I wonder how many times it's possible to be wrong in sequence about the same thing.In my last blog post about tilings and finite-state transducers, I mentioned making a mistake about the Ammann-Beenker tiling. I had a description of it in my software, and I had believed it didn't admit a deterministic transducer, because my code reported failure when trying to construct one. But I later found out that it did: my construction algorithm had an overzealous failure detector, and had given up too soon.But it turns out I was right the first time, if for the wrong reasons. It _doesn't_ admit a transducer after all, because I _also_ got the tiling description slightly wrong!The Ammann-Beenker rhombus tile is 180° symmetric, but its orientation matters nonetheless (it forms part of the proof of aperiodicity). My description of the tiling had the rhombuses wrongly oriented, in a way that – as it turns out – made the system accidentally simpler.The first picture here is the Ammann-Beenker analogue of the Penrose 'cartwheel pattern': a singular instance of the tiling, having the symmetry group of a regular octagon _except_ for the dark-coloured tiles, of which the blue ones vary their shape under rotation by some multiple of 45°, and the red ones keep the same shape but change orientation.The second picture is the same pattern, but generated using my earlier wrong description of the tiling. Because I got some of the rhombus orientations wrong, the red tiles are in much weirder places. _That_ diagram has been puzzling me for a week, until I figured out last night what mistake had caused it!Oh well, at least I got some pretty pictures out of it.#TilingTuesday #oops
