluster counts (2026)
Museums, competitions, and workshops
- UKMT Einstein Mad Hat Awards, competition archive
- Hatfest, Oxford / Grimm Network conference archive
- Cambridge Faculty of Mathematics, Tip of the Hat celebration
- Marcello Seri, Spectres, Hats and Maths workshop kits (CC BY 4.0)
- Bridges 2024, Group activity to build a Spectre tiling
- Numberphile, Discovery of the Aperiodic Monotile
- Quanta, Hobbyist finds maths elusive Einstein tile
Fabrication, installations, and products
- Beach Spectres, public sand-tiling project and how-to guides
- Printables, Einstein Tiles family (Hat, Tile(1,1), Spectre variants)
- Spirko/SpectreOpenSCAD, parametric Spectre STL generator (GPL-3.0)
- vmagnin/hat_polykite, laser-cut Hat and Tile(1,1) sheets
- Nervous System, wooden Spectre puzzle (111 pieces)
Curated quick links
- Kaplan et al., An aperiodic monotile project page, interactive patch builders, source code, and CC BY sample images
- Kaplan et al., A chiral aperiodic monotile project page, Tile(1,1) patch app and Spectre SVG outlines for cutting and printing
- Printables: Spectre chiral aperiodic monotile, parametric 3D-printable tiles with this-way-up orientation grids
- National Museum of Mathematics: The Hat and the Spectre, exhibits and the Einstein Mad Hat competition
- Nan Ma, aperiodic-monotile-4d, original Wolfram Language code and projection experiments (no license; link only)
- Arnaud Chéritat and Nan Ma, interactive 4D projection applet, CC BY-SA
- Simon Tatham, Combinatorial Coordinates for the Aperiodic Spectre Tiling
- Christian Lawson-Perfect et al., multi-format monotile assets, CC0 SVG, DXF, OpenSCAD, and STL
- Infinite Spectres, MIT-licensed Rust/WebGPU deep-zoom viewer
- Tilings Encyclopedia: Spectre, institutional patch and reference record
- Large Spectre Tiling, 488-piece hierarchical group activity
- Terracing with Spectres, built CNC-waterjet limestone terrace and process archive
- Hats in Grout, practical fabrication and installation geometry
- OEIS A363348, recursive turn sequence for drawing an infinite Hat tiling
Task-oriented starting points
- Learn the theorem: start with the official Hat and Spectre papers and project pages, then compare the independent proof and survey.[1][2][3][4]
- Generate patches: choose a repository with an explicit license, documented substitution rules, coordinate conventions, and reproducible export.
- Fabricate: obtain canonical SVG/DXF or a parametric source, verify handedness and units, then follow Materials and fabrication.
- Run research: begin from a cited physical or computational paper, reproduce its geometry and baselines, and archive the exact patch manifest.
- Teach: use museum/workshop kits whose redistribution terms permit classroom copies, and state which observations illustrate rather than prove aperiodicity.
Licenses, geometry, and provenance
A public URL is not a reuse license. Check the repository root, release asset, and individual file for a license; record its version, attribution requirements, and whether commercial use or modification is allowed. If no license is present, link to the source but do not redistribute its files.
Validate geometry against an official outline or paper: units, vertex order, edge lengths, handedness, straight Tile(1,1) versus edge-modified Spectre, and allowed reflections.[2] Record source URL, commit or release, retrieval date, file hash, generator parameters, seed/root, generation, clipping rule, transforms, and any repairs. A screenshot is not adequate data provenance.
Still missing?
No single index can guarantee completeness on the open web. If you maintain a generator, dataset, fabrication guide, or installation archive that belongs here, contact us with a URL, license, and one-line description.
See also
Bibliography, Education, Materials and fabrication
Categories: References · Meta