[The, Commutator] vol.3 issue 1

[The,Commutator] Edition 3 2012 Back Up_Commutator Edition 3 13/08/2012 11:11 Page 1

2 [The, Commuttator] Sept 2012 www.the-commutator.com

Contents

Jaspal PuriAssistant Editor/Graphic Designer

Gillian BowmanWriter

Sean LeaveyWriter

Eamon QuinlanWriter

The book of nature has been

written in thelanguage ofmathematics

Galileo Galilei

Special thanks toDr. Lorna Love, Mrs. Shazia Ahmed,

Dr. Radostin Simitev, Dr. David MacTaggart, Dr.Chris Athorne, Prof. Nicholas Hill, Dr. Marianne

Freiberger, Dr. Rachel Thomas, Emily Wood-house, Prof. Jos-Manuel Rey, Remus Stana,

Sebastian Popescu, Jennifer Petrie.

University ofGlasgowSchool of Mathematics and Statistics

University Gardens, University of GlasgowGlasgow, G12 8QA

http://www.gla.ac.uk/mathematicsstatisticstel: +44 (0) 141 330 5176

email: [email protected]

[email protected]

Editor in ChiefHristo Georgiev

Graphic Designer/Assistant EditorJaspal Puri

3 Letter from the editor

4 Our Special Thanks

6 Diophantine Problems for Garden Gnomes

9 The Riemann Hypothesis

11 The Carol Syndrome

14 To Infinity and Beyond

17 The Birthday Problem; Number Curiosities

18 Cellular Automata and Conways Game of Life

29 Prime Numbers

32 An Epidemic Beginning

34 The Significance of the Number 264-1

36 The Mathematics of Fluid Dynamos

42 Modelling the Spread of Ideas

46 Incircles and Excircles of a Circular Triangle

47 The Look-and-Say Sequence; The WorldsSmallest Football

48 Humour

50 Puzzles Printed in the UK byThe Magazine Printing Company

using only paper from FSC/PEFC supplierswww.magprint.co.uk


Sept 2012 [The, Commutator] 3www.the-commutator.com

Letter from the editor ...

Radko KotevWriter

Fiona MaddenWriter

Karl NordstrmWriter

Dear Reader,

2012 is the year of our thirdbirthday and we would liketo invite you to celebrate withus through our new mathematical adventure. Unlike atraditional birthday, we areextremely happy to be able togive you the copy of the latestissue of our magazine as apresent and we hope that youwill enjoy reading it as muchas we enjoyed preparing it for

you. Now let the trip in the maths wonderland begin.

Maths is fun, dont you think?

Whether for modelling football score predictions, unravelling the genetic code or saving human livesthrough tomography, mathematics remains the mostfascinating subject of all. Not only mathematics is fun,but it is also incommensurably beautiful. The evidences are numerous and the only diculty one mayface is where to start shall it be Eulers identity whichstates that ei + 1 = 0, or his fundamental theorem ofthe innitude of primes. Perhaps the trip could beginfrom the various proofs of the Pythagorean theoremeach one more elegant than the other, EulerMacLaurins formula, or the complex 248dimensional E8structure, which has been used as a basis in variouscandidates for the theory of everything. While somepeople would commence the exploration with EulerMascheronis constant , others might prefer to set upNewtons iteration for matrix inversion as a startingpoint.

So, have we managed to convince you yet?

Well, we at [The, Commutator] believe Maths is quitefun, and shall amplify our meaning of these words inthe subsequent paragraphs. Throughout the pages ofthe current issue we shall discuss, not necessarily inthis particular order, various fascinating topics in

mathematics, such as the beauty in the Riemann Hypothesis, what the universal formula for attraction is,as well as the dierent types of innity, actual and potential.

Our mathematical discussion then continues withthe subject of uid dynamos and how they generatethe magnetic elds of astrophysical objects such as theEarth.

Several dierent aspects of mathematical modellingshall be also concerned and if you keep reading, wepromise that well show you how mathematics helpsus model the spread of ideas, or even the spread of abiological disease. Mathematical models also provideus with a more profound understanding of the particularities of evolution, through idealisations such as cellular automata. We shall also reveal the key behind thequestion why a particular species of insect has developed a mating cycle of a prime number of years, andhow random this choice actually has been. Further,youll learn what the mathematical properties of theworlds smallest football are.

A closer look will be taken at the story of a happygnome couple, Mr and Mrs Magnus, as well as at twopopular ancient legends and what they share in common. In addition, we present a special case of the Apollonius problem and its solution.

Hang tight because apart from the aforementionedtopics, we shall try to interest you with many more.While attendance in class will not be recorded, we shalltest your knowledge in [The, Commutator] 3A coursewith an unexpected examination in the form of twoindependent crossnumber puzzles. You are also invited to participate into the rst Maths & Stats ArticleCompetition 2013 organised by our magazine.

As usual, the editorial sta of [The, Commutator]would like to ask you not to hesitate to share any comments, ideas or suggestions, should you think thismight help the improvement of the magazine, whichhas always been our primary goal.

Warmest Regards, Hristo GeorgievEditor in Cheif


Two of the articles that we are presenting in current issue, Diophantine Problems for Garden Gnomes and The CarolSyndrome, have been kindly provided by the popular Plus magazine (where they were rst published) and theirauthors, Emily Woodhouse an undergraduate student at Durham University, and JosManuel Rey AssociateProfessor 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 Trevelyanis in charge of the graphics design and Owen Smith is computer ocer. The magazine aims to introduce readersto the beauty and the practical applications of mathematics. It provides feature articles, which describe applicationsof 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 thenewspapers; a lucky dip of mathematical curiosities; and opinions on various mathsrelated topics and news stories; a regular interview with someone in a mathsrelated career, showing the wide range of uses maths gets putto in the real world. All past issues are available online, whichis a useful resource for maths school students and teachers.

Plus is part of the Millennium Mathematics Project, a longterm national initiative based in Cambridge and active acrossthe UK and internationally. The MMP aims to help people ofall ages and abilities share in the excitement of mathematicsand understand the enormous range and importance of its applications to science and commerce. It works to change peoplesattitudes to maths, to act as a national focus for renewing andimproving appreciation of the dynamic importance of mathsand its applications, and to demonstrate the vital contributionof 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 andkeep uptodate by subscribing to Plus (on email, RSS, Facebook, iTunes or Twitter) at http://plus.maths.org.

4 [The, Commutator] Sept 2012 www.the-commutator.com

Glasgow Mathematical Journal publishes original researchpapers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory,group theory, functional analysis, combinatorics, dierentialequations, dierential geometry, number theory, algebraictopology, and the application of such methods in appliedmathematics.

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

A Note On TheSponsor

A Note On The Current Issue



Problems ForDiophantine

Garden Gnomes



Suppose you are asked to solve theequation

There are two unknowns and only oneequation, so youll get innitely manysolutions. Rearranging, you get

so you know what shape these solutions take.

But now suppose youre asked to nda particular solution such that both

and are positive whole numbersand their sum is as small as possible.You could use trial and error to nd thesolution (if it exists), but this is mathematics, and in mathematics we like a bitof method and order. Fortunately, onemethod for integer solutions came outof Alexandria in about 250 AD. But, before we get into that, lets add a littlecontext to make things more interesting.

