The knot and the tile
What the Spectre Knot is, the tile it is drawn on, and why straight edges are enough.
What is the Spectre Knot?
It is a Celtic knot drawn on a tile. Every tile carries the same small drawing of ribbon, and when the tiles are laid together the ribbons meet across the edges and carry on, weaving over and under.
The tile is the spectre, a shape that covers the whole plane but only in patterns that never repeat. The surprise is twofold, and each half is a theorem. The Daedalus Theorem: in every spectre tiling whose supertiles fill the plane, no strand ever closes into a loop. The Ariadne Theorem: 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, all the pieces join into one endless strand, with no second strand. The proofs have not been peer reviewed: what has been checked. (Both names come from the Greek myth of the labyrinth: where the names come from.)
What is the spectre, and what does “never repeats” mean?
Squares and hexagons tile the plane periodically: slide the pattern in the right direction and it lands exactly on itself. For a long time nobody knew whether a single shape could force a tiling that never does this. It was called the einstein problem (‘one stone’).
In 2023 David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman‑Strauss found one: first the hat, which needs some copies flipped over, then weeks later the spectre, which needs none. Copies of the spectre cover the whole plane, but every such covering is non-repeating.
Is it really a spectre? The edges are straight.
The tile drawn here is the straight-edged form of the spectre, which the paper calls Tile(1,1). Laid without flipping, its copies tile exactly as spectres do, and only in patterns that never repeat.
Give each edge the paper’s S-curve and you have the strict spectre, which cannot be flipped at all. The knot works the same way on both, because it only depends on how the tiles fit together, not on the exact shape of the edges.
The straight edges make the tile easy to cut out and fold. And because it is printed on one side, a flipped tile would show its blank back, so every tile goes down the same way up. That is what keeps out the mirror-image copies, and it is why straight edges are enough.