Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42). Whether pure or applied, mathematician or not, there will be something interesting for everyone. If you don’t know where to start reading, skim through the pages and have a look at the Number Dictionary at the bottom. If you have any comments, would like to write an article for the next issue, or be part of the next production team, please do not hesitate to contact us or any member of the Archimedeans committee. I hope you will enjoy reading the 61st issue of Eureka! Philipp Legner 38 Painting with Pixels 4 The Archimedeans 30 How Electrons Spin Philipp Kleppmann, Corpus Christi Natasha Kudryashova, Darwin College Schrödinger was the first to explain electron 6 Annual Problems Drive spin using trembling, the Zitterbewegung. Mary Fortune and Jonathan Lee 34 News from the Beginning 10 Bayesian Bets of the World Andrew Pollard, Churchill College Sophie Dundovic, St John’s College What can Bayes tell us about the Monty What are Multiverses? And how can Hall Problem and similar games? they explain why we are alive? 14 Fractal Finance 38 Painting with Pixels Aidan Chan, Peterhouse College Martin Kleppmann, Corpus Christi Alumni Can you really model complex financial Alexander Hess, Universität Bonn markets using mathematical equations? These abstract computer images consist of millions of geometric shapes. 20 Mathematical Origami Philipp Legner, St John’s College 42 How to teach Physics to Trisecting the angle, doubling the cube – Mathematicians paper folding makes it possible! Zhou Min Lim, St John’s College An answer to the never-ending fight: 26 Quantum Entanglement theoretical abstraction or experiments. Elton Zhu, Queens’ College Will entangled quantum states ever 46 The Showroom make teleportation possible? Philipp Legner, St John’s College Smallest and only even prime number. Euler charac- 2 teristic of a polyhedron homeomorphic to a sphere. 14 Fractal Finance 20 Mathematical Origami 34 News from the Beginning of the World 52 The Liar Paradox 74 The Factors of Binomials Prof. Keith Devlin, Stanford Unievrsity Samin Riasat, Queens’ College Is “This sentence is a lie” true or false? Here is an ingenious method for finding Model Theory can help determining it. the prime factors of binomial coefficients. 56 When Logic Meets Geometry 76 Terrifying Transcendentals Tom Avery, St John’s College Aled Walker, Trinity College Zariski structures, linking Set Theory and The proof that e is transcendental has far Algebraic Geometry, have many applications. reaching consequences in mathematics… 60 Graphs, Groups and 80 Impossible Integrals Topology Damian Reding, Trinity College x2 Anja Komatar, Queens’ College Did you ever wonder why e has Graphs can be used to visualise many groups, no elementary antiderivative? and can be embedded in various surfaces. 84 Engage. Explore. Excite. 64 Using Finite Simple Groups Mathematical Collaboration Prof. Cheryl Praeger, U. o. W. Australia Dr Marj Batchelor, DPMMS Insights into one of the greatest discoveries It is the collaboration between scientists of the last century: the classification theorem. from different fields that drives research. 68 Sequences 88 Book and Movie Reviews Trevor Pinto, Queens’ College Forget square numbers and primes – here 92 Solutions are some curious, little-known sequences. 94 Copyrights Number of dimensions we can see. Number of generations of particles in the Standard Model. 3 The Archimedeans Philipp Kleppmann, President 2011 − 2012 he highlight of this year’s Archimedeans’ At the end of term we organised the annual prob- calendar was the black-tie Triennial Dinner. lems drive. The questions were set by last year’s Seventy-five members and guests attended winners, and people from as far away as Warwick Tthe champagne reception and excellent meal in took part in this light-hearted competition. The the Crowne Plaza Hotel to celebrate another step- questions can be seen on the following pages; ping stone in the long life of the society. points were also awarded for the funniest and most creative answers. We hosted several academic talks this year, start- ing with Prof. Siksek’s account of Diophantine The first event in Easter Term was a relaxing punt- equations, their history, and methods of solving ing trip immediately after the exams. A little later these deceptively simple-looking equations. Our we joined in the general madness of May Week main theme were the Millennium Prize Prob- with the Science Societies’ Garden Party. Novel- lems, with five engaging talks throughout the year. ties this year were live music and a cheese bar – These problems where set by the Clay Mathemat- both went down very well! ics Institute in 2000, and only one of them has since been resolved. The talk by Prof. Donaldson This has been a fantastic year, both for our mem- on the Poincaré Conjecture proved to be especial- bers and for the committee, and we are looking ly popular. Prof. Donaldson was on the Advisory forward to another interesting year. I would like Board that recommended Dr. Perelman as prize to thank the committee for their dedication, and winner, and is an expert in the area. the members and subscribers of Eureka for their continued support. The Committee 2011 – 2012 President Treasurer Philipp Kleppmann (Corpus Christi) Lovkush Agarwal (Corpus Christi) Vice-President Events Managers Fangzhou Liu (Sidney Sussex) Pawel Rzemieniecki (Fitzwilliam) Corporate Officer Dana Ma (Newnham) Sean Moss (Trinity) Publicity Officer Internal Secretary Yuhan Gao (Trinity) Colin Egan (Gonville and Caius) Webmaster External Secretary Katarzyna Kowal (Churchill) Laura Irvine (Murray Edwards) Maximum number of colours needed to 4 colour any map in the Euclidean plane. Archimedeans Garden Party 2011 Talk by Prof. Béla Bollobás Party after Prof. Archimedeans Bollobás’ Talk Garden Party 2011 Post-Exams Punting 2011 Fundamental Post-Exams Theorem of Comedy Punting 2011 Number of Platonic solids. Number of vertices of a pyramid. 5 Archimedeans Problems Drive 2011 by Mary Fortune and Jonathan Lee 1 Attitude Adjuster 2 Funny, It Worked Last A velociraptor spots you 40 meters away and Time… attacks, accelerating at 4 m/s2 from a stand You are in the kitchen below. Half black start, up to its top speed of 25 m/s. When it squares are mirrored surfaces, and raptors spots you, you begin to flee, quickly reaching may run on clear areas. If a raptor sees your top speed of 6 m/s. How far can you get you in the corner of its eye, it will turn and before you’re caught and devoured? give chase. Otherwise it will run forward. Raptors are intelligent and do not run into walls. If there is a choice they turn left. There are a number of bear traps scattered in the kitchen. If a raptor runs into any- What Are The Civilian thing, they’re incapacitated and become 3 Applications? someone else’s problem. Do you survive? Microraptors are quite like Raptors physi- cally, except that they are smaller and less cantankerous (and are satisfied with Tuna). Being cheap to hire, they can improve a firm’s profit margin, and are being rolled out in accountancy firms across the country. It is hoped that no one will notice. Their limit- ing factor is that they cannot use computers, and instead use the abacus, making use of a wall chart of all possible multiplications of the numbers from 1 to 10 (an old fashioned multiplication table). A Cambridge mathmo visited the office one day, and remarked that the sum of the entries in the table was a per- fect square (which was true) and that the sum was 2420. Make a conjecture about the anatomy of Microraptors. Smallest perfect number. Order of the smallest 6 non-abelian group. Number of distinct Quarks. The Precise Nature of the Another Victim Of The 4 Catastrophe 5 Ambient Morality There has been a Zombie outbreak on the You and a law student are in the CMS. In Pirate island. Each Zombie infects 1 Pirate front of you are two choices - one of the per day, and are invulnerable to Pirates. rooms contains the answers to Examples The Pirates have some caged Raptors.