Mr and Mrs Magnus are a happygnome couple. Unfortunately, foreseeing that they will be unable to keep upwith their rising mortgage repayments,theyve been forced to move into myback garden (where salaries are muchhigher). However, they have been presented with some trouble in buildingtheir new home. Gnome bylaws statethat the total number of bricks used inany construction project must be 177 orplanning permission will not begranted.

As gnome houses form roughly theshape of a triangular prism and onewall (namely the garden fence) is already in place, only two walls need tobe built. The plot of land they have acquired is shown below with all dimensions measured in bricks.

The question is, what is the smallestnumber of bricks they have to buy tocomply with the local council, yet notwaste any money?

Luckily, Mr and Mrs Magnus son,Zurich, knows a thing or two aboutnumbers. As his parents discuss theirdilemma over tea, Zurich writes out anequation:

where and are the height in bricksof the two respective walls. Chuedwith himself, he shows his parents andreceives a solemn thats nice dear.

Duly encouraged, Zurich sits andthinks about how to solve this equation.The secret here is that you cannot havehalf a brick: and must be integers.(This, by the way, rules out that the twowalls have the same height, ie that

, because

gives that , which isnt awhole number.)

Now, Zurich has recently been reading Euclids Elements before he goes tobed and so has just found out about anifty trick called Euclids algorithm,which nds the highest common factorof two numbers, and . We assume

. First divide by and take theremainder, calling it . Then divide by , and take the remainder, calling it

. Keep repeating this process andeventually you get a remainder of zero.The last nonzero remainder is the highest common factor of the original twonumbers.

So, Zurich lets andfrom our previous equation. He thenperforms the algorithm with some zest.At each step, he keeps track of the calculation by writing .Here is the number thats being divided at that step, is the number itsbeing divided by and is the corresponding remainder:

This tells him that the highest common factor is 1 (if he goes any furtherhe will reach zero). Fortunately this iswhat we would expect, as 11 is a primenumber. Now he uses the equationsthat he has just generated to write out atrue statement equal to 1 that involves25 and 11:

He keeps 25 and 11 as separate factors,so:

Collecting terms:

Lets compare that to the original:

25 11 177.x y+ =

11177 25

,yx

=-

( , )x y

yx25 11 177,x y+ =

yx

yx

x y=

25 11 36 177x x x+ = =

116/36x =

r2r1

r2r1>r r1 2

r2r3

r3

r4

11r2 =r 251 =

r qr ri i i1 2- =+ +

ri

ri 1+

ri 2+

3 2 1 1

(25 11 2) (11 3 3) 1

#

# #

- =

- - - =

25 11 3 3 3 1

25 11 3 3(25 11 2) 1

25 25 3 11 3 11 6 1

# #

# #

# # #

- + =

- + - =

+ - - =

24 4 11 9 1# #- =

Mr and Mrs Magnus are a happy gnome couple.

Unfortunately, forseeing that they will beunable to keep up with their rising mort-gage repayments, theyve been forced to

move into my back garden (where salariesare much higher).

Emily Woodhouse

Durham University



So, Zurich rewrites his sum as:

Now, thats all very well and good, but it needs to equal 177not 1, so still treating 25 and 11 as terms, he multiplies thewhole equation by 177:

This is a perfectly legitimate answer, but sadly it is verydicult to come by negative bricks these days, and it certainly doesnt look like the smallest answer. So, Zurich ponders how to make positive without changing the sum ofthe equation from 177. To make the coecient of 11 positive, he needs to add some multiple of 11, say , to theleft hand side of the equation. To keep the equation true,he needs to take this multiple away again:

To get the equation into the required form, he writes

Since must be a whole number, he nows thatmust be a multiple of 25. Say for some . Thisgives

So now Zurich only needs to nd a number so thatis greater than zero but as small as possible.

Dividing 1593 by 25 using longdivision he gets:

Therefore, the number he is looking for is , giving

Zurich compares this to the original problem and nds thatthe solution is and the problem is solved just as his parentsnish their tea.

Diophantine equations

Generally, equations of the form

where the variables and are only allowed to be wholenumbers are called linear Diophantine equations after thegreat Greek arithmetician Diophantus, who dealt withmany such problems in his works.

If is a multiple of the greatest common divisor of and, then an equation of this form has an innite number of

solutions. This is the case in our example equation, where

, and , with the greatest commondivisor of and being 1. The further constraint that thesolutions must be positive and their sum as small as possible then gave us the particular result. If isnt a multipleof the highest common factor of and , then the equationhas no integer solutions at all.

There are also diophantine equations of higher degree,for example

where , and are the variables and is a constant. Thestatement that no integer solutions exists when is Fermats famous last theorem, which remained unproved forover 400 years. (It was in the margin of his copy of Diophantus book Arithmetica that Fermat wrote his renownedlines. See the Plus article Fermats last theorem and AndrewWiles for more information.)

In terms of practical applications, being able to solve thissort of problem is surprisingly useful even in the modernworld. The classic example is working out how many men,women and children attended a show given only the priceof each ticket and the nal takings. Plus, if you are that wayinclined, certain types of puzzles in magazines and newspapers can suddenly become a whole lot simpler with thisunder your belt. From business to coee breaks, exploringnumber theory to building your house, its denitely a technique worth knowing!

P.S.: In his excitement, Zurich forgot to make an allowance for the front door in one of the walls. Luckily, hisskilled father was able to make a chimney out of the extrabricks at the same time as close the gap that must be created where the two walls and the roof join.

25(708) 11( 1593) 177.+ - =

y

11n

25(708) 11( 1593)

25(708) 11( 1593 ) 11

177

n n

+ -

= + - + -

=

25(708 11 /25) 11( 1593 )

177.

n n- + - +

=

11 /25n n

25n t= t

25(708 11 ) 11( 1593 25 )

177

t t- + - +

=

t

1593 25t- +

64t =

25(708 (64 11)) 11( 1593 (64 25)) 177

25(708 704) 11( 1593 1600) 177

25(4) 11(7) 177.

# #- + - + =

- + - + =

+ =

ax by c+ =

x y

c a

b

a 25= 11b = c 117=

a b

c

a b

x y zn n n+ =

x y z n

n>2

25 11 177,x y+ =

25(4) 11( 9) 1+ - =



The Riemann Hypothesis

The Riemann Hypothesis has been considered one of the most importantunproven conjectures in pure mathematics ever since Bernhard Riemann stated

it in his legendary paper On the Number of Primes Less Than a Given Magni-tude (1859). Unfortunately it is quite dicult to correctly state the hypothesis

in a simple way, but this article will make an attempt to explain Riemannsconjecture by examining some of the historical background to his paper andthe discoveries he made in it, stating the conjecture in its original form, and

give an equivalent conjecture that is easier to understand by intuition. Finallyit will show how the hypothesis, if proven true, would allow for a much better

approximation to the prime counting function (x).

