I have a question about the aperiodic spectre tile (or the hat/turtle).
I know that the proof of aperiodicity works by showing that the tiles must fit together in a hierarchical structure that eventually repeats itself at a larger scale. But the larger units aren't literally scaled copies of the spectre. I also know that there is some freedom as to how you draw the edges of the spectre.
Is there a way you can draw the edges that allows you to literally use spectres to cover a larger copy of themselves? If so, is this way of doing it unique?
@OscarCunningham I'm pretty sure you can transform the hats' HTPF metatile system into a form where each higher-order metatile exactly covers a set of metatiles of the next order down. (Use the 'converged' metatile shapes; use a non-overlapping version of the expansion rules; do some horrible limiting thing that fractalises all the metatile edges.)
But then you still have four different fractally-shaped metatiles, and no way to decompose those into individual hats that are all congruent.
@OscarCunningham in fact, here's the paper I vaguely remembered seeing but couldn't put my hands on yesterday, which does pretty much what I said. https://arxiv.org/abs/2305.05639, diagrams on pages 7 and 8.
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