[The, Commutator] vol.3 issue 1

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Contents 3 4 6 9 11 14 17 18 29 32 34 36 42 46 47 48 50

“The book of nature has been written in the language of mathematics”

Letter from the editor Our Special Thanks Diophantine Problems for Garden Gnomes The Riemann Hypothesis The Carol Syndrome

Galileo Galilei

To Infinity and Beyond The Birthday Problem; Number Curiosities

Editor in Chief Hristo Georgiev

Cellular Automata and Conway’s Game of Life Prime Numbers

Graphic Designer/Assistant Editor Jaspal Puri

An Epidemic Beginning

Special thanks to Dr. Lorna Love, Mrs. Shazia Ahmed, Dr. Radostin Simitev, Dr. David MacTaggart, Dr. Chris Athorne, Prof. Nicholas Hill, Dr. Marianne Freiberger, Dr. Rachel Thomas, Emily Woodhouse, Prof. José-Manuel Rey, Remus Stana, Sebastian Popescu, Jennifer Petrie.

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The Significance of the Number 2 -1 The Mathematics of Fluid Dynamos Modelling the Spread of Ideas

editor@the-commutator.com

Incircles and Excircles of a Circular Triangle

University of Glasgow School of Mathematics and Statistics University Gardens, University of Glasgow Glasgow, G12 8QA http://www.gla.ac.uk/mathematicsstatistics tel: +44 (0) 141 330 5176 email: maths­stats­enquiries@glasgow.ac.uk

The Look-and-Say Sequence; The World’s Smallest Football Humour Puzzles

Jaspal Puri Assistant Editor/ Graphic Designer

Printed in the UK by The Magazine Printing Company using only paper from FSC/PEFC suppliers www.magprint.co.uk

Gillian Bowman

Sean Leavey

Eamon Quinlan

Writer

Writer

Writer

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Letter from the editor ... mathematics, such as the beauty in the Riemann Hy­ pothesis, what the universal formula for attraction is, 2012 is the year of our third as well as the different types of infinity, actual and po­ birthday and we would like tential. Our mathematical discussion then continues with to invite you to celebrate with us through our new mathe­ the subject of fluid dynamos and how they generate matical adventure. Unlike a the magnetic fields of astrophysical objects such as the traditional birthday, we are Earth. Several different aspects of mathematical modelling extremely happy to be able to give you the copy of the latest shall be also concerned and if you keep reading, we issue of our magazine as a promise that we’ll show you how mathematics helps present and we hope that you us model the spread of ideas, or even the spread of a will enjoy reading it as much biological disease. Mathematical models also provide as we enjoyed preparing it for us with a more profound understanding of the partic­ you. Now let the trip in the maths wonderland begin. ularities of evolution, through idealisations such as cel­ lular automata. We shall also reveal the key behind the question why a particular species of insect has devel­ Maths is fun, don’t you think? oped a mating cycle of a prime number of years, and Whether for modelling football score predictions, un­ how random this “choice” actually has been. Further, ravelling the genetic code or saving human lives you’ll learn what the mathematical properties of the through tomography, mathematics remains the most world’s smallest football are. A closer look will be taken at the story of a happy fascinating subject of all. Not only mathematics is fun, but it is also incommensurably beautiful. The evi­ gnome couple, Mr and Mrs Magnus, as well as at two dences are numerous and the only difficulty one may popular ancient legends and what they share in com­ face is where to start – shall it be Euler’s identity which mon. In addition, we present a special case of the Apol­ states that eiπ + 1 = 0, or his fundamental theorem of lonius problem and its solution. Hang tight because apart from the aforementioned the infinitude of primes. Perhaps the trip could begin from the various proofs of the Pythagorean theorem topics, we shall try to interest you with many more. each one more elegant than the other, Euler­MacLau­ While attendance in class will not be recorded, we shall rin’s formula, or the complex 248­dimensional E8 test your knowledge in [The, Commutator] 3A course structure, which has been used as a basis in various with an “unexpected” examination in the form of two candidates for the “theory of everything”. While some independent cross­number puzzles. You are also in­ people would commence the exploration with Euler­ vited to participate into the first Maths & Stats Article Mascheroni’s constant γ, others might prefer to set up Competition 2013 organised by our magazine. Newton’s iteration for matrix inversion as a starting As usual, the editorial staff of [The, Commutator] point. would like to ask you not to hesitate to share any com­ ments, ideas or suggestions, should you think this So, have we managed to convince you yet? might help the improvement of the magazine, which Well, we at [The, Commutator] believe Maths is quite has always been our primary goal. fun, and shall amplify our meaning of these words in the subsequent paragraphs. Throughout the pages of Warmest Regards, the current issue we shall discuss, not necessarily in this particular order, various fascinating topics in Dear Reader,

Hristo Georgiev Editor in Cheif

Radko Kotev

Fiona Madden

Karl Nordström

Writer

Writer

Writer

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A Note On The Current Issue Two of the articles that we are presenting in current issue, Diophantine Problems for Garden Gnomes and The Carol Syndrome, have been kindly provided by the popular Plus magazine (where they were first published) and their authors, Emily Woodhouse ­ an undergraduate student at Durham University, and José­Manuel Rey ­ Associate Professor in the Faculty of Economics at the Universidad Complutense in Madrid. Plus is an internet magazine edited by Marianne Freiberger and Rachel Thomas. Freelancer Charles Trevelyan is in charge of the graphics design and Owen Smith is computer officer. The magazine aims to introduce readers to the beauty and the practical applications of mathematics. It provides feature articles, which describe applications of maths to real­world problems, games, and puzzles; reviews of popular maths books and events; a news section, showing how recent news stories were often based on some underlying piece of maths that never made it to the newspapers; a lucky dip of mathematical curiosities; and opinions on various maths­related topics and news sto­ ries; a regular interview with someone in a maths­related career, showing the wide range of uses maths gets put to in the real world. All past issues are available online, which is a useful resource for maths school students and teachers. Plus is part of the Millennium Mathematics Project, a long term national initiative based in Cambridge and active across the UK and internationally. The MMP aims to help people of all ages and abilities share in the excitement of mathematics and understand the enormous range and importance of its ap­ plications to science and commerce. It works to change peopleʹs attitudes to maths, to act as a national focus for renewing and improving appreciation of the dynamic importance of maths and its applications, and to demonstrate the vital contribution of maths to shaping the everyday world. You can read the latest mathematical news on the site every week, browse the blog, listen to the podcasts and keep up­to­date by subscribing to Plus (on email, RSS, Facebook, iTunes or Twitter) at http://plus.maths.org.

Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An in­ ternational journal, its policy is to feature a wide variety of re­ search areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.

A Note On The Sponsor

“Piled Higher and Deeper” by Jorge Cham www.phdcomics.com

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Diophantine Problems For

Garden Gnomes

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Mr and Mrs Magnus are a happy gnome couple. Unfortunately, forseeing that they will be unable to keep up with their rising mortgage repayments, they’ve been forced to move into my back garden (where salaries are much higher). Emily Woodhouse

Durham University

Suppose you are asked to solve the equation 25x + 11y = 177.

r4 . Keep repeating this process and eventually you get a remainder of zero. The last non­zero remainder is the high­ est common factor of the original two numbers.

There are two unknowns and only one equation, so you’ll get infinitely many The question is, what is the smallest solutions. Rearranging, you get So, Zurich lets r1 = 25 and r2 = 11 number of bricks they have to buy to from our previous equation. He then comply with the local council, yet not performs the algorithm with some zest. 177 - 25x y= , waste any money? 11 At each step, he keeps track of the cal­ Luckily, Mr and Mrs Magnus’ son, culation by writing ri - qri + 1 = ri + 2 . so you know what shape these solu­ Zurich, knows a thing or two about Here ri is the number thatʹs being di­ tions take. numbers. As his parents discuss their vided at that step, ri + 1 is the number itʹs But now suppose youʹre asked to find dilemma over tea, Zurich writes out an being divided by and ri + 2 is the corre­ equation: a particular solution (x,y) such that both sponding remainder: x and y are positive whole numbers 25x + 11y = 177, and their sum is as small as possible. You could use trial and error to find the solution (if it exists), but this is mathe­ where x and y are the height in bricks matics, and in mathematics we like a bit of the two respective walls. Chuffed of method and order. Fortunately, one with himself, he shows his parents and method for integer solutions came out receives a solemn ʺthat’s nice dearʺ. of Alexandria in about 250 AD. But, be­ This tells him that the highest com­ Duly encouraged, Zurich sits and fore we get into that, letʹs add a little context to make things more interesting. thinks about how to solve this equation. mon factor is 1 (if he goes any further The secret here is that you cannot have he will reach zero). Fortunately this is Mr and Mrs Magnus are a happy half a brick: x and y must be integers. what we would expect, as 11 is a prime gnome couple. Unfortunately, foresee­ (This, by the way, rules out that the two number. Now he uses the equations ing that they will be unable to keep up walls have the same height, ie that that he has just generated to write out a true statement equal to 1 that involves with their rising mortgage repayments, x = y , because 25 and 11: theyʹve been forced to move into my back garden (where salaries are much 3-2#1 = 1 25x + 11x = 36x = 177 higher). However, they have been pre­ (25 - 11 # 2) - (11 - 3 # 3) = 1 sented with some trouble in building their new home. Gnome by­laws state gives that x = 116/36 , which isn’t a He keeps 25 and 11 as separate factors, that the total number of bricks used in whole number.) so: any construction project must be 177 or 25 - 11 # 3 + 3 # 3 = 1 Now, Zurich has recently been read­ planning permission will not be ing Euclid’s Elements before he goes to granted. 25 - 11 # 3 + 3(25 - 11 # 2) = 1 bed and so has just found out about a 25 + 25 # 3 - 11 # 3 - 11 # 6 = 1 As gnome houses form roughly the nifty trick called Euclid’s algorithm, shape of a triangular prism and one which finds the highest common factor wall (namely the garden fence) is al­ of two numbers, r1 and r2 . We assume Collecting terms: ready in place, only two walls need to r1 > r2 . First divide r1 by r2 and take the 24 # 4 - 11 # 9 = 1 be built. The plot of land they have ac­ remainder, calling it r3 . Then divide r2 quired is shown below with all dimen­ by r3 , and take the remainder, calling it Let’s compare that to the original: sions measured in bricks.

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25x + 11y = 177, So, Zurich rewrites his sum as: 25(4) + 11( - 9) = 1 Now, that’s all very well and good, but it needs to equal 177 not 1, so still treating 25 and 11 as terms, he multiplies the whole equation by 177: 25(708) + 11( - 1593) = 177.

Zurich compares this to the original problem and finds that the solution is and the problem is solved just as his parents finish their tea.

Diophantine equations Generally, equations of the form ax + by = c where the variables x and y are only allowed to be whole numbers are called linear Diophantine equations after the great Greek arithmetician Diophantus, who dealt with many such problems in his works. If c is a multiple of the greatest common divisor of a and

This is a perfectly legitimate answer, but sadly it is very difficult to come by negative bricks these days, and it cer­ tainly doesnʹt look like the smallest answer. So, Zurich pon­ ders how to make y positive without changing the sum of b , then an equation of this form has an infinite number of the equation from 177. To make the coefficient of 11 posi­ solutions. This is the case in our example equation, where tive, he needs to add some multiple of 11, say 11n , to the a = 25 , b = 11 and c = 117 , with the greatest common left hand side of the equation. To keep the equation true, divisor of a and b being 1. The further constraint that the he needs to take this multiple away again: solutions must be positive and their sum as small as possi­ ble then gave us the particular result. If c isn’t a multiple 25(708) + 11( - 1593) = 25(708) + 11( - 1593 + n) - 11n

of the highest common factor of a and b , then the equation has no integer solutions at all.

= 177 To get the equation into the required form, he writes

There are also diophantine equations of higher degree, for example n

n

x +y = z

n

25(708 - 11n/25) + 11( - 1593 + n) = 177. Since 11n/25 must be a whole number, he nows that n must be a multiple of 25. Say n = 25t for some t . This gives 25(708 - 11t) + 11( - 1593 + 25t)

where x , y and z are the variables and n is a constant. The statement that no integer solutions exists when n>2 is Fer­ mat’s famous last theorem, which remained unproved for over 400 years. (It was in the margin of his copy of Dio­ phantus’ book Arithmetica that Fermat wrote his renowned lines. See the Plus article Fermat’s last theorem and Andrew Wiles for more information.)

In terms of practical applications, being able to solve this sort of problem is surprisingly useful even in the modern So now Zurich only needs to find a number t so that world. The classic example is working out how many men, women and children attended a show given only the price - 1593 + 25t is greater than zero but as small as possible. of each ticket and the final takings. Plus, if you are that way Dividing 1593 by 25 using longdivision he gets: inclined, certain types of puzzles in magazines and news­ papers can suddenly become a whole lot simpler with this under your belt. From business to coffee breaks, exploring number theory to building your house, itʹs definitely a tech­ nique worth knowing! = 177

Therefore, the number he is looking for is t = 64 , giving 25(708 - (64 # 11)) + 11( - 1593 + (64 # 25)) = 177

P.S.: In his excitement, Zurich forgot to make an al­ lowance for the front door in one of the walls. Luckily, his skilled father was able to make a chimney out of the extra bricks at the same time as close the gap that must be cre­ ated where the two walls and the roof join.

25(708 - 704) + 11( - 1593 + 1600) = 177 25(4) + 11(7) = 177.

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Karl Nordström

The Riemann Hypothesis The Riemann Hypothesis has been considered one of the most important unproven conjectures in pure mathematics ever since Bernhard Riemann stated it in his legendary paper On the Number of Primes Less Than a Given Magnitude (1859). Unfortunately it is quite difficult to correctly state the hypothesis in a simple way, but this article will make an attempt to explain Riemann’s conjecture by examining some of the historical background to his paper and the discoveries he made in it, stating the conjecture in its original form, and give an equivalent conjecture that is easier to understand by intuition. Finally it will show how the hypothesis, if proven true, would allow for a much better approximation to the prime counting function Π(x).

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Of central importance to the hypothesis is the Riemann zeta function, defined as

be derived using the Liouville function. It is defined as

m (m) = (- 1) ~(m)

3

g (s) = / 1s n=1 n for a complex variable s . About a century before Riemann wrote his paper, Euler had shown that sums of this form are equivalent to infinite products using prime numbers of the following form: 3

/ n1

s

n=1

%

1 1 - p- s p = prime 1 # 1 # 1 g, = 1 - 2- s 1 - 3- s 1 - 5- s

=

s ! R. The proof for this equality is very beautiful and can be derived from the fundamental theorem of arithmetic, but is also outside the scope of this article. It resulted in estab­ lishing of the connection between the zeta function and prime numbers. Euler also calculated the values of the sum for positive integer values of s up to 16, where ζ(2), in par­ ticular, gives the solution to the Basel Problem. However, Riemann was the first person to consider s as a complex variable, s = σ + it rather than simply a real number. His aim was to discover a way to accurately count the number of primes smaller than a given value. By analytically con­ tinuing the zeta function to all complex numbers, except for s = ­1 ­ an impressive feat in itself ­ he managed to de­ rive an explicit formula for doing this. The formula implied that the real part of the zeros of the Riemann zeta function control how the values of prime numbers vary from their expected values. In order to find the zeros of the function, we can start by realising that since convergent infinite products never be­ come zero, by Euler’s product equality there can not exist any zeros when the real part of s is greater than 1, so ζ(s)≠0 when ℜ(s) > 1. Riemann also noted that all points where the real part of s was a negative even integer, so that ℜ(s) = ­2, ­4, ­6 ..., are zeros. These are called the trivial zeros of the function. From his explicit formula he derived that any non-trivial zero could not take a negative real value, thus any non­trivial s such that ζ(s) = 0 must be in the interval 0 ≤ ℜ(s) ≤ 1. The Riemann Hypothesis can now be stated in its origi­ nal form: any non­trivial zero has ℜ(s) = ½. Riemann did not attempt to give a proof for this statement in his paper and only based the conjecture on his calculations for the values of the first few non­trivial zeros, but despite the nu­ merous attempts of many brilliant mathematicians the proof is just as elusive today as it was in 1859. The range of possible values for ℜ(s) has been narrowed down some­ what over the years. Hardy proved that there exists an in­ finite number of zeros on the critical ℜ(s) = ½ line, and it has been computationally checked to hold true for the first ten trillion zeros or so, but such results are rather incon­ clusive. Since this way of stating the hypothesis is very abstract, a more easily underastandable equivalent statement can

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where ~ (m) is the number of prime factors of m , which are not necessarily distinct. In 1899, Edmund Landau showed that the Riemann Hypothesis is equivalent to the statement that for any fixed ε > 0,

m (1) + m (2) + f + m (n) 0. lim = 1 n"3 n 2 +f Looking at the numerator, this states that every integer has an equal chance of having an either even or odd number of prime factors. This is another way of stating the Rie­ mann Hypothesis, and arguably the easiest to understand by intuition. It also gives some idea of the far­reaching im­ plications of the conjecture. The one presented here is only one of many significant results in number theory which could be reached, if it were assumed that the Riemann Hy­ pothesis was true. Returning to the problem of counting primes, recall that the zeros of the zeta function are related to the oscillations of prime numbers from their expected values. It has been shown that, assuming the conjecture is true, the error term in the prime counting function defined by the prime num­ ber theorem can be very accurately stated. For any real x , x

#

r (x) = 2

dt + O ( x log x) log t

where O is the big-O notation. The x logx error term is derived straight from the Riemann Hypothesis, and is a much better approximation than any other that has been made so far. In fact, it has been proved to be the best pos­ sible approximation. Finally, this is only a very shallow introduction to the Riemann Hypothesis. As for further reading, the book The Riemann Hypothesis - For the Afficionado and Virtuoso Alike by Borwein, Choi, Rooney and Weirathmueller (2007) goes into much more detail and shows how Riemann continued the zeta function step by step. It also contains many of the most important historical papers related to the hypothesis as appendices. References Bombieri, E. (2000), The Riemann Hypothesis, http://www.clay­ math.org/millennium/Riemann_Hypothesis/riemann.pdf Borwein, P., Choi, S., Rooney, B. and Weirathmueller, A. (2007), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, New York: Springer Conrey, J.B. (2003), The Riemann Hypothesis, http://www.ams.no ices/200303/fea­conrey­web.pdf Gourdon, X. (2004), The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height, http://numbers.compu­ tation.free.fr/Constants/Miscellaneous/zetazeros1e13­1e24.pdf Sarnak, P. (2008), Problems of the Millennium: The Riemann Hypothesis, http://www.claymath.org/millennium/Riemann_Hypothe­ sis/Sarnak_RH.pdf

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The Carol Syndrome Words: José­Manuel Rey (left) Universidad Complutense, Madrid

Art: Gianni Peg (right)

My friend Carol is nice and beautiful. Anyone would bet she has plenty of dates. But it turns out that this is not the case. The fact is that Carol has dated no­one for ages. And although she is shy by na­ ture, she is also open to adequate proposals and would love to find someone special. But Carol claims that men do not usually approach her. She thinks she frightens them away. Is it just a matter of bad luck? Or is it something else? Maybe Carol has a distorted perception of reality?