Karl Nordstrm



Of central importance to the hypothesis is the Riemannzeta function, dened as

for a complex variable . About a century before Riemannwrote his paper, Euler had shown that sums of this formare equivalent to innite products using prime numbers ofthe following form:

The proof for this equality is very beautiful and can bederived from the fundamental theorem of arithmetic, butis also outside the scope of this article. It resulted in establishing of the connection between the zeta function andprime numbers. Euler also calculated the values of the sumfor positive integer values of s up to 16, where (2), in particular, gives the solution to the Basel Problem. However,Riemann was the rst person to consider s as a complexvariable, s = +it rather than simply a real number. Hisaim was to discover a way to accurately count the numberof primes smaller than a given value. By analytically continuing the zeta function to all complex numbers, exceptfor s = 1 an impressive feat in itself he managed to derive an explicit formula for doing this. The formula impliedthat the real part of the zeros of the Riemann zeta functioncontrol how the values of prime numbers vary from theirexpected values.

In order to nd the zeros of the function, we can start byrealising that since convergent innite products never become zero, by Eulers product equality there can not existany zeros when the real part of s is greater than 1, so (s)0when (s) > 1. Riemann also noted that all points wherethe real part of s was a negative even integer, so that (s)= 2, 4, 6 ..., are zeros. These are called the trivial zeros ofthe function. From his explicit formula he derived that anynon-trivial zero could not take a negative real value, thusany nontrivial s such that (s) = 0 must be in the interval 0 (s) 1.

The Riemann Hypothesis can now be stated in its original form: any nontrivial zero has (s) = . Riemann didnot attempt to give a proof for this statement in his paperand only based the conjecture on his calculations for thevalues of the rst few nontrivial zeros, but despite the numerous attempts of many brilliant mathematicians theproof is just as elusive today as it was in 1859. The rangeof possible values for (s) has been narrowed down somewhat over the years. Hardy proved that there exists an innite number of zeros on the critical (s) = line, and ithas been computationally checked to hold true for the rstten trillion zeros or so, but such results are rather inconclusive.

Since this way of stating the hypothesis is very abstract,a more easily underastandable equivalent statement can

be derived using the Liouville function. It is dened as

where is the number of prime factors of , which arenot necessarily distinct. In 1899, Edmund Landau showedthat the Riemann Hypothesis is equivalent to the statementthat for any xed > 0,

Looking at the numerator, this states that every integer hasan equal chance of having an either even or odd numberof prime factors. This is another way of stating the Riemann Hypothesis, and arguably the easiest to understandby intuition. It also gives some idea of the farreaching implications of the conjecture. The one presented here is onlyone of many signicant results in number theory whichcould be reached, if it were assumed that the Riemann Hypothesis was true.

Returning to the problem of counting primes, recall thatthe zeros of the zeta function are related to the oscillationsof prime numbers from their expected values. It has beenshown that, assuming the conjecture is true, the error termin the prime counting function dened by the prime number theorem can be very accurately stated. For any real ,

where O is the big-O notation. The error term isderived straight from the Riemann Hypothesis, and is amuch better approximation than any other that has beenmade so far. In fact, it has been proved to be the best possible approximation.

Finally, this is only a very shallow introduction to theRiemann Hypothesis. As for further reading, the book TheRiemann Hypothesis - For the Acionado and Virtuoso Alikeby Borwein, Choi, Rooney and Weirathmueller (2007) goesinto much more detail and shows how Riemann continuedthe zeta function step by step. It also contains many of themost important historical papers related to the hypothesisas appendices.

ReferencesBombieri, E. (2000), The Riemann Hypothesis, http://www.claymath.org/millennium/Riemann_Hypothesis/riemann.pdf

Borwein, P., Choi, S., Rooney, B. and Weirathmueller, A. (2007),The Riemann Hypothesis: A Resource for the Acionado and VirtuosoAlike, New York: Springer

Conrey, J.B. (2003), The Riemann Hypothesis, http://www.ams.noices/200303/feaconreyweb.pdf

Gourdon, X. (2004), The 1013 rst zeros of the Riemann Zeta function,and zeros computation at very large height, http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e131e24.pdf

Sarnak, P. (2008), Problems of the Millennium: The Riemann Hypoth-esis, http://www.claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf

( )sn1s

n 1

=g3

=

/s

s

,

R.

n p1

11

1 21

1 31

1 51

sn

sp prime

s s s

1

# # g

!

=

=

-

- - -

3

= =-

- - -

%/

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

( )m~ m

( ) ( ) ( )0.lim

n

n1 2n

21

f+ + +=

m m m"3 +f

x

( ) ( )log

logxt

dt O x xx

2

= +r #

logx x



Consider a man lets call himGuy who is attracted to Carol andhas the opportunity to talk to her, letssay at a coee bar. Realising thatCarol is shy he considers whether ornot to approach her. Guy considersthe possible outcomes:

(a) He talks to Carol and she re-sponds in a friendly manner. He getsher phone number and a proper datenext week.

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

(like reading the last issue of Plus).

(c) He talks to Carol and she provesuninterested. He will feel mis-evrable for a week.

Guy evaluates the outcomes assigning 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).

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

fact that the outcomes depend dramatically on the actions of other guysindependently considering whetherto approach Carol or not. Assume thatGuy thinks that he will get (a) only ifno one else approaches Carol and (c)when he is not the only one approaching Carol. Of course if he does not talkto Carol, he gets (b). Guys modestyseems understandable if he does nottrust himself as a strong candidate inthe face of competition.

The fact that his choices depend sostrongly on the choices of others is

My friend Carol is nice and beautiful. Anyone would bet she hasplenty of dates. But it turns out that this is not

the case.

The fact is that Carol hasdated noone for ages. And although she is shy by nature, she is also open to adequate proposals and would

love to nd someone special. But Carol claims that men donot usually approach her. She thinks she frightens them away.Is it just a matter of bad luck? Or is it something else? MaybeCarol has a distorted perception of reality?

Luck is an issue thats naturally addressed in mathema-tics. If it is a question of luck, mathematics may shedsome light on Carols problem.

The Syndrome

Carol

Words: JosManuel Rey (left)

Art: Gianni Peg (right)

Universidad Complutense, Madrid



what makes this an interactive decisionproblem. The study of how individualbehaviour is conditioned by the social environment is the objective ofsocial psychology. The interaction involved here is the trademark of gametheory, which was developed in thetwentieth century.

Maths about Carol

The solution of Guys problem,and that of all of the guys, is a bundleof actions deemed rational within thedescribed framework. The assumption that everyone behaves rationallymay seem slightly unrealistic, but itscrucial in the game theoretical approach, since mathematics cant account for someone behavingirrationally or against their own interest. The symmetry of the problem Guy may be anyone impliesthat all will act in the same way, sinceall will make the same rational considerations. Rationality permits todiscard the symmetric solution inwhich no one talks to Carol: giventhat none of the others talk to her,Guy gets a better outcome by approaching her, so rationality wouldimply that Guy does approach her. Asimilar reasoning permits the discarding of the solution in whicheveryone talks to Carol. There are no

more apparent symmetric solutions.

Guy needs to be subtle to proceed.He may think of solving his dilemmaby tossing a fair coin. This amountsto Guy being totally uncertain aboutwhat to do. If we admit that being

uncertain is a possible solution tohis dilemma, is the 5050 chance trulyrational, or would another level ofuncertainty, say 3070 in favour of approaching Carol, be better? Is thereeven a rational way to be uncertain?This seems an interesting idea. Let ussee where it takes us.

Guys uncertainty is described bythe probability p of approachingCarol (so 1 p is the probability of notapproaching her). By symmetry,everyones uncertainty is representedby the same number p everyone in

dependently makes his choice according to p. The goal is to nd thebest value p* of p.

