Aperiodicity

Should we open up ala Chaos? or just aperiodic tiling? e.g. Robert Shaw, The Dripping Faucet as a Model Chaotic System (1984), pp. 1-2

The non-physicist cannot be expected even to grasp, let alone to appreciate, the relevance of the difference in ‘statistical structure’ stated in terms so abstract as I have just used. To give the statement life and colour, let me anticipate what will be explained in much more detail later, namely, that the most essential part of a living cell — the chromosome fibre — may suitably be called an aperiodic crystal. In physics we have dealt hitherto only with periodic crystals. To a humble physicist’s mind, these are very interesting and complicated objects; they constitute one of the most fascinating and complex material structures by which inanimate nature puzzles his wits. Yet, compared with the aperiodic crystal, they are rather plain and dull. The difference in structure is of the same kind as that between an ordinary wallpaper in which the same pattern is repeated again and again in regular periodicity and a masterpiece of embroidery, say a Raphael tapestry, which shows no dull repetition, but an elaborate, coherent, meaningful design traced by the great master.

Erwin Schrodinger, What Is Life? (1944), “Statistical Physics. The Fundamental Difference in Structure”

Notes forthcoming.

There are exactly 26 “sporadic” groups, of which the monster is the largest. These function as a sort of “prime numbers for symmetry.” But why 26? ¯_(ツ)_/¯

In the 1960s, the logician Hao Wang linked aperiodic tilings to Turing completeness. Unaware of the prior work from the 15th century, however, he conjectured that aperiodic tilings should not exist as they would imply computational undecidability of his system. His student Robert Berger later disproved this conjecture by constructing a 20,426 tile aperiodic counterexample.

In the 1970’s Roger Penrose re-discovered a much simpler aperiodic tiling with reflection and five-fold rotational symmetry. The “Penrose tiling” is probably the best-known example of aperiodic tilings. You’ve probably seen them around, or read about them in Neal Stephenson books.

There are many, many more species of aperiodic tilings and they are all strangely beautiful.

I’m enormously grateful to the maintainers of the Tilings Encyclopedia, without which I would not have made it much beyond the Penrose tilings.