Eureka 62 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Jack Williams Cover by Andrew Ostrovsky, Inner Cover by George Hart © The Archimedeans, see page 95 December 2012 Editorial Eureka 62 hen the Archimedeans asked me to edit Eureka for the third Editors time, I was a bit sceptical. Issue 60 was the first to get a pa- Philipp Legner (St John’s) perback binding and issue 61 was the first to be published in Jack Williams (Clare) Wfull colour and with a new design. How could we make this issue special – not just a repeat of the previous one? Assistant Editors Stacey Law (Trinity) Eureka has always been a magazine for students, not a research journal. Carina Negreanu (Queens’) Articles should be interesting and entertaining to read, and often they Katarzyna Kowal (Churchill) are a stepping stone into particular problems or areas of mathematics Douglas Bourbert (Churchill) which the reader would not usually have encountered. Ram Sarujan (Corpus Christi) Every year we receive many great articles by students and mathemati- Subscriptions cians. Our task as editors is often to make them more visually appealing Wesley Mok (Trinity) – and we can do so using images, diagrams, fonts or colours. What we wanted to add in this issue was interactivity, such as videos, slideshows, animations or games. Unfortunately this still is quite dif- ficult on paper, so we decided to publish a second version of Eureka as interactive eBook for mobile devices like iPad. And we hope that this will make reading mathematics even more engaging and fun. The digital version will, for the first time, make Eureka available to a large number of students outside Cambridge. And therefore we have reprinted some of the best articles from previous issues. We spent many hours in the library archives, reading old copies of Eureka, though of course there are many more great articles we could have included. The articles in this issue are on a wide range of topics – from num- ber theory to cosmology, from statistics to geometry. Some are very technical while others are more recreational, but we hope that there is something interesting for everyone. I want to thank the editorial team for all their work, and the authors for their excellent articles. We hope you will enjoy reading Eureka 62! Philipp Legner and Jack Williams 10 Pentaplexity 4 The Archimedeans 32 Fractals, Compression and Contraction Mapping † 6 Talking to Computers Alexander Shannon, Christ’s Stephen Wolfram 38 Stein’s Paradox 10 Pentaplexity † Richard J Samworth, Statslab Cambridge Sir Roger Penrose, Oxford University 42 Optimal Card Shuffling † 16 Squared Squares Martin Mellish Philipp Kleppmann, Corpus Christi 44 Harmonic Game Theory † 20 Tricky Teacups Blanche Descartes Vito Videtta, Trinity Hall 46 The Logic of Logic 24 Annual Problems Drive Zoe Wyatt, Newnham David Phillips and Alec Barnes-Graham 48 Timeline 2012 28 Pi in Fours † Stacey Law and Philipp Legner John Conway and Michael Guy 50 Hopes and Fears † 30 Surface Differences Matter! Paul Dirac Arran Fernandez, Cambridge University First– magic number in nuclear physics. 2 √2 was the first known irrational number. 72 Glacier Dynamics 58 Multiverses 32 Fractals, Compression and Contraction Mapping 54 Quantum Gravity † 88 Archimedes Stephen Hawking, DAMTP Cambridge Tom Körner, DPMMS Cambridge 58 Multiverses 90 Book Reviews Georg Ellis, University of Cape Town 92 Christmas Catalogue † 64 Primes and Particles Jack Williams, Clare 93 Call My Bluff † 68 M-Theory, Duality and Art 94 Solutions David Berman, QMUL 95 Copyright Notices 72 Glacier Dynamics Indranil Banik and Justas Dauparas 76 Finding Order in Randomness Maithra Raghu, Trinity 80 Mathematics in Wartime † G H Hardy 84 Consecutive Integers † Paul Erdős Items marked † are reprinted from past issues of Eureka. Three of the five platonic solids have triangular faces. Trisecting angles is an impossible construction. 3 The Archimedeans Yuhan Gao, President 2012 − 2013 his year was yet another highly successful from as far afield as Oxford came to take part in an one for The Archimedeans. The society engaging and entertaining mathematics competi- welcomed over 150 new members, courtesy tion. Prizes were awarded not only for the teams Tof a very popular Freshers’ Squash. We hosted a with the highest scores, but also for particularly number of talks given by speakers from the uni- creative team names. The questions given can be versity over the course of Michaelmas and Lent. found in this journal, and we welcome you to try These covered a number of different topics, cater- them yourself. ing for those with interests in pure, applied and applicable mathematics. Highlights included The year finished on a high in May Week, courtesy talks by Prof. Grae Worster on Ice, and Prof. Imre of the Science and Engineering Garden Party. Six Leader on Games of Pursuit and Evasion. societies from the university joined together to host a brilliant