The value p* can be obtained indirectly via the following observation.The bundle of common uncertaintiesdened by p* is rational only if Guy(and any other guy) associates thesame level of reward with his twopossible actions, approaching Carolor not approaching her, given that therest are all uncertain with probabilityp*. Otherwise there is no uncertaintyabout what to do: Guy would go forthe action with the higher reward.

But how can Guy assign a rewardvalue to a bundle of uncertainties?One possible answer, given by JohnVon Neumann and Oskar Morgenstern, is to average the reward valuesassigned to all possible bundles of actions with weights equal to theirprobability of occurrence. This iscalled expected valuation. Rationalitybased on expected valuations hasbeen the major paradigm for decisionmaking analysis since the 1950s.

If there are N guys deciding independently, then there are 2N possiblebundles of actions. Guy makes hiscalculation as follows:

(1) The bundle in which only our Guy

This surprising phenomenon

which we call theCarol syndrome is a byproduct of

psychological social interaction.



approaches Carol yields (a) withreward value a and occurs withprobability (1 p)N1, since all decisions are taken independently. Anybundle in which both Guy andsomeone else approach Carol yields(c) and has reward value 0. Thus theexpected value of approachingCarol is a(1 p)N1.

(2) Any bundle of actions in whichGuy does not talk to Carol has valueb whatever the others do. Thus theexpected value of not approachingCarol is clearly b.

Guy is uncertain about what todo when these two values coincide:

Solving for p he gets

Since a >p, p* lies between 0 and 1and therefore denes a proper probability.

The conclusion is that anyoneshould approach Carol with probability p* This is the rational solutionfor the Carol dilemma when she hasN identical admirers.

Since a >b for N not too large it ispossible that a >2N1b, in which casep* > and the odds are that Guywill talk to Carol. However, as soon

as N is large enough, the

possibility of no one approachingCarol becomes more likely. In fact,p* approaches 0 as N gets large, no

matter how stunning Carol may be.

Hence the following (un)veiledmechanism may be inhibiting Guyfrom approaching Carol: (1) Thelarger the number of Carols admirers the more probable it is that Guywill not talk to her; (2) The more attractive Carol is, the more likely it isthat there will be lots of guys considering this decision problem. Guyis thus led to believe thatN is largeand p* correspondingly small. Inconsequence he will very likelychoose curling up with Plus ratherthan risking Carols rejection.

The Carol syndrome

The important point of Carolsfeeling about frightening guys awayis not the probability p* but theprobability pnone of no one talking toher. Since all the guys act independently,

In this expression pnone increases asN increases. The entry of new potential dates adds to the probabilitythat Carol is left alone.

Also, asN gets large, pnone does not

vanish but tends to . Therefore,

pnone always lies between two values:

In particular, as long as a is notmuch larger than b, we have that

pnone > and the odds are that nobody will talk to Carol. This is trueirrespective of the number of guysand it becomes worse for Carol asthat number increases.

Carols perception that she scaresmen away is not a delusion after all.According to the mathematicsabove, she may be justied in thinking that guys stay away from her. Itis not a matter of bad luck but a collateral eect of interactive rationality. A paradoxical consequence isthat Carols attractiveness acts as arepellent. This surprising phenomenon which we call the Carol syndrome is a byproduct ofpsychological social interaction.

Scary evidence

The Carol syndrome is not a theoretical artifact. As striking as it mayappear, the syndrome is often reported by attractive women andmen. A web search brings up somenotorious cases. In an interview forThe Sunday Times in February 2008,American actress Uma Thurmanwas reported as saying that menrarely chatted her up. She considered her bad luck with men a lifelong curse. The same was true ofAmerican star singer Jessica Simpson who declared in a TV show inSeptember 2006: I scare guysaway. Another example, reportedin March 2009 by The Telegraph online, is that of 19year old British actress Emma Watson, who said thather starring character in Harry Potters saga, while bringing her worldwide fame, scared boys away.

Hence Carols luck is not odd, although it obviously diers fromcelebrities like Ms Thurman. Mathematics can explain how. The number of Ms Thurmans admirers is

huge so that the ceiling value is a

good approximation for pnone andthis somehow eases Thurmans sufferings. For Carol,N is not as large,

but the ratio and in turn pnone

is closer to 1. This is what causesCarols Carol syndrome.

2 >b

a1N-

(1 ) .p pb

anone

1N NN

= - =) -` j

a

b

< < .a

bp

a

b2none` j

a

b

a

b

(1 ) .a b b1N

- =-

1 .pa

b 11

N= -) -` j



The concept of innity has captured theimagination of mathematicians andphilosophers for centuries and will

doubtlessly continue to far into the future.This article will discuss the dierent types

of innity, actual and potential, and thepossibility that there are actually dierent

sizes of innity.

Fiona Madden

To Infinity andBeyond



Around 460 BC, Zeno of Elea wroteone of the rst texts about innity andalthough none of his texts have survived, his paradoxes are some of themost quoted on the subject of innity.Probably the most famous is the paradox of Achilles and the tortoise. HereAchilles is running a race against thetortoise and he gives the tortoise a 10metre head start. Say Achilles runs 10times faster than the tortoise. Then bythe time Achilles has run 10m, the tortoise will have moved a further 1mforward. Then when Achilles hasreached the point 1m ahead, the tortoise will be a further 1/10m aheadand so on ad innitum. So the tortoisemust win the race. Another paradoxalong the same lines is that of Achillesrunning a race. Before he can run thefull distance, he must run half the distance. But before he runs half the distance he must run a quarter of thatdistance then an eighth, a sixteenthand so on ad innitum. The paradoxcan then be modied so that Achillescan never actually start the race.

Any mathematician with a basicunderstanding of convergence andlimits can conclude for the series:

But is this really solving the puzzle orsimply ignoring the real problem bythrowing some calculus at it and bypassing the conceptual problem?

A possible solution is to think of asmallest distance, where the processof dividing a distance in half (or intenths) must come to an end. This ismuch the same as the ageold fact thatyou cannot fold a piece of paper morethan seven times (although apparently 12 folds have been achieved butthe idea remains the same). Conceptually, you should be able to keepfolding a piece of paper forever but inpractice this is not the case. Could thisbe the same for distances? Acceptingthis would mean that space is nolonger continuous but discrete andthe same, mutatis mutandis, for time.So the question now is: are the notionsof time and space truly continuousand, therefore, always divisible(sometimes called everywheredense) or are time and space discrete?

This now brings up the issue of po-tential innity and actual innity. Aristotle recognised the confusionsurrounding the concept of innity

and drew a distinction between actualand potential innity. In its simplestform potential innity is the idea thatsomething could go on forever,should enough eort has been applied, such as continuing a geometricline. Actual innity, on the other hand,is something that is completed anddenite, and consists of innitelymany elements. Actual and potentialinnity can be explained using theearlier paradoxes: You could actualisea point by stopping on it, or markingit, otherwise there remain potentialpoints. If you stopped and markedevery point, this would take an innite amount of time, but if you leftmost points as potential, unmarkedpoints, you would arrive at your destination without a problem. This follows Aristotles view that the actuallyinnite is impossible and potential innity is the only plausible idea of innity. Jules Henri Poincar agreedwith this: Actual innity does notexist. What we call innite is only theendless possibility of creating new objects no matter how many objects existalready.