Luck is an issue that’s naturally addressed in mathematics. If it is a question of luck, mathematics may shed some light on Carolʹs problem.

(like reading the last issue of Plus).

(a) He talks to Carol and she responds in a friendly manner. He gets her phone number and a proper date next week.

Guy evaluates the outcomes assign­ ing the numbers a, b and 0 to options (a), (b), and (c), respectively, with a > b > 0. By this he means that he prefers (a) to (b) and (b) to the worst scenario (c).

fact that the outcomes depend dra­ matically on the actions of other guys independently considering whether to approach Carol or not. Assume that Guy thinks that he will get (a) only if no one else approaches Carol and (c) when he is not the only one approach­ ing Carol. Of course if he does not talk to Carol, he gets (b). Guy’s modesty seems understandable if he does not trust himself as a strong candidate in the face of competition.

(b) He does not approach Carol. He can enjoy another rewarding task

Now Guy realises that he is not the only man in town. He is aware of the

The fact that his choices depend so strongly on the choices of others is

Consider a man — let’s call him Guy — who is attracted to Carol and has the opportunity to talk to her, let’s say at a coffee bar. Realising that Carol is shy he considers whether or not to approach her. Guy considers the possible outcomes:

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(c) He talks to Carol and she proves uninterested. He will feel misevrable for a week.

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what makes this an interactive decision problem. The study of how individual behaviour is conditioned by the so­ cial environment is the objective of social psychology. The interaction in­ volved here is the trademark of game theory, which was developed in the twentieth century.

Maths about Carol The solution of Guy’s problem, and that of all of the guys, is a bundle of actions deemed rational within the described framework. The assump­ tion that everyone behaves rationally may seem slightly unrealistic, but it’s crucial in the game theoretical ap­ proach, since mathematics can’t ac­ count for someone behaving irrationally or against their own in­ terest. The symmetry of the problem — Guy may be anyone — implies that all will act in the same way, since all will make the same rational con­ siderations. Rationality permits to discard the symmetric solution in which no one talks to Carol: given that none of the others talk to her, Guy gets a better outcome by ap­ proaching her, so rationality would imply that Guy does approach her. A similar reasoning permits the dis­ carding of the solution in which everyone talks to Carol. There are no

more apparent symmetric solutions. Guy needs to be subtle to proceed. He may think of solving his dilemma by tossing a fair coin. This amounts to Guy being totally uncertain about what to do. If we admit that “being

This surprising phenomenon ­ which we call the Carol syndrome ­ is a by­product of psychological social interaction. uncertain” is a possible solution to his dilemma, is the 50­50 chance truly rational, or would another level of uncertainty, say 30­70 in favour of ap­ proaching Carol, be better? Is there even a rational way to be uncertain? This seems an interesting idea. Let us see where it takes us. Guy’s uncertainty is described by the probability p of approaching Carol (so 1 ­ p is the probability of not approaching her). By symmetry, everyone’s uncertainty is represented by the same number p ­ everyone in­

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dependently makes his choice ac­ cording to p. The goal is to find the best value p* of p. The value p* can be obtained indi­ rectly via the following observation. The bundle of common uncertainties defined by p* is rational only if Guy (and any other guy) associates the same level of “reward” with his two possible actions, approaching Carol or not approaching her, given that the rest are all uncertain with probability p*. Otherwise there is no uncertainty about what to do: Guy would go for the action with the higher reward. But how can Guy assign a reward value to a bundle of uncertainties? One possible answer, given by John Von Neumann and Oskar Morgen­ stern, is to average the reward values assigned to all possible bundles of ac­ tions with weights equal to their probability of occurrence. This is called expected valuation. Rationality based on expected valuations has been the major paradigm for decision making analysis since the 1950s. If there are N guys deciding inde­ pendently, then there are 2N possible bundles of actions. Guy makes his calculation as follows: (1) The bundle in which only our Guy

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pnone > ½ and the odds are that no­ body will talk to Carol. This is true irrespective of the number of guys and it becomes worse for Carol as that number increases.

approaches Carol yields (a) ­ with reward value a – and occurs with probability (1 ­ p)N­1, since all deci­ sions are taken independently. Any bundle in which both Guy and someone else approach Carol yields (c) and has reward value 0. Thus the expected value of approaching Carol is a(1 ­ p)N­1. (2) Any bundle of actions in which Guy does not talk to Carol has value b whatever the others do. Thus the expected value of not approaching Carol is clearly b.

matter how stunning Carol may be. Hence the following (un)veiled mechanism may be inhibiting Guy from approaching Carol: (1) The larger the number of Carol’s admir­ ers the more probable it is that Guy will not talk to her; (2) The more at­ tractive Carol is, the more likely it is that there will be lots of guys con­ sidering this decision problem. Guy is thus led to believe that N is large and p* correspondingly small. In consequence he will very likely choose curling up with Plus rather than risking Carol’s rejection.

Guy is uncertain about what to do when these two values coincide: a (1 - b)

N-1

The Carol syndrome

= b.

Solving for p he gets ) p = 1 -` bj a

1 N-1

.

Since a > p, p* lies between 0 and 1 and therefore defines a proper prob­ ability. The conclusion is that anyone should approach Carol with proba­ bility p* This is the rational solution for the Carol dilemma when she has N identical admirers. Since a > b for N not too large it is possible that a > 2N­1b, in which case p* > ½ and the odds are that Guy will talk to Carol. However, as soon > a the b possibility of no one approaching Carol becomes more likely. In fact, p* approaches 0 as N gets large, no as N is large enough, 2

N-1

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The important point of Carol’s feeling about frightening guys away is not the probability p* but the probability pnone of no one talking to her. Since all the guys act independ­ ently, N ) N pnone = (1 - p ) = ` a j N - 1 . b

In this expression pnone increases as N increases. The entry of new poten­ tial dates adds to the probability that Carol is left alone. Also, as N gets large, pnone does not vanish but tends to b . Therefore, a pnone always lies between two val­ ues: 2 ` ab j < pnone < ab . In particular, as long as a is not much larger than b, we have that

Carol’s perception that she scares men away is not a delusion after all. According to the mathematics above, she may be justified in think­ ing that guys stay away from her. It is not a matter of bad luck but a col­ lateral effect of interactive rational­ ity. A paradoxical consequence is that Carol’s attractiveness acts as a repellent. This surprising phenome­ non ­ which we call the Carol syn­ drome ­ is a by­product of psychological social interaction.

Scary evidence The Carol syndrome is not a the­ oretical artifact. As striking as it may appear, the syndrome is often re­ ported by attractive women and men. A web search brings up some notorious cases. In an interview for The Sunday Times in February 2008, American actress Uma Thurman was reported as saying that “men rarely chatted her up”. She consid­ ered her bad luck with men a life­ long curse. The same was true of American star singer Jessica Simp­ son who declared in a TV show in September 2006: “I scare guys away”. Another example, reported in March 2009 by The Telegraph on­ line, is that of 19­year old British ac­ tress Emma Watson, who said that her starring character in Harry Pot­ ter’s saga, while bringing her world­ wide fame, “scared boys away”. Hence Carol’s luck is not odd, al­ though it obviously differs from celebrities like Ms Thurman. Math­ ematics can explain how. The num­ ber of Ms Thurman’s admirers is huge so that the ceiling value b is a a good approximation for pnone and this somehow eases Thurman’s suf­ ferings. For Carol, N is not as large, but the ratio b ­ and in turn pnone ­ a is closer to 1. This is what causes Carol’s Carol syndrome.

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To Infinity and Beyond The concept of infinity has captured the imagination of mathematicians and philosophers for centuries and will doubtlessly continue to far into the future. This article will discuss the different types of infinity, actual and potential, and the possibility that there are actually different sizes of infinity.

Fiona Madden

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Around 460 BC, Zeno of Elea wrote one of the first texts about infinity and although none of his texts have sur­ vived, his paradoxes are some of the most quoted on the subject of infinity. Probably the most famous is the par­ adox of Achilles and the tortoise. Here Achilles is running a race against the tortoise and he gives the tortoise a 10­ metre head start. Say Achilles runs 10 times faster than the tortoise. Then by the time Achilles has run 10m, the tor­ toise will have moved a further 1m forward. Then when Achilles has reached the point 1m ahead, the tor­ toise will be a further 1/10m ahead and so on ad infinitum. So the tortoise must win the race. Another paradox along the same lines is that of Achilles running a race. Before he can run the full distance, he must run half the dis­ tance. But before he runs half the dis­ tance he must run a quarter of that distance then an eighth, a sixteenth and so on ad infinitum. The paradox can then be modified so that Achilles can never actually start the race. Any mathematician with a basic understanding of convergence and limits can conclude for the series:

and drew a distinction between actual and potential infinity. In its simplest form potential infinity is the idea that something could go on forever, should enough effort has been ap­ plied, such as continuing a geometric line. Actual infinity, on the other hand, is something that is completed and definite, and consists of infinitely many elements. Actual and potential infinity can be explained using the earlier paradoxes: You could actualise a point by stopping on it, or marking it, otherwise there remain potential points. If you stopped and marked every point, this would take an infi­ nite amount of time, but if you left most points as potential, unmarked points, you would arrive at your des­ tination without a problem. This fol­ lows Aristotle’s view that the actually infinite is impossible and potential in­ finity is the only plausible idea of in­ finity. Jules Henri Poincaré agreed with this: ‘Actual infinity does not exist. What we call infinite is only the endless possibility of creating new ob­ jects no matter how many objects exist already.’

10 + 1 + 1 + 1 + f " 11 1 . 10 100 9 But is this really solving the puzzle or simply ignoring the real problem by throwing some calculus at it and by­ passing the conceptual problem? A possible solution is to think of a “smallest distance”, where the process of dividing a distance in half (or in tenths) must come to an end. This is much the same as the age­old fact that you cannot fold a piece of paper more than seven times (although appar­ ently 12 folds have been achieved but the idea remains the same). Concep­ tually, you should be able to keep folding a piece of paper forever but in practice this is not the case. Could this be the same for distances? Accepting this would mean that space is no longer continuous but discrete and the same, mutatis mutandis, for time. So the question now is: are the notions of time and space truly continuous and, therefore, always divisible (sometimes called ‘everywhere dense’) or are time and space discrete?

This stance is very tempting and does seem to solve most of the prob­ lems related to infinity but can we re­ ally accept that something can only be potentially infinite? Georg Cantor firmly disputed this and posed a very good argument in favour of actual in­ finity using cardinal numbers.

This now brings up the issue of potential infinity and actual infinity. Aris­ totle recognised the confusion surrounding the concept of infinity

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resultantly, there would always be room for another infinite number of guests. Infinite cardinals can “absorb” finite and even some infinite cardi­ nals without changing their own car­ dinality. This seems completely counter intuitive though, so let’s start with the basics. Cardinality simply means the size of a set, which is just how many ele­ ments it has. Cardinal numbers are an indication of ‘how many’ so neither order nor magnitude has any effect on cardinality. This can be illustrated by a trivial example: Let A = {1, 1, 2}, B = {2, 1, 1}, C = {96, 1022, 3478}. Let us de­ note the cardinalities of A, B and C by #A, #B and #C respectively. Then #A = #B = #C = 3. Now let us consider the set of natural numbers N and the set of natural number that are divisible by 2 (i.e. even). If the cardinality of N is infinity then surely the cardinality of the set of even numbers is half this. But then what is half of infinity? Per­ haps half of infinity is also infinity just as in Hilbert’s hotel. A fairly basic defi­ nition is that two sets are of the same size if, and only if, they can be placed in a one­to­one correspondence. Let the set of even numbers be called E . It is a proper subset of N but can be placed in a one­to­one correspon­ dence with N : N 1 2 3 4 5 6

7 f

E 2 4 6 8 10 12 14 f To introduce the concept of infinite cardinals, it is useful to think of David Hilbert’s hotel paradox. Suppose there was a hotel with an infinite number of rooms. A large mathematics confer­ ence is taking place and the delegates are to stay in this hotel. An infinite amount of delegates turn up and are accommodated in the hotel starting in room 1, then 2, 3, 4 and so on. Then a tourist arrives at the hotel looking for a room. The hotel has an infinite num­ ber of rooms currently occupied by an infinite number of people. The recep­ tionist could take the tourist to the room at the very end but that would involve walking an infinitely long way. Instead, the receptionist asks all of the delegates to move along one room so that room 1 is now unoccupied and the tourist can have this room. By the same argument there would always be room for one more in this hotel and,

We could carry on pairing indefinitely because both sets are infinite, so N and E are the same size. This leads to Richard Dedekind’s definition of an infinite set: ‘A set is infinite if and only if it can be placed in one-to-one correspondence with one if its proper subsets’ Therefore N is infinite. E is also infi­ nite because a proper subset of E , say the set of even numbers larger than 2, can be placed in a one­to­one corre­ spondence with it. This set can also be proved to be infinite and so on. This definition distinguishes finite from in­ finite cardinal numbers. This idea is fine for natural num­ bers and can easily be extended for in­

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tegers, Z , but what about rational numbers, Q ? A rational number is one of the form m/n where both m and n are natural numbers. For any two ra­ tional numbers, there will always be a third in between them. For example take 1/2 (= 6/12) and 1/3 (= 4/12): there is 5/12 between them. Then take 1/3 (= 8/24) and 5/12 (= 10/24): there is 9/24 between them. And so it follows that there is an infinite number of rational numbers between any two rational numbers. It now looks as if there are far many more rational numbers than natural numbers, but it is actually pos­ sible to place Q in a one­to­one corre­

tional numbers. Irrational numbers are any numbers unable to be denoted

Figure 1. Spiral as m/n, such as 0.1211211121.... or π (= 3.1415926535897...). Irrational num­ bers have an infinite, non­repeating expansion, i.e. the numbers after the decimal point do not form a pattern that is exactly repeated.