afternoon of fun, aided by a jazz The society expanded the range of events which band. Finger food and Pimm’s was served, and we offered to our members this year. We held a there was even a cheese bar on offer. board games evening, which proved to be a thor- oughly enjoyable night for all those who attended. We would like to thank our members for contrib- One of our most anticipated events was the black- uting to an excellent year for the society. I would tie Annual Dinner in the delightful surroundings also like to thank the committee for all of their of the Crowne Plaza Hotel. hard work, and Philipp Kleppmann, last years’ President, along with the previous committee, for A tradition of the Archimedeans is to hold an an- everything which they have done for the society. nual Problems Drive. This time around, teams We look forward to another exciting year ahead. The Committee 2012 – 2013 President Treasurer Yuhan Gao (Trinity) Colin Egan (Gonville and Caius) Vice-Presidents Events Managers Sean Moss (Trinity) Pawel Rzemieniecki (Fitzwilliam) Dana Ma (Newnham) Yuming Mei (Emmanuel) Corporate Officer Publicity Officer Joseph Briggs (Trinity) James Bell (Gonville and Caius) Secretary Webmaster Jacquie Hu (Jesus) Ben Millwood (Downing) Smallest composite number. Number of 4 Nucleobase types in the DNA: A, G, C and T. Archimedeans Garden Party Professor Archimedeans David Tong Problems Drive Archimedeans Annual Dinner Archimedeans Talk in the CMS Archimedeans Problems Drive Archimedeans Garden Party K5 is the smallest complete non-planar graph. The cycle of fifths underlies musical harmonies. 5 Talking to Computers Stephen Wolfram love computer languages. In fact, I’ve spent Wolfram|Alpha responds by computing and roughly half my life nurturing one particular presenting whatever knowledge is requested. But I very rich computer language: Mathematica. programming is different. It is not about gen- erating static knowledge, but about generating But do we really need computer languages to tell programs that can take a range of inputs, and our computers what to do? Why can’t we just use dynamically perform operations. natural human languages, like English, instead? The first question is: how might we represent If you had asked me a few years ago, I would these programs? In principle we could use pretty have said it was hopeless. That perhaps one could much any programming language. But to make make toy examples, but that ultimately natural things practical, particularly at the beginning, language just wouldn’t be up to the task of creat- ing useful programs. we need a programming language with a couple of key characteristics. But then along came Wolfram|Alpha in which we’ve been able to make free-form linguistics The most important is that programs a user might work vastly better than I ever thought possible. specify with short pieces of natural language must typically be short – and readable – in the But still, in Wolfram|Alpha the input is essen- computer language. Because otherwise the user tially just set up to request knowledge – and won’t be able to tell – at least not easily – whether Smallest perfect number. Hexagonal tilings give 6 the densest ‘sphere’ packing in two dimensions. the program that’s been produced actually does The linguistic capabilities of Wolfram|Alpha give what they want. one the idea that one might be able to under- stand free-form natural language specifications A second, somewhat related criterion is that it of programs. Mathematica is what gives one the must be possible for arbitrary program frag- idea that there might be a reasonable target for ments to stand alone – so that large programs can realistically be built up incrementally, much programs generated automatically from natural like a description in natural language is built up language. incrementally with sentences and the like. For me, there was also a third motivating idea – To get the first of these characteristics requires that came from my work on A New Kind of Sci- a very high-level language, in which there are ence. One might have thought that to perform already many constructs already built in to the any kind of complex task would always require language – and well enough designed that they a complex program. But what I learned in A New all fit together without messy “glue” code. Kind of Science is that simple programs can often do highly complex things. And to get the second characteristic essentially requires a symbolic language, in which any piece And the result of this is that it’s often possible to of any program is always a meaningful symbolic find useful programs just by searching for them expression. in the computational universe of possible pro- Conveniently enough, there is one language that grams – a technique that we use with increas- satisfies rather well both these requirements: ing frequency in the actual development of both Mathematica! Wolfram|Alpha and Mathematica.