This stance is very tempting anddoes seem to solve most of the problems related to innity but can we really accept that something can only bepotentially innite? Georg Cantorrmly disputed this and posed a verygood argument in favour of actual innity using cardinal numbers.

To introduce the concept of innitecardinals, it is useful to think of DavidHilberts hotel paradox. Suppose therewas a hotel with an innite number ofrooms. A large mathematics conference is taking place and the delegatesare to stay in this hotel. An inniteamount of delegates turn up and areaccommodated in the hotel starting inroom 1, then 2, 3, 4 and so on. Then atourist arrives at the hotel looking fora room. The hotel has an innite number of rooms currently occupied by aninnite number of people. The receptionist could take the tourist to theroom at the very end but that wouldinvolve walking an innitely long way.Instead, the receptionist asks all of thedelegates to move along one room sothat room 1 is now unoccupied andthe tourist can have this room. By thesame argument there would alwaysbe room for one more in this hotel and,

resultantly, there would always beroom for another innite number ofguests. Innite cardinals can absorbnite and even some innite cardinals without changing their own cardinality. This seems completelycounter intuitive though, so lets startwith the basics.

Cardinality simply means the sizeof a set, which is just how many elements it has. Cardinal numbers are anindication of how many so neitherorder nor magnitude has any eect oncardinality. This can be illustrated bya trivial example: Let A = {1, 1, 2}, B ={2, 1, 1}, C = {96, 1022, 3478}. Let us denote the cardinalities of A, B and C by#A, #B and #C respectively. Then #A =#B = #C = 3. Now let us consider theset of natural numbers and the setof natural number that are divisibleby 2 (i.e. even). If the cardinality of is innity then surely the cardinalityof the set of even numbers is half this.But then what is half of innity? Perhaps half of innity is also innity justas in Hilberts hotel. A fairly basic denition is that two sets are of the samesize if, and only if, they can be placedin a onetoone correspondence. Letthe set of even numbers be called . Itis a proper subset of but can beplaced in a onetoone correspondence with :

We could carry on pairing indenitelybecause both sets are innite, so and are the same size. This leads toRichard Dedekinds denition of aninnite set:

Therefore is innite. is also innite because a proper subset of , saythe set of even numbers larger than 2,can be placed in a onetoone correspondence with it. This set can also beproved to be innite and so on. Thisdenition distinguishes nite from innite cardinal numbers.

This idea is ne for natural numbers and can easily be extended for in

10 110

1

100

111

9

1 ."f+ + + +

N

N

E

N

N

1 2 3 4 5 6 7

2 4 6 8 10 12 14

N

E

f

f

N

E

EN

E

A set is innite if and only if itcan be placed in one-to-one cor-respondence with one if itsproper subsets


16 [The, Commuttator] Sept 2012 www.the-commutator.com

tegers, , but what about rationalnumbers, ? A rational number is oneof the form m/n where both m and nare natural numbers. For any two rational numbers, there will always be athird in between them. For exampletake 1/2 (= 6/12) and 1/3 (= 4/12): thereis 5/12 between them. Then take 1/3 (=8/24) and 5/12 (= 10/24): there is 9/24between them. And so it follows thatthere is an innite number of rationalnumbers between any two rationalnumbers. It now looks as if there arefar many more rational numbers thannatural numbers, but it is actually possible to place in a onetoone correspondence with ? How can anyonestart to write down all the rationalnumbers? Just by writing two rationals you have then faced an inniteamount in between. To avoid thisproblem, consider a tabular representation of the rational numbers shownin Fig. 1. The table can be extended toinnity in all directions, and zero hasbeen left out of the denominator forobvious reasons.

By following the spiral around, wecan list the rational numbers, leavingout any repetitions. Clearly, all rational numbers will be found eventually using this method. Then, byputting them in a list, can be placedin a onetoone correspondence with

as so:So and have the same cardinality.

This cardinality can be denoted bythe symbol 0 where is the capitalof the rst letter of the Hebrew alphabet, aleph, and 0 is simply zero. 0 isspoken of as aleph null or aleph naught.So now we can say that all innite setsthat have been discussed here ( , ,

, ), as well as any other innite setshave cardinality 0. Such sets are also

known as countably innite and it issets with a countably innite cardinality that can be absorbed by other innite sets, as referred to in Hilbertshotel paradox. Now, if a set can becountably innite, surely there mustbe such a thing as an uncountably innite set with a cardinality greaterthan 0. The real numbers, , are theset of all rational numbers and all irra

tional numbers. Irrational numbersare any numbers unable to be denoted

as m/n, such as 0.1211211121.... or (=3.1415926535897...). Irrational numbers have an innite, nonrepeatingexpansion, i.e. the numbers after thedecimal point do not form a patternthat is exactly repeated.

The set of real numbers is called thecontinuum as it is a smooth line withno gaps, whereas the rational numberline has gaps where irrational numbers go. Gregor Cantor used the ideaof diagonalisation to be the rst toprove that has cardinality greaterthan 0. The proof is simple but astonishing, and in fact Cantor himself said

I see it but I dont believe it when writing to a friend about his work. Theproof is a reductio ad absurdum argument where we rst suppose that isthe same size as , show that thisleads to a contradiction, conclude thatthe two sets are of dierent sizes, andthen nally conclude that musthave greater cardinality.

Proof:Suppose and are the same size.

Place the real numbers in a onetoonecorrespondence with the naturals,start with just the reals from 0 to 1,each with an innite expansion(though often trivial, such as 0.2000...).Listing them in an order using ax,ywhere x is the ordering and y is thedigit itself:

So if the third number in the list is0.38594..., then a3,2 refers to 8, a3,4 refersto 9 and so on. Consider a numbermade 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

bers, it should appear somewhere onthe list. Now for each digit in D add 1

to it, unless it is 9in which case turnit to a 1. Call thisnumber CantorsDiagonal. However, Cantors Diagonal will notturn up on the listabove because it isdierent to therst number onthe list by at least

the rst decimal, dierent to the second by at least the second decimaland so on. This means that all the realnumbers between 0 and 1 cannot beput in a onetoone correspondencewith the natural numbers which contradicts the original supposition. Sothe set of real numbers from 0 to 1 isnot of the same size as the naturalnumbers, mutatis mutandis, for thewhole set of real numbers.Since is a proper subset of ,must have larger cardinality. (In facteven (0,1) has greater cardinalitythan .)

So it has been proven that there are,in fact, dierent sizes of innity andthe existence of the continuum makesthe concept of actual innity possible.This, in turn, contradicts the earlieridea that space and time are discreteand there is some smallest value ofthem. Whichever view seems moresensible Aristotle sums up the discussion of innity perfectly:

References

Friend, M. (2007) Introducing Philosophy ofMathematics. Stockseld: Acumen Publishing.

Russell, B. (1956) The Principles of Mathe-matics. London: George Allen & UnwinLtd

Meschkowski, H. (1965) Evolution of Math-ematical Thought. California: HoldenDayInc. (translated by Jane H. Gayl)

Q

N

N

Q

Q

Q

E

N

N

Z

R

R

N

R

R N

1 0

2 0

3 0

a a a a a

a a a a a

a a a a a

.

