Penrose Tiles to Trapdoor Ciphers

"In a way, this book is representative of the many books that Martin Gardner wrote. He wrote a column called “Mathematical Games” for Scientific American magazine for many years. If you talk to 10 mathematicians at random in the US – at least of a certain age – probably half of them got excited about maths by reading Martin Gardner’s column and books. He wrote about recreational mathematics, and encouraged people to participate in mathematics by posing interesting questions. His readers would write in and suggest ideas. I could have picked any of his books, as he has written dozens of them. But this one is of particular interest because it contains the columns that introduced the world to two of the most fascinating mathematical discoveries of the 1970s. The first one was Penrose tiles. These are tiles that cover a plane with pentagonal symmetry, which had long been considered impossible. The trick is that they have a long-range order that is different from simple repetition, as in a crystal. In fact, analogues of the Penrose tilings were later discovered in chemistry and called “quasicrystals”. The discoverer of quasicrystals, Daniel Shechtman won the Nobel prize for chemistry last year. But what the newspapers mostly failed to report was that mathematicians discovered them first. In particular, one mathematician, Sir Roger Penrose, who discovered that if you slightly change the rules you can get a pentagonal symmetry. It’s very similar to what Euler did to evaluate that sum I talked about earlier. Great scientists are not afraid of bending the rules. They were also discovered in the 1970s, and before then no one had any clue that these things existed. Basically they are secret codes that let anyone send a message. Before trapdoor ciphers were discovered, it was assumed that if you wanted to have a secret code the sender and recipient both needed to know the key. But then Ronald Rivest, Adi Shamir and Leonard Adleman discovered a “public key” code where the sender of a message doesn’t need to know the secret for decoding it. Public key cryptography is the more common name for it these days. These codes have enabled technology like ATMs and smart cards. Anyone can make a deposit, which involves sending a coded message to the bank, and yet it can’t be stolen because of a public-key or trapdoor code. Nowadays they have gotten quite a bit more sophisticated, but they are all based on mathematical principles, and the fundamental idea is not that difficult. Gardner explains it very well."