Alan Turing’s bicycle serves as a lead into Enigma machines in Neal Stephenson’s Cryptonomicon and opens up a teaching opportunity in a mathematics classroom.
I have been a huge fan of Neal Stephenson’s since picking up Anathem while browsing in Exclusive Books. After standing there reading for at least 20 minutes, I remember coming back to reality and being not at all sure where I was. At present, my audio book for my daily commute is Neal’s Stephenson’s Cryptonomicon (from Audible, and beautifully read by William Dufris). I have read the e-book twice, and now I am really enjoying listening to it. There is such pleasure in Stephenson’s use of language, his ideas, his characterisation, his tangential rants and his hyperbolic descriptions:
At the currency exchange window of NAIA, Randy had stood behind a Chinese man who, just before he stepped back from the window with his money, unloaded a Sneeze of such titanic force that the rolling pressure wave turbulating outwards from his raw, flapping facial orifices caused the wall of bulletproof glass separating him from the moneychangers to flex slightly, so that the reflection of the Chinese man, Randy behind him, the lobby of NAIA, and the sunlit passenger-dropoff lane outside underwent a subtle warpage. The viruses must have roiled back from the glass, reflected like light, and enveloped Randy. So maybe Randy is the personal vector of this year’s version of the flu-named-after-some-city-in-East-Asia that annually tours the United States, just barely preceded by rush shipments of flu vaccine.
Alan Turing was a real Cambridge mathematician and computer scientist. I first heard of him in Computer Science I, where we learnt about the Turing Machine. The Turing Machine is a Gedankenexperiment which imagines an infinitely long piece of tape divided into cells and a read-write head which can move along the tape. Each cell on the tape may be blank or contain a symbol. The state of the machine and the symbol in the cell currently under the head will determine the next action of the machine: this could be read from a cell, write to a cell or move one cell left or right. Despite the simplicity of this model, any computer algorithm can be simulated using a Turing Machine. The Turing machine was one method of tackling the wh
ole issue of computability, and this issue can lead us into the marshes of lambda-calculus, the Halting Problem, and Gödel’s Theorem.
My next meeting with Alan Turing was when I learned about the work done at Bletchley Park during the war. Bletchley Park was the home of the Allied code-breakers in the European theatre, who over the course of WWII were able to break almost all of the German military ciphers, and Alan Turing was involved in this effort from the beginning of the war.
The weak link in the chain
In Cryptonomicon Alan Turing is usually seen interacting with the (fictional) mathematician Lawrence Waterhouse, whose connection with the world he moves in as tenuous at best since most of his mind is on mathematical fugues and the puzzles that circle continuously in his head. Today on my morning drive I was listening to the chapter in Cryptonomicon where Alan Turing goes off with Lawrence to find a cache of silver bars. Alan’s bicycle has one weak link in the chain, and one bent tooth on the gear that drives the chain. Whenever the weak link goes over the bent tooth, the chain breaks and falls off.
Stephenson begins by describing the degenerate case: for example if n (the number of teeth on the gear) is 20 and L (he number of links in the chain) equals 100 the chain will break every time the weak link comes around. In the non-degenerate case, if we assuming Alan starts one step past where the weak link and bent tooth would break the chain, he can cycle quite a way: the chain will then only break again at the Lowest Common Multiple of L and n. If there are no common factors between L and n, the LCM is the product of these two numbers – the numbers are relatively prime, or co-prime. So if L and n are relatively prime, Alan will be able to ride further before his chain breaks.
I will introduce the idea of this bicycle to my Grade 11 learners after the exams, we shall see how long it takes them to figure out that the LCM and relative-primeness are the important concepts to focus on.
The Enigma Machine
Incidentally, Stephenson takes us back into the main theme of his narrative by relating the idea of the weak link and the bent gear tooth to the precepts of codebreaking in the second world war. The Enigma machine was used by the German military to encipher messages. This machine had slots for three wheels or rotors (well, four wheels for the Shark, the submarine cipher) which could be set to any of 26 positions (one for each letter of the alphabet). In addition a plugboard was added which allowed letters to be connected in pairs, to make the cipher stronger.
The rotors were selected from a set of five possibilities. The cipher was created by pressing a key on the keyboard; each time a key on the machine was pressed, the rotors advanced so that each letter was generated using a different substitution. The military Enigma machine had almost 160 quintillion possible setups; to decode these signals, a machine called a bomb was used which was originally developed by three Polish mathematicians – Marian Rejewski was able to reconstruct the wiring of the Enigma machines without ever seeing one, a feat which certainly played a part in the outcome of the Second World War.
Stephenson moves from the Enigma machine to the Enigma Problem (How to Use What We Know Without Them Realising That We Know It) and by that time I was turning in at school. It was a lovely way to drive to work, oblivious to red lights and slow drivers.