The mathematical physicist and cosmologist Roger Penrose, now professor emeritus at Oxford University, is best known to mathematicians for his discovery of Penrose tiles. These are two four-sided polygons that tile the plane only in a nonperiodic way, that is, without a fundamental region that repeats periodically like the hexagonal tiling of a bathroom floor, or the amazing tesselations of the Dutch artist M. C. Escher. To everyone’s surprise, including Penrose’s, his whimsical tiling turned out to underlie a previously unknown type of crystal. You can read all about this in my book Penrose Tiles to Trapdoor Ciphers.

Computers of the sort we know how to build obviously are no more aware of what they do than a typewriter knows it is typing.

Penrose’s two best sellers, The Emperor’s New Mind and its sequel, Shadows of the Mind, were slashing...


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