Wednesday, September 01, 2021

Quantum Mechanics and the Penrose Rose

The illustration in my previous article was not just to look pretty. It's an oil painting I did a few years ago inspired by Roger Penrose, physicist and creator of aperiodic 'Penrose Tilings'. 

Creating the picture involved cutting out a lot of pieces of card in three shapes of rhombus, with point angles of either 18, 36 or 72 degrees. The cards were then arranged to make a tiling, no gaps or overlaps, to create the design. The pattern has five-fold rotational symmetry (ignoring the colours) but is aperiodic in that the pattern will never repeat itself going outwards, not to the ends of the Universe. It would never work for printing a roll of wallpaper.

But laying out the little pieces of card is no simple matter. It's easy enough to begin with, starting at the centre and adding tiles round the circle to grow the pattern, but one soon finds something going wrong. Add a ring of tiles and the last one turns out not to fit without gap or overlap. One has to undo recent work and start with a tile oriented in a different direction. There is a choice of orientation at each step but it is impossible to know which is the 'correct' way round until one has worked all the way around the growing pattern and discovered whether or not it fits. There's a long process of trial and error involved.

To create the pattern quickly and efficiently one would need to lay tiles on opposite sides of the pattern simultaneously, each 'knowing' the orientation of the other since they are interdependent. There needs to be instant communication. And that's spooky action at a distance, as Einstein may have said.

With a bit of imagination one might conceive a three dimensional version, with each rhombus a rhomboid to fill the space. The pattern then forms a solid, which, if the rhomboids were arrangements of atoms, would form a sort of crystal. For such a crystal to grow, however, there would need to be that spooky action at a distance. Before a new atom settles into position, with the correct orientation, on a growing crystal face it would have to know the orientation of an atom settling down on the opposite side.

Roger Penrose postulated that such things could exist. And then it was found that they do. Read about quasicrystals here.


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