Amir D. Aczel (amirdaczel.com) writes about mathematics and physics and has published 18 books, numerous newspaper and magazine articles, as well as professional research papers.
A Higgs candidate event from the ATLAS detector of the LHC.
Courtesy of CERN
What made me fall in love with theoretical physics many years ago (in 1972, when I first met Werner Heisenberg) was its stunningly powerful relationship—far beyond any reasonable expectation—with pure mathematics. Many great minds have pondered this mysteriously deep connection between something as abstract as mathematics, based on theorems and proofs that seem to have little to do with anything “real,” and the physical universe around us. In addition to Heisenberg, who brilliantly applied abstract matrix theory to quantum physics, Roger Penrose has explored the deep relation between the two fields—and also, to a degree, between them and the human mind—in his book The Road to Reality.
And in 1960, the renowned quantum physicist and Nobel Laureate Eugene Wigner of Princeton wrote a fascinating article that tried to address the mysterious nature of this surprising relationship. Wigner marveled at the sheer mystery of why mathematics works so well in situations where there seems to be no obvious reason why it does. And yet, it works.
There was a time when people thought of playing cards as cosmic instruments. Fortunes were told, fortunes were lost, and the secrets of the universe unveiled themselves at the turn of a card. These days we know better. And yet, a look at the mathematics of card shuffling reveals some startling insights.
Consider, for instance, the perfect, or “faro” shuffle—whereby the cards are divided exactly in half (top and bottom) and then interleaved so that they alternate exactly. Most people think shuffling tends to mix up a deck of cards, and usually that’s true, because a typical shuffle is sloppy. But a perfect shuffle isn’t random at all. Eight consecutive perfect shuffles will bring a 52-card deck back to its original order, with every card in the pack having cycled through a series of predictable permutations back to its starting place. This holds true for any deck, regardless of its size, although eight isn’t always the magic number. If you have 25 cards, it takes 20 shuffles, whereas for 32 cards it only takes 5; for 53 cards, 52 shuffles are needed. You can derive a formula for the relationship between the number of cards in the deck and the number of faro shuffles in one full cycle.
There was a time when people thought of playing cards as cosmic instruments. Fortunes were told, fortunes were lost, and the secrets of the universe unveiled themselves at the turn of a card. These days we know better. And yet, a look at the mathematics of card shuffling reveals some startling insights.