.

.

, , , , ,

, , , , ,

, , , , ,

1 1 1 2 1 3 1 4 1 5

2 1 2 2 2 3 2 4 2 5

3 1 3 2 3 3 3 4 3 5

f

f

f

h

R R

R

N

N

R

Q

Z

The fundamental diculty withthe theory of the innite is this: ei-ther an outright denial or an out-right acknowledgement of thebeing of the innite leads to manyimpossibilities.

Figure 1. Spiral

1 2 3 4 5 6 7

0 1 1 2 2

N

Q 1 2 1 2

f

f- - -



A famous puzzle asks about the minimum number of people in a group so that the probability thatat least two of them have the same birthday isgrater than .

In order to nd a solution, we begin by statingsome assumptions. First, we assume that the birthdays of the people in the group are independent,and that each birthday is equally likely out of 366days in a year, which is our second assumption.

The probability that at least two in group of npeople have the same birthday is 1 pn, where pn isthe probability that all of the people in the grouphave birthdays on dierent days. To calculate thisprobability, we should consider the n people insome xed order, as a sequence of elements. Whatis important for us is to compute how probable itis for the kth person to have a dierent birthday thanthe rst k1 people in this sequence. For instance,the birthday of the rst person certainly does notmatch any other birthday of the people before himin the sequence, because k1, when k=1, equals 0.The probability that the birthday of the second person is dierent that the birthdays of the people before him in the sequence, namely just the rstperson, is 365/366. We can generalise this to

which indicates the probability that the kth personin the sequence has a dierent birthday than thepreceding k1 people, where 2 k 366. Now wecan conclude for pn that:

Then, for 1 pn we have:

Thus, with the aid of a calculator (or a computerprogram), we conclude that the minimum numberof people needed, so that the probability that twohave the same birthday is greater than , is 23(with 1 pn = 0.506323).

One of the Birthday problem applications is nding the probability of a collision in a hashing function, which is the mapping of the keys (of elementsthat are to be stored in a table or database) to storage locations.

References

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

366

366 ( 1)

366

367p

k k ,k =- -

=-

366

365

366

364

366

363

366

367p

n .n f=-

1366

365

366

364

366

363

366

3671 p

n .n f= --

-

The Birthday Problem

NumberCuriosities

How can you determineif a given number is apower of 2?

Convert the number tobase2 numeral system. Ifits binary representationconsists of only one 1, followed by k zeros,then thisnumber is 2 to the powerof k.

This can be generalisedfor determining if a givennumber is a power of n,by converting the number to basen numeralsystem and checking ifthe number of 1s is one.

?A

How can you say with certaintyif a given number is a perfectsquare (without applying thesquare root function)?

By counting the number of itsfactors. All square numbershave an odd number of factors.

?

A55 + 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

Consider the digits from 1to 9, and from 0 to 9.Arrange them in descending order, reverse and subtract. The same nine digitsreappear in the answer:

987654321 123456789

864197532

9876543210 0123456789

9753086421

Hristo Georgiev

Hristo Georgiev



Cellular Automata andConways Game of Life

The subject of Cellular automata wasborn in the early 1950s, with John vonNeumann and Stanislaw Ulam pioneering the eld. By denition, cellular automata are mathematical idealisations ofphysical systems in which both spaceand time are discrete, and physical quantities take on a nite set of discrete values. A cellular automaton consists of anndimensional array of variables andevolves in discrete time steps, with thevalue at each position in the array beingaected by the values of variables at locations in its neighbourhood on the previous time step.

For twodimensional cellular automata, there exist two ways of deningthe relative location of a given cell.

Von Neumann neighbourhood is namedafter John von Neumann, the creator ofthe rst selfreplicating cellular automata. This neighbourhood is diamondshaped and it represents the set of allcells that are orthogonallyadjacent to theregion of interest, which in the simplestcase is the single cell (x0,y0). The vonNeumann neighbourhood of range r isdened by:

Von Neumann neighbourhoods ofranges r=0, 1, 2, 2 and 3 are illustrated inFig. 1. Usually, if no range is speciedwhen a von Neumann neighbourhood isbeing referred to, it is assumed that thegiven case is r=1, by default. The number

{( , ):| | | | }.x y x x y yN r0 0( , )x y0 0 #= - + -o

Hristo Georgiev



I expect the children of 50 yearsfrom now will learn cellular au-tomata before they learn algebra.

Stephen Wolfram,New Scientist, Nov 18, 2006

of cells in a von Neumann neighbourhood of range r is given by the generalterm r2 + (r + 1)2 and the sequence theyform has been recorded in Sloane's OnlineEncyclopaedia of Integer Sequences underthe identier A001844:

1, 5, 13, 25, 41, 61, 85, ...

Moore neighbourhood is named after Edward F. Moore and is a squareshapedneighbourhood that, in the simplest case,comprises the eight cells surrounding acentral cell (x0,y0) on a twodimensionalsquare lattice. The Moore neighbourhoodof range r is dened by:

Moore neighbourhoods of ranges r=0, 1,2, and 3 are illustrated in Fig. 2. As withthe von Neumann neighbourhood, if norange is specied when a Moore neighbourhood is being referred to, it is assumed that the case in consideration isthe default one, i.e. r=1. The number ofcells in a Moore neighbourhood of ranger is the odd square (2r+1)2, and the sequence they form has been given aSloane entry A016754:

1, 9, 25, 49, 81, 121, 169, ...

This neighbourhood type is used inimage editing software in tools such asthe edge nder and magic wand whichare concerned with the proper management and allocation of the boundary andedges of a digital image.

{( , ):| | | | }.x y x x r, y y rN 0 0( , )M

x y0 0 # #= - -

For each cellular automaton, there isa set of rules that dene its evolution. Arule can be viewed as a mapping fromthe space of possible states of the cellular automaton to itself. In the case ofonedimensional cellular automata, thestate space of an automaton with an innite number of cells and its rules correspond to a Cantor set and a contiguousmapping of this Cantor set to itself, respectively.

The three fundamental features of acellular automaton are: uniformity: all cell states are updated bythe same set of rules synchronicity: all cell states are updatedsimultaneously locality: the rules are local in nature

Automata are characterised by havingstates and each cell can exist, at dierenttime steps, in two or more states. In regard to shape, square cells are mostcommon, whereas other shapes, such astriangle and hexagon have also been experimented with but have not gainedmuch popularity.

The cells, of which a given cellular automata consists, all change their ownstates at the same time, according to thestates of their neighbours in the lattice inwhich they are situated.

Albert Einstein once said that Theonly reason for time is so that everythingdoes not happen at once. Consequently, inorder to evolve, our cells need a set oftransition rules as well as an increase intime t: 0, 1, 2, 3, , where each step(from 0 to 1, 1 to 2, 2 to 3, ) can be considered as a clock tick.

The simplest cellular automata havetwo states, which are commonly denoted as 0 and 1, or on and o. In onedimensional space, the number ofneighbour cells, on the left and on theright, that aect a cells state (after everyclock tick) determines the radius of thecellular automaton. In the simplest case,the state of each cell at time t+1 dependson and can be aected only by its stateand the state of its two immediateneighbours at time t, and thus we saythis cellular automaton has a radius of1.