spondence with N ? How can anyone start to write down all the rational numbers? Just by writing two ratio­ nals you have then faced an infinite amount in between. To avoid this The set of real numbers is called the problem, consider a tabular represen­ ‘continuum’ as it is a smooth line with tation of the rational numbers shown no gaps, whereas the rational number in Fig. 1. The table can be extended to line has gaps where irrational num­ infinity in all directions, and zero has bers go. Gregor Cantor used the idea been left out of the denominator for of “diagonalisation” to be the first to obvious reasons. prove that R has cardinality greater than ℵ0. The proof is simple but aston­ By following the spiral around, we ishing, and in fact Cantor himself said can list the rational numbers, leaving ‘I see it but I don’t believe it’ when writ­ out any repetitions. Clearly, all ra­ ing to a friend about his work. The tional numbers will be found eventu­ proof is a reductio ad absurdum argu­ ally using this method. Then, by ment where we first suppose that R is putting them in a list, Q can be placed the same size as N , show that this in a one­to­one correspondence with leads to a contradiction, conclude that N as so: the two sets are of different sizes, and

So N and Q have the same cardinality. then finally conclude that R must have greater cardinality. This cardinality can be denoted by the symbol ℵ0 where ℵ is the capital Proof: of the first letter of the Hebrew alpha­ Suppose R and N are the same size. bet, aleph, and 0 is simply zero. ℵ0 is Place the real numbers in a one­to­one spoken of as aleph null or aleph naught. correspondence with the naturals, So now we can say that all infinite sets start with just the reals from 0 to 1, that have been discussed here ( E , N , each with an infinite expansion Z , Q ), as well as any other infinite sets have cardinality ℵ0. Such sets are also N 1

2

3 4

5

6

7 f

Q 0 -1 1 1 2 -1 2 -2 2 f known as ‘countably infinite’ and it is sets with a countably infinite cardinal­ ity that can be absorbed by other infi­ nite sets, as referred to in Hilbert’s hotel paradox. Now, if a set can be countably infinite, surely there must be such a thing as an uncountably in­ finite set with a cardinality greater than ℵ0. The real numbers, R , are the set of all rational numbers and all irra­

bers, it should appear somewhere on the list. Now for each digit in D add 1 to it, unless it is 9 in which case turn it to a 1. Call this number ‘Cantor’s Diagonal’. How­ ever, Cantor’s Di­ agonal will not turn up on the list above because it is different to the first number on the list by at least the first decimal, different to the sec­ ond ­ by at least the second decimal and so on. This means that all the real numbers between 0 and 1 cannot be put in a one­to­one correspondence with the natural numbers which con­ tradicts the original supposition. So the set of real numbers from 0 to 1 is not of the same size as the natural numbers, mutatis mutandis, for the whole set of real numbers. Since N is a proper subset of R , R must have larger cardinality. (In fact even R (0,1) has greater cardinality than N .) So it has been proven that there are, in fact, different sizes of infinity and the existence of the continuum makes the concept of actual infinity possible. This, in turn, contradicts the earlier idea that space and time are discrete and there is some smallest value of them. Whichever view seems more sensible Aristotle sums up the discus­ sion of infinity perfectly: ‘The fundamental difficulty with the theory of the infinite is this: either an outright denial or an outright acknowledgement of the being of the infinite leads to many impossibilities.’

(though often trivial, such as 0.2000...). Listing them in an order using ax,y where x is the ordering and y is the References digit itself: 1 0.a1, 1 a1, 2 a1, 3 a1, 4 a1, 5 f Friend, M. (2007) Introducing Philosophy of 2 0.a2, 1 a2, 2 a2, 3 a2, 4 a2, 5 f

Mathematics. Stocksfield: Acumen Publish­ ing.

3 0.a3, 1 a3, 2 a3, 3 a3, 4 a3, 5 f h So if the third number in the list is 0.38594..., then a3,2 refers to 8, a3,4 refers to 9 and so on. Consider a number made up of all the diagonals, 0.a1,1a2,2a3,3 and so on, and call this D. Since the list contains all the real num­

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Russell, B. (1956) The Principles of Mathematics. London: George Allen & Unwin Ltd Meschkowski, H. (1965) Evolution of Mathematical Thought. California: Holden­Day Inc. (translated by Jane H. Gayl)

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The Birthday Problem Hristo Georgiev A famous puzzle asks about the minimum num­ ber of people in a group so that the probability that at least two of them have the same birthday is grater than ½. In order to find a solution, we begin by stating some assumptions. First, we assume that the birth­ days of the people in the group are independent, and that each birthday is equally likely out of 366 days in a year, which is our second assumption. The probability that at least two in group of n people have the same birthday is 1 ­ pn, where pn is the probability that all of the people in the group have birthdays on different days. To calculate this probability, we should consider the n people in some fixed order, as a sequence of elements. What is important for us is to compute how probable it is for the kth person to have a different birthday than the first k­1 people in this sequence. For instance, the birthday of the first person certainly does not match any other birthday of the people before him in the sequence, because k­1, when k=1, equals 0. The probability that the birthday of the second per­ son is different that the birthdays of the people be­ fore him in the sequence, namely just the first person, is 365/366. We can generalise this to

?

How can you say with certainty if a given number is a perfect square (without applying the square root function)?

A

By counting the number of its factors. All square numbers have an odd number of factors.

55 + 45 + 75 + 45 + 85 = 54748 212 + 222 + 232 + 242 = 252 + 262 + 272 1111111112 = 12345678987654321

9/5 + √(9/5) = 3.1416+ A remarkable approximation to pi

Hristo Georgiev

366 - (k - 1) = 367 - k , 366 366 which indicates the probability that the kth person in the sequence has a different birthday than the preceding k­1 people, where 2 ≤ k ≤ 366. Now we can conclude for pn that: pk =

pn = 365 364 363 f 367 - n . 366 366 366 366 Then, for 1 ­ pn we have: 1 - pn = 1 - 365 364 363 f 367 - n . 366 366 366 366 Thus, with the aid of a calculator (or a computer program), we conclude that the minimum number of people needed, so that the probability that two have the same birthday is greater than ½, is 23 (with 1 ­ pn = 0.506323). One of the Birthday problem applications is find­ ing the probability of a collision in a hashing func­ tion, which is the mapping of the keys (of elements that are to be stored in a table or database) to stor­ age locations. References

Number Curiosities

? A Consider the digits from 1 to 9, and from 0 to 9. Arrange them in descend­ ing order, reverse and sub­ tract. The same nine digits reappear in the answer: 987654321 ­ 123456789 864197532

How can you determine if a given number is a power of 2? Convert the number to base­2 numeral system. If its binary representation consists of only one 1, fol­ lowed by k zeros,then this number is 2 to the power of k. This can be generalised for determining if a given number is a power of n, by converting the num­ ber to base­n numeral system and checking if the number of 1s is one.

9876543210 ­ 0123456789 9753086421

Rossen, K. H. (2007) Discrete Mathematics and Its Applications 6th ed. New York: McGraw­Hill

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Cellular Automata and Conway’s Game of Life

Hristo Georgiev

The subject of Cellular automata was ‘born’ in the early 1950s, with John von Neumann and Stanislaw Ulam pioneer­ ing the field. By definition, cellular au­ tomata are mathematical idealisations of physical systems in which both space and time are discrete, and physical quan­ tities take on a finite set of discrete val­ ues. A cellular automaton consists of an n­dimensional array of variables and evolves in discrete time steps, with the value at each position in the array being affected by the values of variables at lo­ cations in its neighbourhood on the pre­ vious time step. For two­dimensional cellular au­ tomata, there exist two ways of defining the relative location of a given cell. Von Neumann neighbourhood is named after John von Neumann, the creator of the first self­replicating cellular au­ tomata. This neighbourhood is diamond­ shaped and it represents the set of all cells that are orthogonally­adjacent to the region of interest, which in the simplest case is the single cell (x0,y0). The von Neumann neighbourhood of range r is defined by:

No(x ,y ) = {(x, y):| x - x |+| y - y | #r}. 0 0

0

0

Von Neumann neighbourhoods of ranges r=0, 1, 2, 2 and 3 are illustrated in Fig. 1. Usually, if no range is specified when a von Neumann neighbourhood is being referred to, it is assumed that the given case is r=1, by default. The number

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‘I expect the children of 50 years from now will learn cellular automata before they learn algebra.’ Stephen Wolfram, New Scientist, Nov 18, 2006

of cells in a von Neumann neighbour­ hood of range r is given by the general term r2 + (r + 1)2 and the sequence they form has been recorded in Sloane's Online Encyclopaedia of Integer Sequences under the identifier A001844: 1, 5, 13, 25, 41, 61, 85, ... Moore neighbourhood is named after Ed­ ward F. Moore and is a square­shaped neighbourhood that, in the simplest case, comprises the eight cells surrounding a central cell (x0,y0) on a two­dimensional square lattice. The Moore neighbourhood of range r is defined by:

N M (x ,y ) = {(x, y):| x - x | #r, | y - y | #r}. 0 0

0

0

Moore neighbourhoods of ranges r=0, 1, 2, and 3 are illustrated in Fig. 2. As with the von Neumann neighbourhood, if no range is specified when a Moore neigh­ bourhood is being referred to, it is as­ sumed that the case in consideration is the default one, i.e. r=1. The number of cells in a Moore neighbourhood of range r is the odd square (2r+1)2, and the se­ quence they form has been given a Sloane entry A016754: 1, 9, 25, 49, 81, 121, 169, ... This neighbourhood type is used in image editing software in tools such as the edge finder and magic wand which are concerned with the proper manage­ ment and allocation of the boundary and edges of a digital image.

Figure 1. Von Neumann neighbourhoods of ranges r = 0, 1, 2, and 3

Figure 2. Moore neighbourhoods of ranges r = 0, 1, 2, and 3

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One­Dimensional Cellular Automata For each cellular automaton, there is a set of rules that define its evolution. A rule can be viewed as a mapping from the space of possible states of the cellu­ lar automaton to itself. In the case of one­dimensional cellular automata, the state space of an automaton with an in­ finite number of cells and its rules cor­ respond to a Cantor set and a contiguous mapping of this Cantor set to itself, re­ spectively. The three fundamental features of a cellular automaton are: ­ uniformity: all cell states are updated by the same set of rules ­ synchronicity: all cell states are updated simultaneously ­ locality: the rules are local in nature Automata are characterised by having states and each cell can exist, at different time steps, in two or more states. In re­ gard to shape, square cells are most common, whereas other shapes, such as triangle and hexagon have also been ex­ perimented with but have not gained much popularity. The cells, of which a given cellular au­ tomata consists, all change their own states at the same time, according to the states of their neighbours in the lattice in which they are situated. Albert Einstein once said that ‘The only reason for time is so that everything does not happen at once’. Consequently, in order to evolve, our cells need a set of transition rules as well as an increase in time t: 0, 1, 2, 3, …, where each step (from 0 to 1, 1 to 2, 2 to 3, …) can be con­ sidered as a clock tick. The simplest cellular automata have two states, which are commonly de­ noted as 0 and 1, or on and off. In one­ dimensional space, the number of neighbour cells, on the left and on the right, that affect a cell’s state (after every clock tick) determines the radius of the cellular automaton. In the simplest case, the state of each cell at time t+1 depends on and can be affected only by its state and the state of its two immediate neighbours at time t, and thus we say this cellular automaton has a radius of 1.

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.

.

ci-1(t) ci(t) ci+1(t)

.

.

Figure 3. The eight possible neighbourhood­ states, and the cell states of the central cell ci and its two immediate neighbours ci-1 and ci+1 at time step t.

With the simple rules presented above, we can now see that there are eight (23 = 8) possible configurations of a neighbourhood of three cells (which is the maximum scope of each cell: radius of 1 on each side, plus the cell itself), each of which can be in one of two states at a time (Fig. 3). ci(t) denotes the state of the ith cell at time t, whereas ci1(t) and ci+1(t) denote the state of its immediate neighbours. After one tick of the clock, the cell state will be ci(t+1), and its neighbours will have states ci-1(t+1) and ci+1(t+1) respec­ tively. Thus, the dependence of a cell’s state can be ex­ pressed by the relation: ci(t+1) = φ[ci-1(t), ci(t), ci+1(t)], where φ is the local transition function. For example, con­ sider the simple transition rule:

1952 ­ Von Neumann formulated a kinematic model of reproduction. He pictured the aggre­ gate as a robot, in a lake, collecting compo­ nents as it jostled against them and assembling them into a copy of itself. Among the components were kinematic elements – artificial arms that could be used to move other objects. John von Neumann decided that his automaton would have to consist of two parts: one flexible construction unit being able to build things out of the elements stored in the warehouse, given their proper specifi­ cations; and one instruction unit telling the machine how to construct a copy of itself. Both of these units corresponded to the dual­ ity between computer and program and, of course, to the duality between cell and genome. Von Neumann decided to use a universal computer in order to realise his automa­ ton, which consisted of 29­states and was presented in 1952. It would automatically make a copy of its initial configuration of cells.

1936 ­ Alan Turing presented his semi­ nal paper on com­ putable numbers and the Turing machine.

ci(t+1) = [ci-1(t) + ci(t) + ci+1(t)] mod 2. We can represent this rule as a transition table for the eight possible initial values (Fig. 4). Since there are eight possible neighbourhood­states of three cells and each of these results in two possible state outcomes for the middle cell, there are 28 = 256 possible transition rules. This has been described by Stephen Wol­ fram as elementary cellular automata. He identified each rule by its rulestring, which is formed by the values of the eight output states. Then each rule is given a unique decimal identifier. The rulestring corresponding to the example we consid­ ered above is 10010110, which is the binary representation of the decimal number 150. Thus, the rule is referred to as Rule 150 and its evolution can be visualised in a two­di­ mensional form, as a plot in which the ith row records the configuration of the automaton at step i (Fig. 5). Other one­dimensional rules that we are going to pres­ ent here are Rule 90 and Rule 110 (Fig. 6 and Fig. 7). The fractal produced by Rule 90 was described by Sierpiński in 1915 and is known today as the Sierpiński sieve, Sierpiński gasket, or Sierpiński triangle. It first appeared in Italian art in the 13th century. In Rule 90, each cell is the exclusive-or (XOR) of its two neighbours. This is equivalent to modulo2 addition and generates the modulo­2 version of Pascal’s triangle, which is a discrete version of the Sierpiński trian­

20 [The, Commutator] Sept 2012

1948 ­ At the Hixon symposium in Pasadena, John von Neumann addressed the following question: ‘Can one build an aggregate out of such elements in such a manner that if it is put in reservoir, in which float all these elements, each of which will at the end turn out to be another automaton exactly like the original one?’

gle. The number of live cells in the ith row of this pat­ tern is 2k, where k is the number of non­zero digits in the binary representation of the number i. The se­ quence of these numbers of live cells is known as Gould’s sequence or Dress’s sequence, and is given a Sloane identifier A001316: 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, ... The rule 110 cellular automaton is universal, which was first conjectured by Stephen Wolfram in 1986, and subsequently proven by Wolfram and his assis­ tant Matthew Cook. The evolution of Rule 110 for a specific initial condition is depicted on the cover of Wolfram’s A New Kind of Science.

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How The Game Of Life Was Born

1957 ­ Lionel Penrose de­ signed an artificial self­ reproducing object. In his model, the environment contained only two kinds of solid objects which served as units. In certain restricted circumstances, two dissimilar units could be linked together mechanically and form a copy of a structure which (was had already been defined) consisted of the same two objects.

Instead of using a chessboard for testing his game, Conway turned his large en­ trance hallway into a finite two­dimen­ sional grid where he placed dishes to indicate the cells which were ‘alive’. He quickly became tired of this because he literally began to step on the pieces.

1966 ­ During the 1950s, von Neumann worked on a proof that a machine can re­ produce, searched for pat­ terns in the digits of pi, and busied himself with more arcane mathematics.

Richard K. Guy later recalled: ‘... only after the rejection of many pat­ terns, triangular and hexagonal lattices as well as square ones, and of many other laws of birth and death, including the introduction of two and even three sexes. Acres of squared paper were cov­ ered, and he and his admiring entourage of graduate students shuffled poker chips, foreign coins, cowrie shells, Go stones or whatever came to hand, until there was a viable balance between life and death.’

He did not finish his proof. The manuscript trailed off into a series of notes about the remainder of the proof, written at various times when other projects did not interfere. His student and colleague Arthur W. Burks completed the proof in von Neumann’s spirit and had it published as Theory of SelfReproducing Automata (Urbana and London: Uni­ versity of Illinois Press, 1966).

1968 ­ Edgar F. Codd presented his simplified cellular automata (8 states).

early 1960s ­ Stanislaw Ulam and co­workers J. Holladay and Robert Schrandt at the Los Alamos Scientific Laboratory (the sprawling largely military research centre in New Mexico) began using comput­ ing machines to investigate various two­dimensional cellular au­ tomata. An infinite plane was considered and divided up into identical squares, while the transition rules, using a von Neumann neighbourhood, were eclectic and the results were mostly empirical.

ci-1(t) 1 1 1 1 0 0 0 0

ci-1(t) 1 1 0 0 1 1 0 0

ci+1(t) 1 0 1 0 1 0 1 0

ci(t+1) 1 0 0 1 0 1 1 0

Figure. 4 The transition table formed by adding together [ci-1(t) + ci(t) + ci+1(t)] mod 2

1970 ­ John Horton Conway cre­ ated his Game of Life (2 states over a Moore neighbourhood).

