I feel like any discussion of aperiodic tiling that doesn’t mention de Bruijn is missing the mark. He showed that aperiodicity results from projecting wireframes in 4 or more dimensions onto a plane. The non-repetitive patterns appear for the same reason that irrational numbers appear in Euclidian geometry.
Isn’t de Bruijn’s method specific to Penrose tiles? The aperiodic monotile paper says in section 2 that it is an open question whether the cut-and-project method can construct hat tilings. So de Bruijn would have been no help to solve Simon Tatham’s problem of how to generate hat tile puzzle grids. His two algorithms are the old one his Loopy puzzle uses for Penrose grids, and the new one it uses for hat grids.
yes, agree, but you’re proving my point because as you say the interesting thing about the hats is that they don’t fit into the simple geometric explanation of other aperiodic tilings.
When I tried to give a quick explanation of what an aperiodic was to a friend recently, calling it a geometric version of irrational numbers was the easiest way to do it. Glad to hear my explanation had some substance to it, and wasn't just one of those "you can think of it like this, but that's not what it really is" explanations.
I'd be surprised if the pinwheel tilings — which have tiles that take on an infinite number of relative angles — could be reproduced in this manner. It seems like projecting a periodic set in high dimensions should give at most a finite number of cell orientations, although I can't prove it offhand.
