One endless strand
How many strands the pieces make, why they keep joining, why no loop can appear anywhere, and why a repeating tiling could never do it.
So how many strands does the whole tiling make?
One, in the nest tiling drawn here: that is the Ariadne Theorem. (The spectre tiles the plane in infinitely many ways, and the answer may differ: computer checks suggest that some other spectre tilings have three strands. That is evidence, not a proof. In every one whose supertiles fill the plane, though, each strand is endless, by the Daedalus Theorem: none closes up.) Look at a small patch and you see many separate pieces, because each piece soon runs out of the patch. Look at a bigger patch and the pieces join up.
Follow the 14 pieces of one tile: in a block of 9 tiles they lie on 5 strands, in 71 tiles on 2, and from 559 tiles on they are one strand. In every block computed, up to 133 million tiles, every tile’s pieces join up like this within a few levels. Why it goes on for ever is on the Prove page.
Why does it keep joining, instead of stopping at a few strands?
The spectre tiles group into supertiles of 9 tiles, these into bigger supertiles of 71, then 559, and so on for ever, every level built by the same rule. Seen from outside, every supertile turns out to look the same: the ribbon makes a hook round each corner of its outline, except at the two ends of one edge, where two ribbons cross.
Because every supertile looks the same from outside, gluing supertiles together depends only on whether each shared run of edges is even or odd: a run two edges longer behaves exactly like the shorter one. The sides of the supertiles at any level are those at the smallest level made longer by even amounts, so one finite check at the smallest level speaks for all levels. The same idea shows that the gluing keeps merging strands until a given patch lies on one. That every level is glued by the same rule comes from the spectre paper’s theorems.
Could a closed loop appear somewhere far away?
None can, in the tiling drawn here or in any spectre tiling whose supertiles grow to fill the plane: that is the Daedalus Theorem. Every supertile, of every kind, was assembled by computer up to level 9 (about 133 million tiles) without a single closed loop, and the argument on the Prove page carries this to every level, using the spectre paper’s theorems. So every strand runs on for ever in both directions. The page marks what is computed, what is quoted and what is argued, and which parts have also been checked by the Lean proof assistant.
It also says plainly what is not covered: tilings with ‘fault lines’, where two infinite supertiles meet. And no closed loop is not the same as one strand: one strand, the Ariadne Theorem, is claimed only for the nest tiling drawn here. The proofs have not been peer reviewed: what has been checked.
Why can’t a repeating tiling do this?
In a repeating pattern, there are two different slides, a and b, that each bring the whole pattern back onto itself. A slide moves every strand onto a strand.
If there were only one strand, both slides would have to move that one strand onto itself, so each would run it along its own length: a by some number of pieces, p, and b by some other number, q. However cleverly the strand winds, this fails. Do a q times, then b backwards p times. Along the strand that comes to q×p − p×q = 0, so every piece is back where it started. But in the plane the two slides point different ways, so the combined slide is not zero and moves every piece. Both can’t be true.
So it takes a tiling that never repeats. The spectre’s is one.