Figure 6. The fractal produced by Rule 90 was described by Sierpiński in 1915 and is known today as the Sierpiński sieve, Sierpiński gasket, or Sierpiński triangle. Figure 5. The evolution of Rule 150 repre­ sented in a two­dimensional form

Figure 7. The Rule 110 cellular automaton has been proven to be universal.

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John von Neumann and the Turing machine Conway’s Game of Life A big boost to the popularisation of the subject of cellular automata came from the famous British mathematician John Horton Conway at the Gonville and Caius College, University of Cambridge and his highly addictive Game of Life. It was invented over a period of two years, offi­ cially presented in 1970, and widely popularised through Martin Gardnerʹs monthly column ‘Mathematical Recreations’ in Scientific American. It quickly be­ came extremely popular among computer enthusiasts. Programmers at numerous computer centres set out to explore the game with machines at their disposal. Many other scientists and mathemati­ cians found pencil and paper adequate to obtain sig­ nificant results. The game is quite simple to understand, easy to program, fun to play, and quickly generates a feeling of being a co­creator of the uni­ verse of possibilities that unfolds in such a cellular au­ tomaton. One can play God in one’s own universe.

Zoo of Life Forms In this section, we are going to have a look at some basic but fundamental types Life forms: still lives, oscillators and spaceships. A still life is a pattern that does not change its shape and position over time. Still lives can also be consid­ ered as oscillators with a period of 1. Some examples of still lives are shown below. From left to right, they are called: block, beehive, loaf, and boat:

An oscillator is a pattern that returns to its original state, in the same orientation and position, after a fi­ nite number of generations (or steps). Thus the evo­ lution of such a pattern repeats itself indefinitely. Depending on context, the term may also include spaceships. The smallest number of generations it takes before the pattern returns to its initial condition is called the period of the oscillator. An oscillator with a period of 1 is usually called a still life, as such a pat­ tern never changes. Some examples of oscillators in­ clude: blinker, toad, and beacon:

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John von Neumann was born in Budapest, Hungary, on December 3, 1903. As a child, he could divide two eight­digit numbers in his head. He entertained family guests by memorising columns from phone books, then reciting names, addresses, and phone numbers unerringly. At the age of twenty, von Neumann published a for­ mal definition of ordinal numbers. He later noticed a star­ tling connection between quantum physics and vector theory. He discovered that the states of quantum systems could be represented by vectors in an abstract, infinite­di­ mensional space. In 1931 von Neumann became professor of mathematics at Princeton University, and in 1933 he was appointed to the nascent Institute for Advanced Study. Von Neumann was famous for working in odd places at odd times – in taxis, during nightclub floor shows, or while waiting for breakfast. He would even slip away to his study during parties. Through the 1930s and early 1940s, von Neumann worked on game theory, and from 1940 on, he was in demand as a consultant to industry and government, as a result of which he became extremely wealthy. In May 1954, for instance, he was a consultant to twenty­one organi­ sations, including the air force, the CIA, IBM, Los Alamos Laboratories, the Pentagon, and Standard Oil. Von Neumann super­ vised the design of unprece­

A spaceship is a finite pattern which reappears after a certain number of generations (or steps) in the same orientation but in a different position. Here, we pres­ ent two types of spaceships: glider (top row), and lightweight spaceship (bottom row):

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powerful computers for the American military during World War II and after. While wrestling with practical problems, he became interested in the potential abilities of au­ tomatons. He was particularly impressed with the work of British mathematician Alan Turing. who had showed that a relatively simple computer can perform any possible calculation, given the right pro­ gramming. This computer was the so called Turing machine.

Sufficient interest existed to maintain a quarterly newslet­ ter, called Lifeline, published by Robert T. Wainwright for almost three years, from 1971 to 1973.

Although usually intended as a mathematical abstrac­ tion, a Turing machine could be designed and built from usual electrical components. The only catch was that it had to have an infinite memory capacity – or what amounted to the same thing. It had to be able to add on to its memory as needed. This infinitely extensible memory would be outside the main body of the computer, thus its being called external memory. Turing pictured his universal computer hovering over an infinite strip of paper tape marked off into squares, each containing 0 or 1. Turing believed that any type of idea, object, action or computer program that can be ex­ pressed in language can be coded by a string of 0s and 1s on the tape of this machine, which of course, first had to be given a complete set of instructions on how to perform the desired calculation. Likewise, the computer could erase and print its own 0s and 1s to keep track of intermediate results of calcula­ tion, or in other words, use the tape as scratch paper. In long computations, the Turing machine might as well use light­years of tape. At the end of a calculation the ‘Anything that we answer would be expressed, in can describe [in our the same manner as the input own world] can happen had been given, as a in the Life world’ coded string of 0s Bill Gosper and 1s. The Glider gun was devised by a young group of enthusiasts from MIT, with Bill Gosper, arguably the most verbally profound in the group, as one of its leading figures. The hackers would spend all night sitting at the PDP­6’s (the computer they used at the MIT in the 1970s) display, trying different patterns logging each ‘discovery’ they made in this artificial universe in a large black sketchbook. What resulted in was a carefully orchestrated col­ lision of 13 Gliders which was able to produce this Gun.

Figure. 8 Bill Gosper’s Glider gun

One interesting fact about Bill Gosper and his

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One can speculate why digital life is so fascinating. If the dream of life’s creation lies within the cellular automaton, what is this dream made of? Perhaps it arises from a Pygmalion desire. An ancient Greek leg­ end tells of a Cypriot king named Pygmalion. He fell in love with a statue of a beautiful woman ­ perhaps the goddess of love herself ­ whom he himself had sculpted. Pygmalion became so enamoured of his work that he embraced it. He begged and pleaded with Aphrodite for a wife of the same appearance, and at last she took pity on him. She answered his prayer by making the statue come to life.

Bill Gosper and the MIT hackers MIT friends is that they created a robot arm which was able to catch a Ping­Pong ball lobbed toward it, by moving itself in the necessary position. As for Gosper’s love for the Game of Life, a friend of his later recalled: ‘Gosper sort of imagined the world as being made out of all these little pieces, each of which is a little machine which is a little independent local state. And [each state] would talk to its neighbours.’ To Gosper, Conway’s simulation was a form of genetic creation, without the vile secretions and emotional complications associated with the real worldʹs ver­ sion of making new life. Later, in the 1970s, Bill Gosper moved to Califor­ nia to study and help the preparation of the 2nd vol­ ume of the seminal work The Art of Computer Programming by Professor Donald Knuth (who gave a BCS and IET Turing Lecture at the University of Glasgow on 10th February 2011) at Stanford. Gosper has also worked as a consultant for Xerox PARC (the inventors of the “windowed” graphical user interface and the mouse device) and Wolfram Research (whose founder and CEO is Stephen Wol­ fram ­ one of the other great contributors to the field of Cellular Automata).

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Conway’s Rules ‘It is possible, given a large enough “Life” space, initially in a random state, that after a long time, intelligent self-reproducing animals will emerge and populate some part of the space.’ John Horton Conway The Game of Life is a two­dimen­ sional zero­player game, meaning that any interaction with it is expressed by setting up an initial configuration (or state) and observing how it evolves in time, according to the pre­defined set of rules. Conway’s automaton is played on an infinite matrix of cells where each cell can be in one (at a time) of two possible states: ‘alive’ and ‘dead’, or simply ‘on’ and ‘off’. Given an initial configuration of living cells, the following rules, which depend of the number of its Moore neighbours at given time t, are applied resulting in a sequence of patterns: Birth: A cell that is dead at time t becomes alive at time t+1 if exactly three of its neighbours are alive at time t. Death by overcrowding: A cell that is alive at time t will die at time t+1 if four or more of its neighbours are live neighbours at time t. Death by exposure: A cell that is alive at time t will die at time t+1 if it has one or no live neighbours at time t. Survival: A cell that is alive at time t will remain alive at time t+1 only if it has either two or three live neighbours at time t. Formally, these rules can be sum­ marised by first defining the set M of coordinates of all Moore neighbours of range r=1: M(i,j) = {(i-1, j-1), (i, j-1), (i+1, j-1), (i-1,j), (i+1, j), (i-1, j+1), (i, j+1), (i+1, j+1)}. Then the dependence of a cell’s state in any two­dimensional cellular automa­ ton can be expressed by the relation:

Figure 9. Link to the Glider gun and UTM demonstrations: http://the­commutator.com/?p=51

Figure 10. Rule 0 is the simplest possible Gar­ den of Eden. Its evolution depicts the trivial set of rules that evolve every cell into 0.

c(i,j)(t+1) = φ[M(i,j)]

world. For instance, Adam P. Goucher created patterns in the Game of Life that can calculate the decimal digits of phi and pi.

In particular, for the Conway’s Game of Life, the rules can be formalised as:

where

Both John von Neumann and Codd provided their artificial worlds with laws intended to facilitate self-reproduction, whereas in contrast to this, the simple laws of Life were not at all tai­ lored to the task of permitting patterns to propagate. Among all possible worlds of cellular automata, Life was much more plausible than John von Neumann’s or Codd’s artefacts. The Life­universe itself is designed as an automaton: a cellular automaton defined by its transition rules. A sufficient number of artefacts were found to enable Conway to demonstrate a universal constructor, which has been one of the principal goals of the theory ever since John von Neumann became interested in auto­ matic factories. It has been shown that any of the wide range of computations that can be performed by practical computers can also be done by cellular automata, and that it is theoretically possible to implement von Neumann’s complicated self­reproduction at higher levels in Life using various pat­ terns such as the Glider gun (Fig. 8). This Turing machine then can im­ plement anything that is at all computable. This means that anything that we can describe can happen in the Life

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The first Life design for an Univer­ sal Turing Machine was completed and realised by Paul Rendell in Febru­ ary 2010 (link [2]). A demonstration of its action can be viewed through the QR code and link in Fig. 9. An alterna­ tive design for a universal computing machine is Marvin Minsky’s register machine, which stores arbitrary large numbers by pushing blocks into empty space. A design for the registers was constructed in Conway’s Game of Life by Dean Hickinson in 1990 (link [3]). Later, it was used by Paul Chap­ man to implement a complete register machine that demonstrates universal capability in 2002 (link [4]).

Gardens of Eden For most cellular automata, there are configurations (states) that are un­ reachable: no state will produce them by the application of the evolution rules. These states are called Gardens of Eden, because they can only appear as initial states. As an example, con­ sider a trivial set of rules that evolve every cell into 0. The one­dimensional Rule 0 exhibits this functionality (Fig. 10). For this automaton, any state with non­zero cells is a Garden of Eden. Considering the Life universe, Mar­ ijn Heule, Christiaan Hartman, Kees Kwekkeboom and Alain Noels from the Delft University of Technology systematically searched the entire space of 10­by­10 patterns with four­ fold rotational symmetry, finding a Garden of Eden with 92 specified cells

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Figure 11. The smallest known Garden of Eden in the Life universe, found at the TU Delft

(56 live, 36 dead) (Fig. 11). Moreover, they proved the non­existence of Gar­ dens of Eden within a 6­by­6 box.

Figure 12. Rule 30 is known to have many of the properties desirable for practical cryptography

state at a given time t, will remain in the same state at time t+1 as well) are not acceptable. Given restrictions such

The surprising power of re­ cursive rules is best illustrated with a few specific cases presenting dif­ ferent approaches that have been examined throughout the years. The simplest recursive rule allow­ ing growth is: any cell touching an occupied cell experiences a birth in the next generation. Then even a single occupied cell acts as a seed for unlimited growth. All its neigh­ bours become occupied in genera­ tion 1, forming a 3­by­3 square, leading itself to a 5­by­5 square, then a 7­by­7 square, then a 9­by­9 square

Figure 13. The first two graphics represent the initial configuration (normal view and zoomed­in), while the third graphic represents the terminal configura­ tion, after the 7,769­step evolution of the pattern

Multidimensional Life and Other Variations Numerous variants have been consid­ ered over the years but none has ever had the same success or has been ac­ cepted as broadly as Conway’s Life. The apparent, vast number of choices of rules, 229 to be accurate, is rather il­ lusory. The rules not permitting rota­ tional and reflective symmetry, or the ones not providing a quiescent state (a state such that if all cells are in that

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as totality, thousands of millions of candidates remain – still many, but not entirely impossible to sort through. Some promising three­dimensional analogues of Life have been found, and although possible, working with higher dimensions is very slowly com­ putable, clumsier and difficult to grasp, since the front surfaces of the objects prevent everything that is hap­ pening in the background to be seen in its entirety. The ‘regular’ two­di­ mensional Life has been arguably found sufficient to allow for the dis­ covery of undreamed.

and so on. It expands outward in all directions at one cell per generation. However, the growth is predictable

Figure 14. Link to the University of Glagow logo demonstration: http://the­commutator.com/?p=43

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and therefore not interesting enough. Hungarian­born mathematician Stanislaw Ulam found ways to elimi­ nate some of the would­be occupied cells so that the pattern acquires richer structure. It grows in complexity as well as size, producing an ever­grow­ ing ‘coral reef’ from a starting pattern of a single occupied cell and may be even played in three dimensions. One other approach was developed in the early 1960s by Edward Fredkin.

four copies of itself. The quadrupling time is a power of 2 and varies with the size and complexity of the initial pattern. Another important specificity of Fredkin’s game is that there is no way of working out which was the original pattern from a given further state of the same pattern. On the neg­ ative side, the type of self­reproduc­ tion is trivial because every pattern reproduces. Therefore, it is not a spe­ cial feature of a pattern’s organisation that allows reproduction.

His variation uses two states, on and off, just like Conway’s Life. Only the four orthogonally connected neigh­ bours count, i.e. Fredkin’s variation operates over a von Neumann neigh­ bourhood. A cell will be on in the next generation if, and only if, one or three (an odd number) of its neighbours are on presently. Otherwise, if zero or two (an even number) of its neighbours are on, the cell will be off. A specific prop­ erty of this variation is that no pattern of live cells can ever die away as any configuration whatsoever becomes

Edward Fredkin’s universe Edward Fredkin became wealthy running a computer company that specialises in image­processing. It was perhaps there that he learnt how to view reality as bits of information. At the age of thirty­four, without even having a bachelorʹs degree, Fredkin became a full pro­ fessor at the Massachusetts Institute of Technology. More recently, he has been a Distinguished Career Pro­ fessor at Carnegie Mellon University, at Boston Univer­ sity and a Visiting Professor at MIT. There is a notion that the laws of physics are discrete and the universe evolves as the result of deterministic computation, such as a cellular automaton, and it dates back to pioneering German computer engineer Konrad Zuse. His proposal, referred to as ‘Zuse’s thesis’, has led to the development of the field of digital physics. One interesting consequence of Zuse’s thesis is that entropy does not increase. Zuse is also credited for developing and building the first fully functional program­con­ trolled digital computer, the Z3, in 1941 (in Berlin). The thesis was elaborated upon by Edward Fredkin, who proposed the related notion of finite Nature which is ‘the assumption that, at some scale, space and time are discrete and the number of possible states of every finite volume of space­time is finite’. Fredkin and others have suggested that space­time it­ self is granular ­ composed of discrete units ­ and that our entire universe is a cellular automaton run by an enormous computer, and what we call motion is only simulated motion. According to his idea, a given cellular automaton can imitate life and it will depict reality more and more precisely when applied to smaller bits of mat­ ter. A moving particle at the ultimate microlevel may be essentially the same as the one of Conway’s Life gliders, appearing to move on the macrolevel, whereas actually there is only an alteration of states of basic space­time cells in obedience to transition rules that have yet to be discovered. An electron is nothing more than a pattern of information, and an electron in a path is a pattern that

26 [The, Commutator] Sept 2012

is moving. At the most fundamental level the automaton will describe the physical world’s forms of movement with perfect precision because at that level the universe is a cellular automaton in three dimensions. Fredkin be­

Stanislaw Ulam and the 3N+1 Problem The Los Alamos National Laboratory was estab­ lished in 1943 and one of the pioneers there, John von Neumann recruited Ulam from Princeton. The first computers at the laboratory were enormously huge, unlike their today’s personal desktop and mobile cousins. Ulam posed a very interesting problem for Los Alamos’s computers, which at the time were used mainly for mathematical computations: Think of any positive whole number. If it is even, divide it by two. If it is odd, triple it and add one. Keep applying this same rule over and over. What happens to the number? The histories of the initial values are surprisingly unpredictable. Say you choose 10. Ten is even, so you halve it and get 5. Five is odd, so you triple it and add one to get 16. Sixteen is halved to 8, then 4, then 2, then 1. One is again odd, so it is tripled and added to one to get 4. Then number enters the endless loop 4­2­1­4­ 2­1­4­2­1­ … Formally, in modular arithmetic notation, we define the function f as follows: f (n) = '

n/2

if n / 0 (mod2)

3n + 1

if n / 1 (mod2)

.

