Names and people

Both come from the Greek myth of the labyrinth. King Minos of Crete had the craftsman Daedalus build a labyrinth to hold the Minotaur, half man and half bull: a maze so tangled that no one who went in could find the way out. Minos’s daughter Ariadne gave Theseus a ball of thread. He tied one end at the entrance, unwound it as he went in, killed the Minotaur, and followed the thread back out.

The Daedalus Theorem is about the labyrinth itself, the way the passages are built. In every spectre tiling whose supertiles fill the plane, no strand ever closes into a loop: every passage of ribbon runs on for ever in both directions.

The Ariadne Theorem is about the thread. In the nest tiling drawn on this site, all the pieces of ribbon join into one thread that runs through every tile, so from any tile you can follow it to any other, however far away. The proofs have not been peer reviewed: what has been checked.

The story carries on into two words used for each tile. Its thread is the shortest stretch of the strand that holds all 14 of its pieces, and its labyrinth is the set of tiles that stretch passes through: the smallest patch in which the tile’s pieces all join up. For the starting tile the thread is 112 pieces long and the labyrinth is 19 tiles. Across the 4,401-tile block, half the tiles have a labyrinth of about 60 tiles or fewer, but about one in ten needs thousands. (These are measurements, not part of the proof.)

The spectre was discovered in 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman‑Strauss. 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).

The search, the computations, the proofs and this website were made with AI: Made with AI says who did what. Also full credits and references.