One tile · one endless strand

The Spectre Knot

A single‑strand Celtic knot on an aperiodic monotile

Draw the same ribbon on every tile of a spectre tiling, which never repeats. Two theorems say what happens.

The Daedalus TheoremIn every spectre tiling whose supertiles fill the plane, no strand ever closes into a loop.

The Ariadne TheoremIn the nest tiling shown here, grown outwards from a single tile, each block sitting in the same place (child number 1) inside the next, all the pieces of ribbon join into one endless strand.

The proofs have not been peer reviewed: what has been checked.

Explore it yourself →What am I looking at?

Each colour is one strand through the outlined middle tile. When strands join, the new strand keeps the colour of its biggest part.The rest of the ribbon.

The idea in three pictures

Same drawing on every tile. One strand overall.

One tile, one drawingEvery tile carries the same 14 ribbon pieces, running edge to edge, with one crossing inside.

The pieces join upWhere tiles meet, the ribbons line up. Nine tiles together: the outlined tile’s 14 pieces are now on 5 strands.

Until it is one strandIn 559 tiles they are a single strand. Zoom out further and it stays one strand, all the way out.

Four ways in

Where would you like to start?

See

Zoom out one block at a time and watch the strands join. Touch any ribbon to light it up.

Watch it join up →

Understand

What the spectre is, what is drawn on each tile, and why a repeating pattern could never do this.

Questions and answers →

Make

Print the tile, cut it out and fold it. Lay a few side by side and the ribbons meet.

Print and fold →

Prove

Why the ribbon never closes up in any spectre tiling whose supertiles fill the plane, and why, in the tiling drawn here, it is one endless strand: the two theorems, from a few easy steps down to the Lean proof.

How we know →

The tile is the straight-edged form of the spectre, Tile(1,1), discovered in 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman‑Strauss. Read their paper: A chiral aperiodic monotile (Combinatorial Theory, 2024).

The knot, the ribbon drawn on it: 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 proofs, too, were made with AI: how this was made.