We now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next: ai = '

n

for i = 0

f(a i - 1)

for i 2 0

.


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Of all simple modifications of Life’s rules, the most widely played is a ver­ sion called 3-4 Life, whose name comes from the MIT Life group. Perhaps 3­4 Life was one of the variations Conway himself was experimenting with be­ fore settling on the version of Life that we know today. As for 3­4 Life’s rules, they are even simpler than the conventional Life’s ones: a cell will be on in the next time step if, and only if, it has three or four Moore neighbours that are on now,

without considering its own present state. Most of what is known about the universe of 3­4 Life has been discov­ ered by computer hacking, just as with Conway’s Life. In this article, as it used to be some 30 or 40 years ago, the so­ cial group which is referred to as hack­ ers is the one that collectively describes those computer program­ mers who regard crafting beautiful code as the most important thing in the world, and use their wizardry to solve difficult and fascinating prob­

lieves that this giant lattice, which we all refer to simply as the world, consists of logical units composed of funda­ mental grains, each computing ­ at a completely local level ­ its state in the next picosecond as a function of the

The smallest i such that ai = 1 is called the total stopping time of n. The conjecture asserts that every n has a well­defined total stopping time. For instance, the stopping time of 5 is 5 because: 5 ⇒ 16 ⇒ 8 ⇒ 4 ⇒ 2 ⇒ 1 ⇒ …

lems in, usually but not only restricted to, maths and logic. Unfortunately, nowadays the term ‘hacker’ is imme­ diately associated with an antisocial computer geek, often using his ex­ traordinary programming powers to write malicious programs and ‘hack’ entire networks, websites or comput­ ers. Back to our 3­4 Life variation, what turns out to be a crucial difference is that it is more pro­growth, and thus it fails to match the vast richness of con­

states of its neighbouring cells. The information processed at this basic level is the factory for our reality. Yet if the universe is indeed a computer, and the physical natural laws are its software, then we will always be cut off from knowing what the hardware is like. Turing cer­ tainly showed that computers are equivalent to universal machines, so that a program does not in any essential sense depend on being realised by a single specific kind of hardware. Since in Fredkin’s universe we ourselves would already be virtual meta­entities in such a cosmic program, we could never know anything about the pri­ mary machine. We could hope that it never crashed, for we would go down with it. The Fredkin Prize

Ulam found that every number he tested eventually entered the 4­2­1 cycle. Of course, some numbers take longer. For instance, twenty­seven takes 109 steps, at one point reaching a maximum value of 9232. The in­ teresting fact is that no one has ever found a number that does not enter the 4­2­1 cycle. By the same token, no one has ever been able to prove that all numbers indeed enter this very same cycle. The problem, ini­ tially posed in the 1930s, is known today as Collatz problem, Collatz conjecture, Kakutani’s problem, Ulam’s problem, Syracuse problem, Thwaites conjecture and the 3N+1 problem and remains unsettled. If there are any numbers that do not enter the loop, they must be very large. All numbers up to 1016 have been tested and found to enter the loop.

In 1980, Carnegie Mellon University announced the establishment of a $100,000 prize (by Edward Fredkin himself), called Fredkin Prize, for the first computer pror­ gram to become World Chess Champion and the begin­ ning of annual computer versus human competition. The prize was three­tiered: 1) The first award of $5,000 was given to Ken Thompson and Joe Condon from Bell Laboratories, who in 1981 de­ veloped the first chess machine to achieve master status.

The 3N+1 problem can be viewed as a 7­state cellu­ lar automaton, with the digits of n written in base­6. Then the question of whether a persistent structure can exist in such cellular automaton arises, which if answered, would be a proof to the Collatz conjecture.

3) The $100,000 third tier of the prize was awarded to the IBM team, who built the first computer chess machine that beat a world chess champion at the Fourteenth National Conference on Artificial Intelligence (AAAI­97, held from 27th July to 31st July 1997, at the convention centre in Providence, Rhode Island).

2) Seven years later, the intermediate prize of $10,000 for the first chess machine to reach international master sta­ tus was awarded in 1989 to five Carnegie Mellon gradu­ ate students who built the chess­playing computer Deep Thought, the precursor to Deep Blue, at the university.

In May 2011, Gerhard Opfer of the Hamburg Uni­ versity released a paper claiming that the Collatz con­ jecture is true (link [1]). Eventually some flaws in the proof were found and he corrected his statement.

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ventional Life. Very small patterns are more likely to sur­ vive under Conway’s rules than under those of 3­4 Life. In the latter, any big random pattern seems to grow for­ ever, whereas in regular Life unlimited growth is possible but only with ingenious constructions such as the Gun, Stabilised switch engine, or Breeder.

Applications of Cellular Automata Cellular automata have applications in many diverse branches of science, such as biology, chemistry, physics and astronomy. Cellular automation models can be used for modelling of traffic flow, where the road is discretised into cells, each either being empty or containing a car moving with a given speed. As the cars interact, collective phenomena such as traffic jams can be modelled. The Nagel-Schreckenberg model, Biham-Middleton-Levine model and Rule 184 are examples of such models. Another one­dimensional cellular automaton of special interest is Rule 30 because it is chaotic, with central col­ umn given by 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, ... (Fig. 12). In fact, this rule is used as the random number generator for large integers in the Mathematica software (developed by Wolfram Research). Moreover, Rule 30 is known to have many of the properties desirable for practical cryp­ tography. It does have a short period of repetition nor it shows any obvious structure for almost any given key. Small changes in the key typically lead to large changes in the encrypted sequence.

(in Belgium), Dale Millen (in the US) and Iannis Xenakis (a Greek composer living in Paris) were also experiment­ ing with cellular automata music at the time. The fundamental challenge of generating music with the Game of Life was to design a suitable method that em­ ploys the corresponding parts of the patterns generated by the automata and their musical and visual representa­ tion. Miranda developed several representation schemes, notably two­dimensional and three­dimensional Carte­ sian implementations, as well as a two­dimensional radial alternative.

Conclusion As a conclusion, we are going to have a look at how (and into what) a Game of Life pattern, formed by the pix­ els of the University of Glasgow logo, evolves through 7,769 generations, with initial and terminal population of 50,905, and 5,905/5,907 respectively (Fig. 13). The terminal value is not constant due to the creation of numerous os­ cillators, which continuously return to their original state. A demonstration of the pattern’s evolution can be viewed through the link or its corresponding QR code in Fig. 14. References Poundstone, W. (1984) The Recursive Universe, US: William Mor­ row & Co Schiff, J. L. (2008) Cellular Automata, Chicester: John Wiley and Sons Emmeche, C. (1996) The Garden in the Machine, New Jersey: Princeton University Press Berlekamp, E. R., Conway, J. H. and Guy R. K. (1982) Winning Ways: for your mathematical plays vol. 2, Natick: Taylor & Francis

Cellular Automata and Music Essentially, music is a time­based art form, where se­ quences of notes and rhythms form configurations of sonic structures organised in time. In this context, Con­ way’s Life is appealing because it produces sequences of coherent patterns, some of which can be very complex, and yet controlled by remarkably simple rules. As a post­graduate student Eduardo R. Miranda, one of the pioneers of Game of Life music, used to ‘play’ with the equipment in the electronic music studio at the Uni­ versity of York in the late evenings until dawn. To aid the preparation of his thesis, Miranda created a system, named CAMUS (short for Cellular Automata Music), which could render music from the behaviour of the Game of Life cellular automata. He soon tested his pro­ gram in the music studio by connecting his Atari 1040ST computer to a synthesizer. The music that was produced by the software sounded remarkably interesting, which was an awesome experience for young Miranda. It marked the beginning of his enduring interest in compos­ ing music with cellular automata. He soon learned that three other composers, Peter Beyls

28 [The, Commutator] Sept 2012

McIntosh, H. V. (4 July 1988) ‘Conway’s Life’ in Adamatzky, A. (ed.) (2010) Game of Life Cellular Automata, London: Springer Lon. Miranda, E. R. and Kirke, A. (2010) ‘Game of Life Music‘ in Adamatzky, A. (ed.) Game of Life Cellular Automata, London: Springer London Guy, R. K. (1982) ‘John Horton Conway: Mathematical Magus’, The Two-Year College Mathematics Journal, Vol. 13, No. 5, November, pp. 290­299 Wolfram, S. (1983) ‘The statistical mechanics of cellular automata’, Review of Modern Physics, 55:601­643 Weisstein, Eric W. ‘Von Neumann Neighborhood.’ From Math­ World ­ A Wolfram Web Resource. http://mathworld.wolfram. com/vonNeumannNeighborhood.html Weisstein, Eric W. ‘Moore Neighborhood.’ From MathWorld ­ A Wolfram Web Resource. http://mathworld.wolfram.com/Moor eNeighborhood.html ‘A community for Conway's Game of Life and related cellular automata’, http://www.conwaylife.com/ ‘Golly’, http://golly.sourceforge.net

Links 1. http://preprint.math.uni­hamburg.de/public/papers/hbam/h bam2011­09.pdf 2. http://rendell­attic.org/gol/utm/index.htm 3. http://www.radicaleye.com/lifepage/patterns/sbm/sbm.html 4 .http://www.igblan.free­online.co.uk/igblan/ca/index.html

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Prime Numbers Prime numbers are the fundamental building blocks of mathemat­ ics, just as atoms are the fundamental building blocks of the Uni­ verse. As we shall see, they occur in nature as well as in the abstract world of mathematics, and they provide a useful form of encryption.

Sean Leavey ‘Every natural number greater than one is either a prime number or it can be written uniquely as a product of primes’ ­ a statement proven by Euclid in his Fundamental Theorem of Arithmetic. The number 93, for example, can be writ­ ten as 31 x 3, both of which are prime numbers. In the same manner, the number 86 can be written as 43 x 2, both of which are prime. Similarly, 76 = 19 x 4, where 19 is a prime, and in 4 = 2 x 2, 2 is a prime. This theorem can be proven through the use of contradiction ­ we first make the assumption that the theory is always true, then look for a contradiction which would make the assumption false. If the contradiction makes no sense, then the original the­

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ory holds. We can assume that there exists a number N which is the first number when counting from 1 that is not prime and cannot be written as a product of primes. As N is not prime, it can be written as a product of two or more factors, A and B, which are less than N. By definition, N is the first number which cannot be written as a product of primes, therefore, it must be possible for A and B to be written as products of primes, because they occur earlier in the sequence. Since N can be written in terms of A and B, N itself can also be written as a product of primes. However, this contradicts the original assumption that N exists and thus proves the theorem. Euclid used a similar method to prove that there are infinitely many prime num­

bers.

Primes in Nature The abstract world of mathematics is not the only place where prime numbers make an appearance. Prime numbers are also apparent in nature. The magicicada, an insect native to North America, has developed a mat­ ing cycle on a 13­ and 17­year basis. The insects live underground for 13 or 17 years before emerging en­masse to mate. The best guess as to why this happens the way it does is because there is a predator to the insect which has a 7 year mating cycle, and by emerging only after a prime number of years ­ 13 or 17 ­ the insect can min­ imise the risk of their emergence coin­

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ciding with the predator’s. A 13­year­ cycle magicicada’s emergence will co­ incide with a 7­year­cycle predator’s only every 91 years. A 17­year­cycle magicicada’s emergence will coincide with a predator’s emergence that has a 7­year cycle only every 119 years. Similarly, the 13­ and 17­year magici­ cadas sharing the same forest and the same food supply would only emerge simultaneously every 221 years. Imagine if the magicicada emerged to mate, for example, every 4 years (4 being non­prime). This would coin­ cide with the predatorʹs 7 year cycle every 28 years. The 13­ and 17­ year cycles give the magicicadas a crucial survival edge, with the help of prime numbers.

Cryptography Prime numbers are also crucial to modern cryptography algorithms, the importance of which is readily appar­ ent: it is in your own interest that your credit card details are encrypted by websites which you use for on­line shopping, so that potential snoopers might not use them for illicit purposes. Similarly, sensitive information that individuals and governments would not want to be made publicly available would need to have a means of en­ cryption in order to prevent it from being revealed. It is therefore impor­ tant to employ a form of cryptography which allows a plaintext to be rela­ tively easily encrypted, while keeping decryption without the relevant key extremely difficult. Prime numbers offer a convenient way to do this.

large number would be unable to be factorised on a typical desktop com­ puter before the heat­death of the uni­ verse. An algorithm that does not involve simply brute forcing the an­ swer has eluded mathematicians ever since the ancient Greeks discovered that all numbers are products of primes. Such an algorithm would have huge ramifications not only for mathematics but also beyond ­ current cryptographic methods, upon which banks, websites and governments put faith, would be rendered useless. If an encryption method is to be hard to break, it needs to be rooted in a mathematical calculation with an easy­to­check but hard­to­calculate so­ lution. The problem of factorising very large numbers into constituent primes turns out to be ideally suited to RSA (named after Ron Rivest, Adi Shamir and Leonard Adleman, 1978), the form of encryption upon which most internet transaction security is based.

Fermat’s Little Theorem At this point, it is necessary to intro­ duce Fermat’s Little Theorem. This theorem regards the use of a modulo p system, where p is a prime number. Fermat found that if one were to take a number, then raise it to the power p, and take the modulo p of the product, one would get back the original num­ ber. Specifically: xp (mod p) = x.

Raising x to the power p using modu­ lus p results in x appearing again. For example, choosing p to be the prime As stated earlier, every number can number 7, and choosing x arbitrarily 7 be written as a product of primes. to be equal to 3, we find that 3 = 2187, Finding these prime number factors is and applying modulus 7 we get back fairly straightforward for small num­ 3 again, our initial value for x. As if by bers. However, when it comes to huge magic, our original value for x falls out numbers, containing one hundred as a remainder of the result divided by digits or more, it becomes very com­ our prime number p. Note, however, putationally expensive to find the con­ that this does not work for non­prime 4 stituent prime factors. Although still p. For example, if p = 4, then 2 (mod mathematically possible, a sufficiently 4) = 0, not 2.

30 [The, Commutator] Sept 2012

Later on, Euler managed to prove that the same result was true for cal­ culations derived from semiprimes ­ products of two primes, say u and v. He found that the value for x would show up when x was raised to a spe­ cial number derived from a pair of primes, when a special semiprime modulus was taken, specifically: xpq (mod r) = x, where r = uv, and pq [mod ((u­1)(v­1) + 1)] = 0. The values p and q are chosen such that their product is zero when ap­ plied modulo (u-1)(v-1) + 1, for u and v prime. If one wishes to encrypt a plaintext x, one can choose two prime numbers and calculate values for r, p and q, knowing that x can be re­ob­ tained using these values. Websites whose services demand high levels of security can therefore send a user’s web browser the values for r and p (but not q), which can be used to en­ crypt the sensitive data, sending back the resulting value xp (mod r). Upon receiving this value from the user, the website can then obtain the original transaction details (for example, the user’s credit card details to perform a sale) by raising the received number to the power q and performing a modulo r operation on the value. By doing this, the website is effectively performing the originally defined xpq (mod r) in order to get back x. All that has been sent over the public domain are the values for r, p and xp, which with cur­ rent computing power cannot be used to work out the value for x without knowing q. This sequence of transfer­ ring information back and forth is known as the Diffie-Hellman key exchange. The values for p and r effec­ tively make up the public key, used to encrypt the text, whereas the values for q and r make up the private key used to decrypt the text. An attacker would first need to factorise r into its separate prime numbers u and v in order to find the value of p and q to

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perform the decryption. Finding the factors u and v that make up r is an example of a mathe­ matical problem which fits the bill as outlined earlier ­ a relatively simple operation to perform one way, i.e. by calculating r, p and q knowing u and v, but extremely difficult to reverse, i.e. working out u and v knowing only r, p and xp, for a sufficiently large pair of primes. A brute force attack to find the values of u and v when they are over one hundred digits long would in­ volve checking numbers bigger than the number of fundamental particles in the observable universe. This kind of challenge is completely beyond the means of an individual desktop com­ puter, and most supercomputers. For a carefully chosen key, it would take longer to compute the prime factors than the time in which most sensitive data would remain useful: credit cards expire, government information is eventually outdated or no longer con­ sidered secret.

