Roger Penrose makes his own rules. He is one of the world’s most distinguished mathematical physicists and most inventive thinkers. Penrose’s work on the theory of general relativity in the 1960s led to the discovery that the gravity of collapsing stars can produce black-hole “singularities” in space-time. This, in turn, set Stephen Hawking on his course to rewrite black-hole physics. The research established Penrose’s name in science, but his thought continued to range much further. In The Emperor’s New Mind (1989) he proposed that the human mind can handle problems that are “non-computable,” which is to say that any computer trying to solve them by executing a set of logical rules (as all computers do) would chunter away forever without reaching a conclusion. This property of the mind, Penrose argued, might stem from the brain’s use of a quantum-mechanical principle, perhaps involving quantum gravity. In collaboration with anaesthetist Stuart Hameroff, he suggested in Shadows of the Mind (1994) what that principle might be, involving quantum behaviour in protein filaments called microtubules in neurons. Neuroscientists scoffed, glazed over, or muttered “Oh, physicists…”

So when I remarked that he is known for ideas that most others couldn’t even imagine, let alone dare voice, introducing a talk by Penrose yesterday, I didn’t expect that I would hear new ones that evening. Penrose was speaking about the discovery for which he is perhaps best known among the public: the so-called Penrose tiling, a pair of rhombus-shaped tiles that can be used to tile a flat surface ad infinitum without the pattern ever repeating itself. It turns out that this pattern is peppered with objects that have five- or ten-fold symmetry; like a pentagon, they superimpose on themselves when rotated a fifth of a full turn. That is very strange, because fivefold symmetry is known to be rigorously forbidden for any two-dimensional packing of shapes. (Try it with ordinary pentagons and you quickly find that you get lots of gaps). The Penrose tiling doesn’t have this “forbidden symmetry” in a perfect form, but it almost does.

These tilings – there are other shapes that have an equivalent result – are strikingly beautiful, with a mixture of regularity and disorder that is somehow pleasing to the eye. This is doubtless why, as Penrose explained, many architects have made use of them. But they also…

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