Credits, and how to cite
How this was made
Justin Roughley came up with the original idea of looking for a mono‑stranded aperiodic knot, and then used AI (Claude, by Anthropic) to search for it. The search produced 34 candidate designs; he chose design 0.1:01 and drew its ribbons by hand in a tile editor.
AI was also used, at his direction, for the proofs, the programs and this website: Made with AI sets out who did what; the proof site sets out what has been checked.
What is checked, and how
The two theorems, the Daedalus Theorem (no strand ever closes into a loop, in any spectre tiling whose supertiles fill the plane) and the Ariadne Theorem (all the pieces join into one strand, in the nest tiling drawn on this site, grown outwards from a single tile, each block sitting in the same place (child number 1) inside the next), rest on a written argument that uses theorems from the spectre paper, on computer checks, and on a formal proof of their combinatorial part checked by Lean. The Prove page sets out each step, marks what is computed, what is cited and what is argued, tags what the Lean version also checks, and states what is not covered. The proofs have not been peer reviewed. In full, on the proof site: what has been checked.
Credits and references
The spectre was discovered in 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman‑Strauss.
- D. Smith, J. S. Myers, C. S. Kaplan, C. Goodman‑Strauss, A chiral aperiodic monotile, Combinatorial Theory 4(2) (2024), arXiv:2305.17743.
- The same authors, An aperiodic monotile, Combinatorial Theory 4(1) (2024), arXiv:2303.10798 (the hat).
- Craig Kaplan’s page on the spectre: cs.uwaterloo.ca/~csk/spectre.
The knot, design 0.1:01: Justin Roughley came up with the original idea of looking for a mono‑stranded aperiodic knot, and then used AI to search for it (2026).
Related work. An online project, bohemian-miser’s Spectre (2026), describes a single curve, without crossings, through every tile of one nested spectre tiling. Other related work is in the paper.
Typefaces: Figtree on this site (SIL Open Font License); TeX Gyre Pagella in the handout and tile (GUST Font License).
Cite
Please cite the spectre paper for the tile, this site for the knot, and the Spectre Knot paper for the theorems:
Justin Roughley, The Spectre Knot: a single-strand Celtic knot on an aperiodic monotile (2026). https://cortado-j.github.io/SPECTRE_KNOT/ Justin Roughley, The Spectre Knot: one endless strand on an aperiodic monotile, draft (2026). https://cortado-j.github.io/SPECTRE_KNOT_PROOF/
Licence
The text and pictures on this site, the handout, the cut-out tile and the paper are licensed under CC BY 4.0: you may reuse and adapt them, with credit. The programs are under the MIT licence. The typefaces keep their own licences.
Contact
Justin Roughley: cortado.just@gmail.com