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However, with the right equipment, it is not as daunting a task as perhaps it initially seems to be. The RSA-129 number, which is 129 digits long, took just 6 months in 1994 to factorise using a network of approximately 1600 com­ puters connected over the internet. While still a long time, this shows that with sufficiently large computing power and some determination, it can become relatively easy to factorise numbers of such magnitude. Nowa­ days, it is typical to find values for r in excess of 600 digits long being used in order to protect the encryption from a brute force attack using coordinated networks of computers. Primes of over a million digits are readily available but not favoured due to the exponen­ tial increase in computational time they require in order to encrypt and decrypt data, especially with the prevalence of low­power mobile phones as internet­enabled devices.

using prime numbers is a perpetual game of cat and mouse played be­ tween the speed of computers and the size of the primes involved. This will remain the case until something quite different altogether eventually su­ percedes the use of primes for encryp­ tion in this way. It also seems that primes, though an abstract mathemat­ ical principle, have a genuine root in the natural world around us. References du Sautoy, M. (2004) The Music of the Primes, London: HarperPerennial Bentley, P. J. (2008) The Book of Numbers, US: Firefly Books Singh, S. (2000) The Code Book, London: Harper Collins Publishers Raiter, B. How RSA works, http://www.mu ppetlabs.com/~breadbox/txt/rsa.html BBC, In Our Time: Prime Numbers, http:// www.bbc.co.uk/programmes/p003hyf5

It seems that secure encryption

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An Epidemic Beginning In the late nineteenth century Bombay was a victim to several epidemic outbreaks in the suburbs. The living environment consisted of open sewers, poor ventilation and rat infestation. In 1906 the absence of hygiene amongst the Hindu outcastes who populated the city’s underbelly caused them to suffer out­ breaks of pneumonic and abdominal plague. These epidemics provided a case study for two accomplished Scottish mathe­ maticians ­ Anderson Gray McKendric and William Ogilvy Kermack.

Gillian Bowman

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Together, the two scientists pub­ lished ground­breaking papers in Mathematical Modelling, and the case of the pneumonic plague in Bombay both challenged and con­ firmed their SIR model (Susceptible, Infective, Removed). This biological model, also known as the McKendrick–Kermack model, still heavily influences modern mathematical bi­ ology. The model takes the disease statis­ tics and, using a system of Ordinary Differential Equations (ODEs), creates a diagram showing whether the dis­ ease is declining, growing or at risk of becoming an epidemic. The model also states the population size needed for a disease to survive through a variable known as the Threshold Effect. In the case of the Bombay epidemic, the model’s effec­ tiveness was questioned due to the vague understanding of the plague. The SIR Model assumes that each class is equally healthy or equally sick, while the contagiousness and mortality rates associated with each form of plague actually differed. Some forms were far more infectious than others and the SIR model as­ summed the disease was spread by human contact, but research con­ structed by the Geological Society of Manchester concluded that the epi­ demic was influenced by climate and rainfall: the epidemic of 1896 had fol­ lowed the warmest summer in Bom­ bay for fifty­one years. Mathematical models in biology are often solved through ODEs but these equations may differ depend­ ing on the particular nature of each biological system. Bacterial and viral diseases are known as microparastic diseases and worm diseases are known as macroparastic. The SIR model for the Bombay epi­ demic divided the population into three sections: those who were healthy ­ Susceptible (S), those were sick ­ Infective (I), and those who were recovered or immune ­ Removed (R).

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For this model it was assumed that each section had individuals who were equally healthy or sick and no or deaths occurred in the births population because the its change was very negligible. This is called an SIR Model without vital dynamics. What made a disease model different from other biological models, such as predator­prey and parasite models, was that the rate of transmission of dis­ ease depended on the rate of encounter between susceptible and infectives. This dependence gives the product: bSI.

The Removed class is eliminated and we can work with two variables. The equations for S and I become: dS = bSI + c (N - S - I) / f (S), I)), dT dI = bSI - oI / g (S), I)). dt We solve these to find the steady states, the points where the Suscepti­ ble class and the Infective class meet on the phase­plane: dI = 0 & I = 0 or S = o , b dt dS = 0 & bSI = c (N - S - I). dt

With this product we can construct a set of Ordinary Differential Equa­ tions, similar to those encountered in first and second year Mathematics, keeping in mind that in this model the recovered can lose immunity and become susceptible again: dS = - bSI + cR, dt dI = bSI - oI, dt dR = oI - cR. dt

We arrive at two steady states: (S1, I2) = (N,0) All population is suseptible and there is no disease

(S2, I2) = c o , b

Differential equations represent how the disease changes in time, thus we can begin solving these with a definite conclusion: that when the growth rate of the classes remains the same, all the differentials must equal zero. We also know that the total population N is the sum of the Susceptible, Infective and Removed classes:

N = S+I+R , R + N - S - I.

N - (o/b) m o+c

Population contains constant proportions provided S2, I2 is positive

For the Infected class to be positive we need: bN >1. N> o & o b Discovered by Kermack and McK­ endrick, the Threshold Effect is the value needed for the disease to be es­ tablished in the population. There­ fore, a population must be over a

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The Significance of the number 264-1

certain size for a disease to estab­ lish itself and become endemic. It is popularly denoted: Nb R0 = . o This variable represents the aver­ age number of secondary infec­ tions caused by introducing a single infected individual into a host population of N Suscepti­ bles. In vector calculus, a Jacobian matrix (sometimes simply called “the Jacobian”) is the matrix of all first­order partial derivatives of a vector­ or scalar­valued function, with respect to another vector. We introduce one such matrix in order to linearise the equations and get the local picture of flow patterns around the steady states:

J df K ) ) J (S , I ) = K dS K dg L dS )

=e

bI - c ) bI

df N O dI dg O O dI P ) - (bS + c) o. ) bS - c

We are most interested in the sta­ bility of the disease steady state. For this, we can examine the trace and the determinant of the Jaco­ bian matrix: ) Tr( J) = - (bI + c)

is always negative ) det(J) = bI (o + c)

is always positive When the trace of the matrix is negative and the determinant is positive, we have a stable steady state. On a graph, the direction of the curves will face the point as the rate of infection remains the same.

Hristo Georgiev

The accuracy of the McK­ endrick­Kermack model has been debated because it predicted that the end of the epidemic was due to a decreasing number of Sus­ ceptibles as the Municipality dis­ infected streets and crowded chawls, moving inhabitants to specialised plague hospitals. The 1906 epidemic ending, as the sea­ son turned to winter, was a sup­ porting evidence that it was the climate which was the greatest in­ fluence on the pneumonic plague in Bombay. Regardless of whether this is true or not, the Mackendric­Kermack Model is certainly one of the earliest con­ tributors to the evolution of mathematical modelling. Since its creation, epidemic models which examine the effect of vaccination on population, the growth of can­ cer tumours and cell survival have been developed. Its impact is profound yet derives from knowledge acquired in Applied Maths in first and second year of university.

References Edelstein­Keshet, L. (2005) Mathematical Models in Biology, US: Society for Industrial & Applied Mathematics

Abu­l ‘Abbas Ahmad ibn Khal­ likan (1211­1282 CE) was a fa­ mous historian, jurist, theologian and grammarian born in Arbela, Iraq. Ibn Khallikan’s most famous work is The Obituaries of Eminent Men, often referred to as The Biographical Dictionary. It is a work of enormous scope ­ the English translation by Mac Guckin de Slane occupies over 2,700 pages ­ and has always been considered of highest importance to the civil and literary history of the Mus­ lim people. It is noticeable that he preferred to relate anecdotes il­ lustrating the humanistic charac­ ter of his subjects rather than describing their lives in full. It is these features that made his work of wider interest to the world out­ side of Islam. In 1256, based on an old legend about the creation of chess, he proposed the following problem: The inventor of Chess, grand vizier Sissa Ben Dahir, presented his work to King Shirham of India as a gift. The ruler was so pleased that he gave the inventor the right to name his prize for the achievement. Sissa, kneeling in front of the king asked if he could be given a grain of wheat to put on the first square of the chessboard, two grains to put on the second square, four grains to put on the third, eight grains to put on the fourth, and so on, doubling the number for each successive square until all 64 squares of the board have been covered. The king, whose strong side, evidently, was not mathematics, surprised but glad that the gift would not cost him much of his treasure, easily accepted the offer and ordered the wise man to be handed over the required amount of wheat. A bag of

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wheat was brought to the throne and the counting immediately began. With 1 grain for the first square, 2 for the second, 4 for the third and so forth, it was emptied before the twentieth square was accounted for. More bags of wheat were brought before the king, but the number of grains needed for each succeeding square increased so rapidly that it soon became clear that even with all the crop of India the king could not fulfil his promise to Sissa Ben.

ing by the constant factor minus 1:

Actually, the amount requested by the grand vizier was that of the world’s wheat production for the pe­ riod of two thousand years! If Sissa were to store his reward he would require a building with the following dimensions: 25 miles (40 km) long, 25 miles wide, and 984 feet (300 me­ ters) high.

Thus, if we added an imaginary 65th square to the chessboard, the num­ ber of grains that were supposed to be laid on it, would be equal to the sum of the grains on the preceding 64 squares minus 1, which gives us the required answer.

The number of grains of wheat on the nth square is 2n­1. This is because the first square has 20 = 1 grain, the second has 21 = 2 grains, and the nth square has twice as many as the pre­ vious. Thus, the total number of grains of wheat is S = 1 + 2 + 4 + 8f + 2

63

or 0

1

2

3

S = 2 + 2 + 2 + 2 + f2

/2

i

i=0

where all terms form what is called a geometric series. In general terms, it can be expressed as 2

3

4

a + ar + ar + ar + ar + f, for r ! 0 , where a is the first term (or the scale factor) of the sequence, and r is the common ratio. We see that the series whose sum gives the total number of grains fol­ lows a regular pattern: each of the numbers is progressively increased by the same factor, which in the par­ ticular case is 2. The sum of all terms can be found by raising the constant factor (in this case 2) to the power represented by the number of steps in the progres­ sion (in this case 64), subtracting the first term (in this case 1), and divid­

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S=

64

/ 2 = 22 --11 = 2 i

64

-1

same thickness but of different diameters, with each disc resting on top of the next bigger one.

i=0

= 18,446,744,073,709,551,615. If we take a closer look at the terms of the sequence, we see that each consecutive number is equal to the sum of all preceding terms, minus 1: n-1

n

ar - 1

= / ar k k=0

A modern day example of the leg­ end asks whether one would accept the following offer: If your employer offered you a million dollars for a month of work, or a penny a day doubled - which would you take? Moreover, the number is a Mersenne number. It is a number of the form:

63

63

=

63

Mn ≡ 2n ­ 1, where n is a positive integer. The Mersenne numbers consist of all 1s in base­2, and are therefore binary repunits (numbers, consisting of copies of the single digit 1). The Tower of Hanoi (sometimes re­ ferred to as the Tower of Brahma or The End of the World Puzzle) was in­ vented by the French mathematician Édouard Lucas in 1883. He was in­ spired by an old legend that tells the following: At the beginning of the world the god Brahma gave this puzzle to the priests in a temple in the Indian city of Benares. The base was a brass plate holding three diamond needles, each as thick as the body of a bee. Around one of the diamond needles, were 64 discs made of gold, all of the

Brahma instructed the priests to work on moving the discs one at a time, never placing a disc on top of a smaller one, and to work efficiently and diligently, until the whole stack of 64 discs was transferred from Tower One to Tower Three in the minimum possible number of moves. When all the discs were finally in Tower Three, this would mark the end of the world, and the known universe would vanish into nothingness. The priests have been working incessantly, day and night, since the beginning of time, following Brahma’s instructions. The minimum number of moves for n discs is the number 2n – 1, hence for 64 discs mentioned in the legend, the required number of minimum number of moves is 264 ­ 1. Even if the discs were moved at the rate of one per second, using the smallest number of moves, it would take 18,446,744,073,709,551,615 seconds, or almost 585 billion years for the priests to finish the task. Scientists currently believe that the universe is 13.7 ± 0.13 billion years old, so by these estimates, and ac­ cording to the legend of the Tower of Brahma, the universe will still go on for at least another 41 times its cur­ rent age before vanishing into noth­ ingness when the priests finish moving the whole stack of 64 discs.

References Gamow, G. (1988) One, Two, Three .. Infinity, New York: Dover Publications Inc. Rouse Ball, W.W. and Coxeter, H.S.M. (1974) Mathematical Recreations and Essays, New York: Dover Publications Inc. ‘Ibn Khallikan’, http://www.humanis­ tictexts.org/ibn_khallikan.htm ‘How Old is the Universe?’ http://map.gsf c.nasa.gov/universe/uni_age.html

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The Mathematics of Fluid Dynamos David MacTaggart

Radostin Simitev

University of Abertay Dundee

University of Glasgow

School of Engineering, Computing and Applied Mathematics

School of Mathematics and Statistics

In this article we discuss fluid dynamos and how they generate the magnetic fields of astrophysical objects such as the Earth. As an illustration, we sketch one well­known mathematical model of Earth’s fluid dynamo and comment on some of its typical features and applications. The exposition will be accessible to undergraduate students in Science and Mathematics with some knowledge in Differential Equations and Physics.

Introduction Magnetic fields are ubiquitous in the Cosmos. They permeate galaxies, cause stellar eruptions and shield planets from cosmic radiation. These fields, however, are not due to permanent magnets but are sustained dy­ namically through processes driven by the motions of electrically con­ ducting fluids within the astrophysical objects. These processes are sometimes called fluid dynamos or, more precisely, homogeneous or magnetohydrodynamic dynamos. Their study is an important topic in geo­ and as­ trophysics and gives rise to a host of interesting mathematical and computational questions.

Self-excitation in the disk dynamo The basic principles of fluid dynamos are best illustrated by analogy to a familiar and simple electro­mechanical de­ vice ­ the bicycle disk dynamo, otherwise known as the Faraday homopolar disk dynamo. The disk dynamo can be eas­ ily constructed at home and understood with some basic knowledge of Classical Mechanics and Electrodynamics. A disk dynamo consists of a metallic disk of radius a that can rotate on its axis with an angular velocity ω, and a wire connecting the rim and the axis, as sketched in Fig.1.

36 [The, Commuttator] Sept 2012

In the same way in which Newton’s second law governs the motion of a particle on a plane, the rigid rotation of the disk in the absence of a magnetic field is gov­ erned by the equation

I~o = - f~ + T,

(1)

which states that the change in angular momentum of the disk with a moment of inertia I is equal ot the sum of the frictional torque, fω, assumed to be growing linearly with ω, and the externally applied torque T. When the disk is placed in an external magnetic field B = Bzt , a micro­

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Figure 1. Disk dynamo and associated notation

Figure 2. Schematic geometrical configuration of Earth’s outer core and associated notation. Electrically conducting fluid is confined between the two concentric spheres.

scopic Lorentz force (r # ~) # B = r~Brt acts on the charges in the disk as they move with speed rω perpendicular to the field. The work done by this force in displacing charges from the axis to the rim is equal to an electric potential f associated with an effective electromotive force

f=

#

a

dr $ ((r # ~) # B) =

0

# r~Bdr

f

(3)

By Ampère’s law this current generates a second magnetic field. If we now identify (i.e. replace) the initial externally applied field by this new field, an interesting dy­ namical feedback occurs. This is due to the processes described above, where the mu­ tual inductance f the magnetic field between coil and disk may be sustained by further rotation of the disk. This feedback effect can be characterised by the mutual inductance M between the coil and the disk, which, by Faraday’s law, is given by Bπr2 = MJ. Hence,

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M~J . 2r

(4)

Inserting this expression in equation (3) and solving the resulting linear first­order ODE, we find the solution J = J0 exp `` ~M - Rj t j . 2r L

(5)

(2)

An alternative way to derive this ex­ pression is to apply Faraday’s law of induction to a portion of the disk. The potential f drives an electric cur­ rent J through the coil. The coil has a self­inductance L and an ohmic resist­ ance R, and thus J is given by the equa­ tion of a RL­circuit LJo + RJ =

f=

a

0

2 = Ba ~ . 2

expression (2) can be written in the form

Note that if the condition that the angular speed is greater than some critical value ωc ~ > ~c = 2rR M

(6)

is satisfied then the current J is exponen­ tially increasing with time. It, t h e r e f o r e , generates an increasingly strong magnetic field. This process is called a self-exciting dynamo action. The exponential growth of the magnetic field B corresponding to solution (5) be­ comes unphysical as soon as the torque of the volumetric Lorentz force

#

a

r # (Jdr # B) = Jzt

0

# rBdr a

0 2

2 (7) = JBr zt = M J zt 2 2r acting on the disk reaches a significant level. Accordingly, equation (1) must be replaced by an equation of the form

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Figure 3. Kinetic (black line) and magnetic (red line) energies of a typical dynamo solution as functions of time.

matical model of the geodynamo. 2 I~o = - f~ + T - gJ ,

(8)

The dynamo equation. where g is a positive constant. The dynamics of the system is now described by the combined equations (3), (4), and (8). Since equation (8) is nonlinear in J, the system is diffi­ cult to solve exactly. Fortunately, we can find, easily, the equilibrium states of the system. These are of interest as they characterise the system’s long­term behaviour. Obviously, the nonmagnetic state J=0 is one possible equilibrium cor­ responding to angular speed ω = T/f. However, as soon as condition (6) is satisfied and the torque T exceeds 2πfR/M, the nonmagnetic state becomes unstable and after a short transient is replaced by a new magnetic equilibrium J = `T 2

2rfR 1 j . M g

(9)

Such a qualitative change of behaviour, as a function of a control parameter like T, is called a bifurcation.