OneDimensional CellularAutomata

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

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


1936 Alan Turingpresented his seminal paper on computable numbers andthe Turing machine.

1948 At the Hixon symposium in Pasadena,John von Neumann addressed the followingquestion:Can one build an aggregate out of such elements insuch a manner that if it is put in reservoir, in whichoat all these elements, each of which will at the endturn out to be another automaton exactly like theoriginal one?

1952 Von Neumann formulated a kinematicmodel of reproduction. He pictured the aggregate as a robot, in a lake, collecting components as it jostled against them andassembling them into a copy of itself. Amongthe components were kinematic elements articial arms that could be used to moveother objects. John von Neumann decidedthat his automaton would have to consist oftwo parts: one exible construction unit beingable to build things out of the elements storedin the warehouse, given their proper specications; and one instruction unit telling themachine how to construct a copy of itself.Both of these units corresponded to the duality between computer and program and, ofcourse, to the duality between cell andgenome. Von Neumann decided to use a uni-versal computer in order to realise his automaton, which consisted of 29states and waspresented in 1952. It would automaticallymake a copy of its initial conguration ofcells.


With the simple rules presented above, we can now seethat there are eight (23 = 8) possible congurations of aneighbourhood of three cells (which is the maximum scopeof 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 ci-1(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), andits neighbours will have states ci-1(t+1) and ci+1(t+1) respectively. Thus, the dependence of a cells state can be expressed by the relation:

ci(t+1) = [ci-1(t), ci(t), ci+1(t)],

where is the local transition function. For example, consider the simple transition rule:

ci(t+1) = [ci-1(t) + ci(t) + ci+1(t)] mod 2.

We can represent this rule as a transition table for the eightpossible initial values (Fig. 4).

Since there are eight possible neighbourhoodstates ofthree cells and each of these results in two possible stateoutcomes for the middle cell, there are 28 = 256 possibletransition rules. This has been described by Stephen Wolfram as elementary cellular automata. He identied each ruleby its rulestring, which is formed by the values of the eightoutput states. Then each rule is given a unique decimalidentier.

The rulestring corresponding to the example we considered above is 10010110, which is the binary representationof the decimal number 150. Thus, the rule is referred to asRule 150 and its evolution can be visualised in a twodimensional form, as a plot in which the ith row records theconguration of the automaton at step i (Fig. 5).

Other onedimensional rules that we are going to present here are Rule 90 and Rule 110 (Fig. 6 and Fig. 7). Thefractal produced by Rule 90 was described by Sierpiski in1915 and is known today as the Sierpiski sieve, Sierpiskigasket, or Sierpiski triangle. It rst appeared in Italian artin the 13th century. In Rule 90, each cell is the exclusive-or(XOR) of its two neighbours. This is equivalent to modulo-2 addition and generates the modulo2 version of Pascalstriangle, which is a discrete version of the Sierpiski trian

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

Figure 3. The eight possible neighbourhoodstates, and the cell states of the central cell ciand its two immediate neighbours ci-1 and ci+1 attime step t.

gle. The number of live cells in the ith row of this pattern is 2k, where k is the number of nonzero digits inthe binary representation of the number i. The sequence of these numbers of live cells is known asGoulds sequence or Dresss sequence, and is given aSloane identier A001316:

1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, ...

The rule 110 cellular automaton is universal, whichwas rst conjectured by Stephen Wolfram in 1986,and subsequently proven by Wolfram and his assistant Matthew Cook. The evolution of Rule 110 for aspecic initial condition is depicted on the cover ofWolframs A New Kind of Science.


1957 Lionel Penrose designed an articial selfreproducing object. In hismodel, the environmentcontained only two kindsof solid objects whichserved as units. In certainrestricted circumstances,two dissimilar unitscould be linked togethermechanically and form acopy of a structure which(was had already beendened) consisted of thesame two objects.

early 1960s Stanislaw Ulam and coworkers J. Holladay and RobertSchrandt at the Los Alamos ScienticLaboratory (the sprawlinglargely military research centre in New Mexico) began using computing machines to investigate various twodimensional cellular automata. An innite plane was considered and divided up intoidentical squares, while the transition rules, using a von Neumannneighbourhood, were eclectic and the results were mostly empirical.

1966 During the 1950s, vonNeumann worked on aproof that a machine can reproduce, searched for patterns in the digits of pi, andbusied himself with morearcane mathematics.

He did not nish his proof.The manuscript trailed ointo a series of notes aboutthe remainder of the proof,written at various timeswhen other projects did notinterfere. His student andcolleague Arthur W. Burkscompleted the proof in vonNeumanns spirit and had itpublished as Theory of Self-Reproducing Automata(Urbana and London: University of Illinois Press,1966).

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

1970 John Horton Conway created his Game of Life (2 states overa Moore neighbourhood).

Instead of using a chessboard for testinghis game, Conway turned his large entrance hallway into a nite twodimensional grid where he placed dishes toindicate the cells which were alive. Hequickly became tired of this because heliterally began to step on the pieces.

Richard K. Guy later recalled:... only after the rejection of many pat

terns, triangular and hexagonal latticesas well as square ones, and of manyother laws of birth and death, includingthe introduction of two and even threesexes. Acres of squared paper were covered, and he and his admiring entourageof graduate students shued pokerchips, foreign coins, cowrie shells, Gostones or whatever came to hand, untilthere was a viable balance between lifeand death.

How The Game Of Life WasBorn


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

ci-1(t) ci-1(t) ci+1(t) ci(t+1)1 1 1 11 1 0 01 0 1 01 0 0 10 1 1 00 1 0 10 0 1 10 0 0 0

Figure 5. The evolution of Rule 150 represented in a twodimensional form

Figure 6. The fractal produced by Rule 90 wasdescribed by Sierpiski in 1915 and is knowntoday as the Sierpiski sieve, Sierpiski gasket,or Sierpiski triangle.

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


A big boost to the popularisation of the subject of cellularautomata came from the famous British mathematicianJohn Horton Conway at the Gonville and Caius College,University of Cambridge and his highly addictive Gameof Life. It was invented over a period of two years, ocially presented in 1970, and widely popularisedthrough Martin Gardners monthly column Mathemat-ical Recreations in Scientic American. It quickly became extremely popular among computerenthusiasts. Programmers at numerous computercentres set out to explore the game with machines attheir disposal. Many other scientists and mathematicians found pencil and paper adequate to obtain signicant results. The game is quite simple tounderstand, easy to program, fun to play, and quicklygenerates a feeling of being a cocreator of the universe of possibilities that unfolds in such a cellular automaton. One can play God in ones own universe.

In this section, we are going to have a look at somebasic but fundamental types Life forms: still lives, os-cillators and spaceships.

A still life is a pattern that does not change its shapeand position over time. Still lives can also be considered as oscillators with a period of 1. Some examplesof still lives are shown below. From left to right, theyare called: block, beehive, loaf, and boat:

An oscillator is a pattern that returns to its originalstate, in the same orientation and position, after a nite number of generations (or steps). Thus the evolution of such a pattern repeats itself indenitely.Depending on context, the term may also includespaceships. The smallest number of generations ittakes before the pattern returns to its initial conditionis called the period of the oscillator. An oscillator witha period of 1 is usually called a still life, as such a pattern never changes. Some examples of oscillators include:blinker, toad, and beacon:

A spaceship is a nite pattern which reappears after acertain number of generations (or steps) in the sameorientation but in a dierent position.Here, we present two types of spaceships:glider (top row), andlightweight spaceship (bottom row):

Zoo of Life Forms

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, andphone numbers unerringly.