We begin with the remark that the disk dynamo of Fig.1 is characterised by a simple form of motion but a complex distribution of the electrical conductivity. In fact, were the winding sense of the coil around the axis the other way around, the dynamo would not function. The opposite is true for fluid dynamos ­ their simple distribution of con­ ductivity requires a complicated velocity field as a condi­ tion for dynamo action, in a similar way in which expression (6) must be satisfied in the disk dynamo. To find the magnetic field generated by an arbitrarily complicated velocity field u, we need an equation analogous to (3). Such an equation can be easily derived according to the follow­ ing scheme from Ohm’s law for a moving conductor and Maxwell’s equations of Classical Electrodynamics, in their so­called magnetohydrodynamic approximation, where the displacement current is neglected, j = v (u # B + E)

A mathematical model of the geodynamo The disk dynamo demonstrates that the motion of a con­ ductor in a small initial magnetic field may amplify this field and sustain it indefinitely against dissipation. By the same basic principles, the motions of electrically conduct­ ing fluids, such as liquid metals or electrically charged plasmas, can induce strong magnetic fields. One such fluid dynamo generates the magnetic field of the Earth. The Earth has a layered structure with an outer core being a spherical shell full of liquid metal as shown in Fig.2. The liquid metal in the outer core of the Earth plays the role of both the RL coil, since it is electrically conducting, and the rotating disk, since it is in a state of convective motion. We will now sketch, as an illustrative example, one well­known mathe­

38 [The, Commuttator] Sept 2012

d # B/n = j

^ nvh- 1 dB = u # B + E

2t B = - d # E

2t B = d # (u # B) - d # (md # B)

d$B = 0

^2t + u $ dh B = B $ du + md2 B The left hand side equation in the third row is called the magnetic induction equation or, simply, the dynamo equation. It is the central equation of Magnetohydrodynamics ­ a disci­ pline that combines Fluid Dynamics and Electrodynamics. In the last row the general form of the dynamo equation has been simplified by the assumption that the velocity field u satisfies ∇ · u = 0. This is a good approximation in

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Figure 4. Examples of dynamo symmetry types. (a) Dipolar dynamo, (b) Quadrupolar dynamo, (c) Hemispherical dy­ namo. The plots in the left column show azimuthally­averaged longitudinal (toroidal) magnetic field lines (to the left) and latitudinal (poloidal) magnetic field lines (to the right) in a meridional cross section of the shell. The plots in the right column show the radial component of the magnetic field in the volume of the shell.

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Figure 5. Structures of the convective velocity field of a typical dynamo solution. The left plot shows stream lines of convection in the equatorial cross section, the middle plot shows radial velocity (downflow ­ dark, upflow ­ bright) at the middle of the shell, and the right plot shows the tempera­ ture field in the equatorial cross section.

Earth’s outer core. The inverse of the product of the elec­ trical conductivity σ and the permeability μ is called the magnetic diffusivity λ and is also assumed constant. When formulating mathematical models it is useful to put the equations in non-dimensional form, so that (a) the number of parameters in the model is minimised and (b) physical quantities are measured in problem­specific characteristic units. For instance, when the thickness of the outer core d is used as unit of length, d2/ν as a unit of time, and ν(μϱ)1/2/d as a unit of magnetic flux density, where ν denotes the kinematic viscosity of the fluid, ϱ its density and μ its magnetic permeability, the dynamo equation can be rescaled to take the form

^2t + u $ dh B = B $ du + P m- 1 d2 B,

(10)

where the dimensionless parameter Pm = λ/ν is called the magnetic Prandtl number. This equation plays the same role in the theory of the geodynamo as equation (3) in the description of the disk dynamo.

The equations of fluid motion and heat. To follow the analogy of the disk dynamo, we now need an equation to play the role of equation (8) and describe the motion of the fluid, i.e. to provide a velocity field u that we can plug into the dynamo equation (10). The equa­ tion that describes the motions of fluids is called the Navier-Stokes equation. This is the fundamental equation of Fluid Dynamics and, in the case of the geodynamo, can be

40 [The, Commuttator] Sept 2012

written in the form

^2t + u $ dh u 2 = - dr - xk # u + Hr + d u + B $ dB.

(11)

In fact, this it is nothing else than Newton’s second law applied to a continuous fluid medium rather than to a sin­ gle point particle. It states that the rate of change of mo­ mentum of a fluid element with time (LHS terms) is equal to the sum of forces acting on it (RHS terms) ­ the pressure, the Coriolis, the buoyancy, the viscous and the Lorentz forces, from left to right respectively. The analogy to equa­ tion (8) is clear, right up to the Lorentz term. Pressure π and temperature Θ are thermodynamic quantities that are undetermined by equations (10) and (11). They are gov­ erned by the laws of Thermodynamics which, in this case, can be reduced to a simple heat equation

^2t + u $ dh H = ^ Rr $ u + d2 Hh 1 . P

(12)

Together with the conditions for the solenoidality of u and B and suitable boundary conditions, equations (10) and (11) and (12) provide one possible mathematical model of Earth’s outer core fluid dynamo. The model is controlled by four non­dimensional parameters ­ the Rayleigh, Cori­ olis and Prandtl numbers 6

R=

2 acbd , x = 2X d , ol o

P = o, l

Pm = o . m

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These are all derived by a dimensional analysis simi­ lar to the magnetic Prandtl number introduced above. In that, ν2/γαd4 is taken as the unit for temperature and κ is the fluid’s thermal diffusivity, α is the thermal expansion coefficient, γ is the gravitational accelera­ tion, β is the density of internal heat sources and Ω is the angular speed of Earth’s rotation. The replacement of the many original parameters by few clearly demonstrates the advantages of non­dimensionalisa­ tion.

Properties of the geodynamo and its model A mathematical model is only useful if it is a true representation of the properties of the respective nat­ ural system, and if it helps to better understand and predict them. Below we illustrate some solutions of our model and make a rudimentary comparison to the magnetic field of the Earth. Equations (10) and (11) and (12) are rather complicated and they can only be solved numerically. Similar to condition (6), after the velocity of the fluid exceeds a certain threshold measured by the parameter R, self­excited dynamo action is achieved. Fig. 3 shows the magnetic and ki­ netic energies of a typical dynamo solution as func­ tions of time. They are not constant but vary in a nearly chaotic fashion, so that discernable patterns are difficult to distinguish. This behaviour is similar to that of the Earth’s magnetic field which is not constant but features numerous temporal variations. Such vari­ ations occur on a broad range of time scales from tens of years (secular variation impulses), to thousands of years (reversals), to billions of years (superchrons). Despite the variations, the paleomagnetic records in­ dicate that the Earth’s magnetic field has remained ap­ proximately dipolar throughout its existence. The structure of a typical dipolar solution of our model is shown in Fig. 4. A dipole is characterised by equatori­ ally antisymmetric radial magnetic field. Other fea­ tures of the true geomagnetic field include magnetic field polarity reversals and excursions, the westward drift of magnetic structures, flux patches, oscillations of the dipole moment and others. While the state­of­ the­art models, such as ours, still fall short of resolv­ ing the full details of the geodynamo, many of these features can be captured in the numerical simulations. The model, however, is capable of generating behav­ iour not observed in the geodynamo. Fig.4(b) shows an example of a so­called quadrupolar dynamo in which the radial magnetic field is symmetric with respect to the equatorial plane and where more than one pair of North/South poles might exist. Fig. 4 also shows a cu­ rious example of a so­called hemispherical dynamo in which the magnetic field is concentrated in only one

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of the hemispheres while the other one is nearly field­ free. Such behaviour is not observed in the case of the Earth but may well be relevant to other planets or stars or in fact occurs in the geodynamo sometime in the future. This illustrates how a mathematical model is useful in predicting unknown behaviour. Similarly, a mathematical model can help to investigate aspects of phenomena which cannot be directly observable. For instance, the motions of the liquid metal in the outer core are inaccessible to observations as they are hidden 2800 km bellow Earth’s surface. However, they can be visualised easily in numerical simulations as shown in Fig. 5. For this reason, self­consistent nu­ merical models are one of the very few methods avail­ able to obtain key insights into the convection­driven dynamo process as well as the myriad of other prob­ lems in stellar and planetary magnetism, where many questions still remain open.

Conclusion Cosmic magnetism plays an important and control­ ling part in many astrophysical objects, ranging from planets to accretion disks to pulsars. The Earth’s mag­ netic field is sustained by fluid dynamo action. This field protects us from the harmful effects of space weather ­ magnetically driven eruptions produced by the Sun’s fluid dynamo. These are important and in­ teresting areas of research. They are studied through mathematical models similar to the one outlined above. We hope that this article has given the readers a flavour of this exciting area and will prompt them to learn more. References E. Bullard. “The stability of a homopolar dynamo.” Math. Procs. Cambr. Phil. Soc., 51:744–760, 1955. P. A. Davidson. An Introduction to Magnetohydrodynamics. CUP, 2001. E. Dormy, J.­P. Valet, and V. Courtillot. “Numerical models of the geodynamo and observational constraints.” Geochem. Geophys. Geosyst., 1:2000GC000062, 2000. R.P. Feynman, R.B. Leighton, and M. Sands. The Feynman Lectures on Physics. Addison­Wesley, 1965. R. Simitev and F. H Busse. “Prandtl number dependence of convection driven dynamos in rotating spherical fluid shells.” J. Fluid Mech., 532:365, 2005. A. Tilgner. “Spectral methods for the simulation of incompressible flows in spherical shells.” Int. J. Num. Meth. Fluids, 30:713,1999.

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Modelling the Spread

of Ideas Eamon Quinlan The aim of this article is to describe mathematically how ideas spread through a community, and to try to derive the necessary conditions for an idea to become successful (that is, to be adopted by all the members of the community). First of all, we must take into ac­ count that every idea has its own op­ posite idea. For instance, the idea ‘I like rock music’ has its opposite ‘I don’t like rock music’, and we as­ sume that every person either has the idea or its opposite, and never both at the same time. From this point on, we shall refer to this oppo­ site idea as a counter-idea.

cle can undertake, and the probabil­ ities for these changes to occur. We shall refer to a single change of a cir­ cle as a toggling of its state (from filled to unfilled or vice versa). Note that the probability that a circle will toggle its state plus the probability that it will not must sum up to one, thus the 1­q and 1­p notations. See Figure 1.

In the above equation, n is the num­ ber of filled circles, N ­ the total num­ ber of circles, P and Q ­ factors that reflect the quality of the idea and its counter­idea (they must be between 0 and 1, with 0 depicting a very bad idea, whereas 1 ­ a very good one), and γ is the factor that would reflect the dependence of the probability on the density.

Now we have to describe what a community is and what the rules for idea spreading through a commu­ nity are. A community is a collection of persons, that we shall represent here as a set of filled or unfilled cir­ cles. Each circle will be filled if the corresponding person has a particu­ lar idea of interest, and unfilled if that person has its counter­idea.

These probabilities, as mentioned above, depend on two factors: the quality of the idea (or the counter­ idea) and the number of people that believe in the same idea (or counter­ idea). Thus the ‘density’ of people that believe in a particular idea equals the number of people that be­ lieve in the idea divided by the total number of people. However, we can­ not predict how the probabilities will depend on this density (since they could be dependent on the den­ sity, or the density squared, or the density to the power 1.33). Moreover, every idea, probably, has a different dependence on the density, but we shall assume here that the depend­ ence for an idea is the same as its counter­idea. Given this, we can write the probabilities as follows:

Now let us have a look at what happens when we have a number of filled circles at a given point in time, and we let these probabilities work on them for an instant of time. This means that every circle can have its state toggled or not but cannot be toggled twice. Since we have as­ sumed that we are dealing with a sufficiently large number of people, the law of large numbers holds, which in our case says that out of all filled circles, a fraction given by q will change and a fraction given by 1­q will not. After this instant of time, the number of filled circles n be­ comes n’ ­ the number of filled cir­ cles that have not changed their state plus the number of circles that have changed it from unfilled to filled. That is:

c p = P` n j , N c N q = Q` - n j . N

nl = n (1 - q) + (N - n) p.

As for the rules which govern idea spreading, we shall consider that for each instant of time, there is a proba­ bility that an unfilled circle will be filled, and a filled circle will be un­ filled, respectively, in the next in­ stant of time. These probabilities depend on two factors: the ‘quality’ of an idea (or the counter­idea), and the number of people that believe in that idea. We shall try to represent in some way the different changes that a cir­

42 [The, Commuttator] Sept 2012

If we subtract n from both sides of the equation, we get the difference in

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Figure 1 the number of filled circles that oc­ curred for an instant of time, that is the derivative of the number of filled cir­ cles with respect to time. If we expand p and q as indicated above, we get: c

dn = P n c 1 dt N c c+1 (N - n) -Pn c . nQ c N N This is a very easily solvable differen­ tial equation for some given value of γ. Now, we shall consider some solu­ tions. 1) Our first case is γ=0. It means that there is no dependence between the probabilities and the density. The solu­ tion is: (P Q) t n (t) y = 0 = ` n0 - NP j e- + P+Q + NP . P+Q We introduce n0, the initial number of filled circles (at time t=0). However, we are more interested in the steady state of this function, i.e. its behaviour as time tends to infinity. We see that: nc= 0 "

NP , as t " 3 . P+Q

2) We now consider the case when γ=1, which is a linear dependence be­ tween the probabilities and the den­ sity. The solution is: N n (t) c = 1 = . N - 1 e- (P + Q) t + 1 `n j 0 Again, we are more interested in the steady state of this function. Because the answer depends on the relative values of P and Q, this case is more in­ teresting: P > Q: nc = 1 " N P < Q: nc = 1 " 0 P = Q: nc = 1 " n0 as t " 3.

will only coexist if they are of the same quality, that is, if the factors P and Q are the same. In a sense there begins to emerge a bit of ‘unfairness’ in that if both probabilities were of the same quality, the idea would not spread to half of the population. This is so be­ cause if the ideas were not dependent on the density the idea would not spread. Instead, the number of believ­ ers remains constant. The equation is difficult to solve for other values of γ. However, it is possible to determine the necessary conditions needed in order for the idea to be able to spread. From the initial differential equation we see that the number of filled circles increases when the left hand side is positive: n grows , c c c 1 (N - n) n + >0. P nc - 1 - nQ P c c N N N We can now get the minimum number of people necessary for the idea to grow for some fixed P, Q, γ and N (this calculation is done by taking P as a common factor). We get the following result: N n> . 1 P c-1 + 1 `Qj This is a very important result as it proves that, as soon as n grows at any instant of time, it will keep growing in­ definitely because the condition will still be satisfied. Analogically, if it de­ creases at any instant of time, it will keep decreasing. Therefore we can say that the condition above is a condition on the initial number of people in order for the idea to spread. Now, let us suppose we are not in­ terested in the number of people needed in order for an idea to spread, but in the quality of the idea (given some initial conditions). This can be easily derived from our last condition: P > Q ` N - 1j n0

c

Here we see that the best idea will be spread to everyone, and both ideas

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.

Again we see that the condition is true for every instant of time, because if we substitute n0 with n, we see that as n grows the condition still holds be­ cause the right hand side becomes smaller. If the condition is not satisfied then the right hand side gets bigger, and therefore it does not hold for every instant of time. Also note that if we set γ to be equal to one, we shall get the same condition that we derived at the beginning. It might be argued that this model does not represent the real behaviour of ideas since it predicts that ideas will either be adopted by everyone or by no one, while in the real world there are ideas that have not been adopted by an entire population for centuries and yet are still alive. It is in fact difficult to think of an idea that is adopted by everyone on the planet. The reason for this is that in the beginning we assumed that our community was strongly intercon­ nected: this means that every circle was able to influence every other circle in exactly the same way. However, in the real world this is not possible to happen. We have proven here that in a strongly interconnected community ideas either spread entirely or decay. This opens the following possibility: we can now consider every commu­ nity as a circle, that is either filled or unfilled, and we can extend our con­ cept to communities of communities. But if we do this we have to define a new parameter that would reflect the interconnection between the different communities. Some communities might be more interconnected than others, and this might be the reason ex­ plaining why certain ideas can survive even if they have not been adopted by everyone. This interconnection has been grow­ ing very fast in the last decades, due to the advances in technology, and now we can see that the world has begun to behave like a big, intercon­ nected community. The University of Glasgow is a very good example, since there are many international students, and it is noticeable that their behav­ iour and ideas are not so different than those of the people from the UK, whereas perhaps a few decades ago, the difference would have been very vast.

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The problem of constructing the incircle of a given triangle is very simple but what happens if we replace the sides of the triangle with arcs of circles? We now need to solve a spe­ cial case of the Apollonius problem ­ when all three given circles have a common area. Apparently, the con­ struction is similar to the one The main problem here is to construct the incircle and excircles of a circular triangle. We shall use an analogical used to construct the incir­ construction to the one used to construct the incircle and ex­ cle of a regular circles of a regular triangle. In the following section some prop­ erties of the circular angle bisector are presented. Then a circular triangle.

Incircles and Excircles of a Circular Triangle

triangle construction is given and proven, followed by a hypothesis that it can also be feasibly applied in higher dimensions.

Radko Kotev

The Circular Angle Bisector The circular angle is an angle defined by two intersecting circles. The angle itself is the angle between the tangent lines to both of the circles at one of their common points (Fig. 1). The circular angle bisector is a circle that bisects the angle of two given circles. There are two circular angle bisectors for any two intersecting circles. The two given circles are inverted images of each other with respect to the circular angle bisector. Another impor­ tant fact is that the centres of the circular angle bisectors are the centres of homothety of the two given circles.

Figure 1. The circular angle.

Main Result A given circular triangle ­ in other words three intersect­ ing circles k1(O1,r1), k2(O2,r2) and k3(O3,r3) ­ defines eight dif­ ferent areas in the plane: one is common for all of the circles, another is external to all of them, three are common for the three different pairs of circles and the rest are con­ tained by only one circle of the given circular triangle.

44 [The, Commutator] Sept 2012

Figure 2. First step ­ constructing the angle bisec­ tors. They intersect at two points, A and B. First, we choose one of those areas. Let it be the one that is inside all of the circles. Considering a regular triangle, the first thing we do is constructing the angle bisectors, which effectively results in the construction of circular angle bi­ sectors of all three angles of the circular triangle. There are three angle bisectors which intersect in that area. The other point of intersection is outside the circles. The main pur­ pose of this first step is finding those two points, denoted A and B (Fig. 2). In the case of a given regular triangle, we construct per­ pendicular lines to the sides of the triangle through the in­ tersection point of the bisectors. Once we have determined the intersection points of the angle bisectors of the circular triangle ­ A and B ­ we construct another three circles through them that are perpendicular to the given three ones ­ k1, k2 and k3. The perpendicular to the respective cir­ cle intersects that circle at two points. At this stage we have got six points (Fig. 3). These points define two circles that

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Figure 3. Second step ­ constructing perpendicular circles through A and B to each of the given circles.

are tangent to k1, k2 and k3 (Fig. 4). In this way, we can construct all seven excircles and the incircle of the cir­ cular triangle. Now we have to prove the above­pre­ sented construction. Proof: Let us have two given circles k1(O1,r1) and k2(O2,r2). The circle kb(O,r) is their circular angle bisec­ tor, and the points A and B are on kb. We shall prove that when the perpendicular circles are constructed through A and B to k1 and k2 respectively, these circles intersect k1 and k2 at such points that we can construct a circle through two of these points, tangent to both k1 and k2. Let ϕ be an inversion with respect to kb. Then ϕ(k1)=k2, ϕ(A)=A and ϕ(B)=B. We construct the circles k1ʹ and k2ʹ, perpendicular to k1 and k2 respectively. Consequently, ϕ(k1ʹ)=k2ʹ. Let k1ʹ intersect k1 at P1 and k2ʹ intersect k2 at P2, where P1 and P2 are in the same half­plane defined by O1O2. Then we know that ϕ(P1)=P2, and therefore, O, P1 and P2 lie on the same line. Since O is the centre of homothety of k1 and k2, there is a circle through P1 and P2 that is tangent to k1 and k2 (Fig. 5). Now we have proven that for any two points on the circular angle bisector of two circles, if we construct the perpendicular circles through those two points, they intersect the given circles at such points that there is a circle through them tangent to the two given cir­ cles. We have got three given circles k1(O1,r1), k2(O2,r2) and k3(O3,r3) that intersect each other and a common area. Their outer circular angle bisectors intersect each other at points A and B. Now construct circles k1ʹ, k2ʹ and k3ʹ which are perpendicular to k1, k2 and k3 respectively. Circle kiʹ intersects ki at points Pi and Ti as shown in Fig.6. We shall prove that the circle defined by points P1, P2 and P3 is tangent to the three given circles.

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Figure 4. Third step ­ constructing the solution cir­ cles through the previously constructed points.

Construct the tangent lines through Pi to circle ki. These tangent lines define the triangle A1A2A3. We know that there is a circle through P1 and P2 tangent to k1 and k2. Consequently A1P1=A1P2. Analogically, we can prove that A2P1=A2P3 and A3P2=A3P3. But these are the points at which the incircle touches the sides of the triangle. This means that the circle defined by P1,P2 and P3 is tangent to A1A2, A2A3 and A1A3 at the points Pi. There­ fore, this circle is tangent to k1, k2 and k3 (Fig. 7).

High Dimensional Hypothesis The presented solution is applicable in three dimen­ sions through further work and research aided by spe­ cialised computer software. However, a mathematical article has to provide concrete proofs and, at the mo­ ment, the above statement is just a hypothesis and one possible opportunity for future development on this topic.

References Gisch, D. and Ridbando, J. (2003) Apollonius' Problem: A Study of Solutions and Their Connections, Department of Mathemat­ ics Northern Iowa Kunkel, P. (2007) ‘The tangency problem of Apollonius: three looks’, Journal of the British Society for the History of Mathematics, vol. 22 http://www.ics.uci.edu/~eppstein/junkyard/tangencies/bisec­ tor.html Weisstein, Eric W. ‘Apollonius’ Problem.’ From MathWorld ­ A Wolfram Web Resource. http://mathworld.wolfram.com/ ApolloniusProblem.html

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Figure 5. Two circles, their circu­ lar angle bisector, perpendicu­ lars through two points on kb to k1 and k2

Figure 6. Three given circles and perpendicular circles through the intersections of the angle bi­ sectors.

Figure 7. Tangent lines through P1, P2 and P3 define the triangle A1A2A3 and they are the tangent points of the incircle.

46 [The, Commutator] Sept 2012

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Hristo Georgiev

The World’s Smallest Football

Figure 1. (left) buckminsterfullerene (C60); (right) truncated icosahedron

Hristo Georgiev

The Look-and-Say Sequence The look-and-say sequence, also known as the Morris sequence, is a sequence of integers begin­ ning with a single digit, in which the next term is obtained by describing the previous term. It is also known as the simplest mathematical pea pattern. Such sequence grows indefinitely and starting with 1, is defined by: 1, 11, 21, 1211, 111221, … (or 1, one 1, two 1s, one 2 one 1, one 1 one 2 two 1s, …) In general terms, starting the sequence with the digit d for 1 ≤ d ≤ 9 gives:

A football is tessellated with 20 hexagonal and 12 pentag­ onal faces. A simple polyhedron made up entirely of pen­ tagons and hexagons is called a fullerene. It was discovered by Robert F. Curl Jr., Sir Harold W. Kroto and Richard E. Smalley, who were awarded the 1996 Nobel Prize in chem­ istry for their discovery. Certain fullerenes have important applications in chemistry. A famous example resembling a football is buckminsterfullerene, C60, also known as buckyballs (Fig. 1 (a)), which was first synthesized in 1985 by Robert F. Curl. It has a carbon atom at each vertex and a bond along each polygon edge. It was named after the American archi­ tect, engineer and inventor, Richard Buckminster Fuller who developed the famous geodesic dome. Long before the first buckyballs were studied in the lab, H. S. M. Coxeter raised the question of what values were possi­ ble for the number of hexagons in a simple polyhedron (made up of only pentagons and hexagons). In 1963, Branko Grün­ baum and Theodore Motzkin proved that there is no fullerene with exactly one hexagon, but any other number of hexagons is permissible. Rufus P. Isaacs proved that if a sphere is tessellated with hexagons and pentagons, there must be exactly 12 pentagons. Fullerenes have 12 pentagons and v/2 – 10 hexagons, which in the case of C60 is 60/2 – 10 = 20. One of the distinctive features of the buckyball is its highly symmetric structure which is obviously mathematical. A fa­ mous Euler formula says that the surface of a polyhedron in three­dimensional space is made up of two­dimensional faces, f, one­dimensional edges, e, and zero­dimensional vertices, v: f – e + v = 2.

The sequence generated by d = 3 is also known as Conway’s sequence (named by Ilan Vardi in 1991) and begins as follows:

For example, the surface of a pyramid has five faces (four triangle and one square), eight edges, and five vertices. It can be easily checked that those numbers satisfy the above equa­ tion: 5 – 8 + 5 = 2. The regular icosahedron has twelve vertices (of degree 2), thirty edges (five edges emanating from each vertex) and twenty triangular faces, 20 – 30 + 12 = 2. A buck­ yball can be obtained by symmetrically truncating the icosa­ hedron at each vertex, thus replacing the vertex by a pentagonal face. The effect is that for each vertex replaced, one face, five edges, and five vertices have been added, thus resulting in thirty two faces (twelve pentagons and twenty hexagons), ninety edges and sixty vertices, 32 − 90 + 60 = 2. This structure is mathematically known as truncated icosahedron (Fig. 1 (b)), which is also one of the Archimedean solids.

3, 13, 1113, 3113, 132113, 1113122113, 311311222113, ...

References

d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, 1321132132211d … The sequences generated by d = 1, 2, 3 have been recoded in the Online Encyclopedia of Integer Sequences (OEIS) and given identifiers A005150, A006751, and A006715 respectively.

References Weisstein, Eric W. ‘Look and Say Sequence.’ From Math­ World ­ A Wolfram Web Resource. http://mathworld. wolfram.com/LookandSaySequence.html

Chung, F. R. K. and Sternberg, S. (1993) Mathematics and the Buckyball. American Scientist 81, No. 1., 56­71 Gardner M. The Colossal Book of Marthematics Hopkins B. (ed.), Resources for Teaching Discrete Mathematics

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Humour

TEXT "The number you have dialed is imaginary. Please, rotate your phone by 90 degrees and try again..." A topologist is a person who doesn’t know the dif­ ference between a coffee cup and a doughnut.

A biologist, a statistician, and a

mathematician are sitting in a street­ cafe watching people going in and com­ ing out of the building on the other side of the street. First they see two people going into the house. Ten minutes later, they come out ac­ companied by a child. ‘They have reproduced’, says the biologist. ‘No’, says the statistician. ‘It’s an observa­ tional error. On average, two and a half people went each way.’ ‘No. No. No.’, says the mathematician. ‘It’s perfectly obvious. If someone goes in now, the building will be empty.’

Three Functions Walk into a Bar...

The functions are sitting in a bar, chatting (how fast they go to zero at infin­ ity, you know, the usual). Suddenly, one cries “Beware! Derivation is com­ ing!”. All immediately hide themselves under the tables, only the exponential sits calmly in his chair. The derivation comes in, sees a function and says “Hey, you don’t fear me?” “No, for I am e to the x”, says the exponential self­confidently. “Well”, replies the derivation, “ who says I differentiate with respect to x?”

Mathematics Revisited Life is complex. It has real and imaginary components. What keeps a square from moving? Square roots, of course. The law of the excluded middle either rules or does not rule. In the topological hell the beer is packed in Klein’s bottles.

To a mathematician, real life is a special case. I heard that parallel lines actually do meet, but they are very discrete. In modern mathe­ matics, algebra has become so important that numbers will soon only have sym­ bolic meaning.

48 [The, Commutator] Sept 2012

How does a mathematician

induce

good behavior in her children?

`Iʹve told you n times,

Iʹve told you n+1 times...!`

Three statisticians go hunting. When they see a rabbit, the first one shoots, missing it on the left. The second one shoots and misses it on the right. The third one shouts: “Weʹve hit it!”

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“Piled Higher and Deeper” by Jorge Cham www.phdcomics.com


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Little Pigley Farm

Little Pigley is the name of a small farm owned by the Dunk family and is located in an idyllic country setting in the south of England. The land on which is stands is a rectangular-shaped meadow known as Dog’s Mead. Find: 1. The year in which the puzzle is set.

This puzzle was created by W.T. Williams in 1935 and first appeard in The Strand Magazine. It has also been known as Dog’s Mead, Little Pigley, Little Piggly Farm, Little Pigsbt, Pilgrims’ Plot and Dog Days.

2. The prime factor of the sum of the puzzle’s digits that is also a factor of 1D Notes: ­ Answers are positive integers and do not contain leading zeroes. ­ Measurements are imperial: A rood = 1/4 acre; An acre=4840 square yards; A mile=1760 yards;

ACROSS

A pound sterling=20 shillings.

DOWN

1. Area (in square yards) of Dog’s Mead.

1. Value (in shillings per acre) of Dog’s Mead.

5. Age of Farmer Dunk’s daughter Martha, who is more than 12 months older than Mary.

2. Square of the age of Mrs Grooby, Farmer Dunk’s mother­inlaw. 3. Age of Mary.

6. Difference (in yards) be­ tween the length & breadth of Dog’s Mead.

4. Value (in pounds sterling) of Dog’s Mead.

7. Number of roods in Dog’s Mead multiplied by 9D.

6. Age of Ted, Farmer Dunk’s first born (next year, Ted will be twice as old as Mary is then).

8. The year that the Dunk family acquired Little Pigley. 10. Farmer Dunk’s age (he married when he was out of his teens and later started a family). 11. The year of birth of Mary, Famer Dunk’s youngest. 14. Perimeter (in yards) of Dog’s Mead.

7. Square of the breadth (in square yards) of Dog’s Mead. 8. Number of minutes Farmer Dunk takes to walk 1 1/3 times around Dog’s Mead. 9. See 10D. 10. 10A times 9D.

15. Cube of Farmer Dunk’s walking speed in miles per hour.

12. 1+ the sum of the digits in the second column.

16. 15A - 9D.

13. Length of tenure (in years) of Little Pigley by the Dunk family.

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[The,Commutator] Edition 3 2012 Back Up_Commutator Edition 3 13/08/2012 11:17 Page 51

The Editor’s

The puzzle’s theme is ‘Glasgow University 2012’ and thus the frequent use of 20, 12 and 2012. The answer to the puzzle is in the shape of ‘GU’, with the digits positioned in the indicated order. Rules: Unless stated otherwise, all numbers and operations over them are carried out in decimal number system. Answer to the puzzle: the first 20 digits after the decimal point in the base­12 expansion of π. The positions of the digits are indicated in the bottom right corner of each shaded cell.

ACROSS 1. The first two­digit prime gap (the difference between two successive prime numbers) multiplied by 2012. 2. The product of the smallest positive integers x and y, such that 27x + 13y = 213. 3. The hexadecimal representation of the first three digits after the decimal point of the 20th root of the 12th number in the look­and­say number sequence generated with ini­ tial digit 1.

8. The minimum number of people in a group so that the probability that at least two of them have the same birth­ day is grater than 1/2. 9. 1 + the sum of all digits in the first column. 10. The doubled number of 1s, in the binary representa­ tion of the 20th Mersenne number, multiplied by 12.

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4. Total number of vertices that a regular football (one that has 20 hexagonal and 12 pentagonal faces) has.

11. The last four digits of the 2012th prime number.

5. The ceiling of the cube root of the total number of rice grains placed on a 8x8 chessboard, if one grain of rice were placed on the first square, two on the second, four on the third, eight on the fourth and so on.

12. Considering the Collatz problem: the 4th, 5th, and 6th digits, converted to hexadecimal, after the decimal point of the stopping time of 20 raised to the power of the stop­ ping time of 12.

6. The total number of cells in hexadecimal that form the following two independent neighbourhoods of a single cell in a two­dimensional grid: a von Neumann neigh­ bourhood of range 20 and a Moore neighbourhood of range 12.

13. The perfect square of the first digit of the answer to question 2.

7. A number that is equal to the sum of each of its digits raised to the power of 5.

15. The reversed first two digits of: the sum of all digits in the second row multiplied by 2012.

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14. 12 muliplied by the sum of all digits in the seventh col­ umn.

Sept 2012 [The, Commutator] 51


[The,Commutator] Edition 3 2012 Back Up_Commutator Edition 3 13/08/2012 11:17 Page 52