At the age of twenty, von Neumann published a formal denition of ordinal numbers. He later noticed a startling connection between quantum physics and vectortheory. He discovered that the states of quantum systemscould be represented by vectors in an abstract, innitedimensional space.

In 1931 von Neumann became professor of mathematicsat Princeton University, and in 1933 he was appointed to thenascent Institute for Advanced Study.

Von Neumann was famous for working in oddplaces at odd times in taxis, during nightclub oorshows, or while waiting for breakfast. He would evenslip away to his study during parties.

Through the 1930s and early 1940s, vonNeumann worked on game theory, and from 1940

on, he was in demand as a consultant to industryand 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 supervised the design of

u n p r e c e


John von Neumann and the Turing machine

Conways Game of Life


Sucient interest existed to maintain a quarterly newsletter, called Lifeline, published by Robert T. Wainwright foralmost three years, from 1971 to 1973.

One can speculate why digital life is so fascinating.If the dream of lifes creation lies within the cellularautomaton, what is this dream made of? Perhaps itarises from a Pygmalion desire. An ancient Greek legend tells of a Cypriot king named Pygmalion. He fellin love with a statue of a beautiful woman perhapsthe goddess of love herself whom he himself hadsculpted. Pygmalion became so enamoured of hiswork that he embraced it. He begged and pleadedwith Aphrodite for a wife of the same appearance,and at last she took pity on him. She answered hisprayer by making the statue come to life.

Anything that wecan describe [in our

own world] can happenin the Life world

Bill Gosper

The Glider gun was devised by a younggroup of enthusiasts from MIT, with Bill Gosper,

arguably the most verbally profound in the group,as one of its leading gures. The hackers wouldspend all night sitting at the PDP6s (the computerthey used at the MIT in the 1970s) display, tryingdierent patterns logging each discovery theymade in this articial universe in a large blacksketchbook.

What resulted in was a carefully orchestrated collision of 13 Gliders which was able to produce thisGun.

One interesting fact about Bill Gosper and his

MIT friends is that they created a robot arm whichwas able to catch a PingPong ball lobbed toward it,by moving itself in the necessary position.

As for Gospers love for the Game of Life, a friendof his later recalled:Gosper sort of imagined the world as being madeout of all these little pieces, each of which is a littlemachine which is a little independent local state.And [each state] would talk to its neighbours. ToGosper, Conways simulation was a form of geneticcreation, without the vile secretions and emotionalcomplications associated with the real worlds version of making new life.

Later, in the 1970s, Bill Gosper moved to California to study and help the preparation of the 2nd volume of the seminal work The Art of ComputerProgramming by Professor Donald Knuth (who gavea BCS and IET Turing Lecture at the University ofGlasgow on 10th February 2011) at Stanford.

Gosper has also worked as a consultant for XeroxPARC (the inventors of the windowed graphicaluser interface and the mouse device) and WolframResearch (whose founder and CEO is Stephen Wolfram one of the other great contributors to the eldof Cellular Automata).

Bill Gosper and the MIT hackers

powerful computers for the Americanmilitary during World War II and after.While wrestling with practical problems, hebecame interested in the potential abilities of automatons. He was particularly impressed with thework of British mathematician Alan Turing. whohad showed that a relatively simple computer canperform any possible calculation, given the right programming. This computer was the so called Turing ma-chine.

Although usually intended as a mathematical abstraction, a Turing machine could be designed and built fromusual electrical components. The only catch was that it hadto have an innite memory capacity or what amounted tothe same thing.It had to be able to add on to its memory asneeded. This innitely extensible memory would be outsidethe main body of the computer, thus its being called externalmemory.

Turing pictured his universal computer hovering overan innite strip of paper tape marked o into squares,each containing 0 or 1. Turing believed that any type ofidea, object, action or computer program that can be expressed in language can be coded by a string of 0s and1s on the tape of this machine, which of course, rsthad to be given a complete set of instructions on howto perform the desired calculation. Likewise, thecomputer could erase and print its own 0s and 1sto keep track of intermediate results of calculation, or in other words, use the tape as scratchpaper. In long computations, the Turingmachine might as well use lightyears oftape. At the end of a calculation theanswer would be expressed, inthe same manner as the inputhad been given, as acoded string of 0sand 1s.


Figure. 8 Bill Gospers Glider gun


It is possible, given a large enough Lifespace, initially in a random state, thatafter a long time, intelligent self-reproduc-ing animals will emerge and populatesome part of the space.

John Horton Conway

The Game of Life is a twodimensional zeroplayer game, meaning thatany interaction with it is expressed bysetting up an initial conguration (orstate) and observing how it evolves intime, according to the predened setof rules. Conways automaton isplayed on an innite matrix of cellswhere each cell can be in one (at atime) of two possible states: alive anddead, or simply on and o. Givenan initial conguration of living cells,the following rules, which depend ofthe number of its Moore neighbours atgiven time t, are applied resulting in asequence of patterns:

Birth: A cell that is dead at time t be-comes alive at time t+1 if exactlythree of its neighbours are alive attime t.

Death by overcrowding: A cell that isalive at time t will die at time t+1 iffour or more of its neighbours arelive neighbours at time t.

Death by exposure: A cell that is aliveat time t will die at time t+1 if it hasone or no live neighbours at time t.

Survival: A cell that is alive at time twill remain alive at time t+1 only if ithas either two or three live neigh-bours at time t.

Formally, these rules can be summarised by rst dening the set M ofcoordinates of all Moore neighboursof 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 cells state inany twodimensional cellular automaton can be expressed by the relation:

c(i,j)(t+1) = [M(i,j)]

In particular, for the Conways Game of Life, the rules can be formalised as:

where

Both John von Neumann and Coddprovided their articial worlds withlaws intended to facilitate self-reproduc-tion, whereas in contrast to this, thesimple laws of Life were not at all tailored to the task of permitting patternsto propagate. Among all possibleworlds of cellular automata, Life wasmuch more plausible than John vonNeumanns or Codds artefacts.

The Lifeuniverse itself is designedas an automaton: a cellular automatondened by its transition rules.

A sucient number of artefactswere found to enable Conway todemonstrate a universal constructor,which has been one of the principalgoals of the theory ever since John vonNeumann became interested in automatic factories. It has been shown thatany of the wide range of computationsthat can be performed by practicalcomputers can also be done by cellularautomata, and that it is theoreticallypossible to implement von Neumannscomplicated selfreproduction athigher levels in Life using various patterns such as the Glider gun (Fig. 8).

This Turing machine then can implement anything that is at all com-putable. This means that anything thatwe can describe can happen in the Life

world. For instance, Adam P. Gouchercreated patterns in the Game of Lifethat can calculate the decimal digits ofphi and pi.

The rst Life design for an Universal Turing Machine was completedand realised by Paul Rendellin February 2010 (link [2]). A demonstration ofits action can be viewed through theQR code and link in Fig. 9. An alternative design for a universal computingmachine is Marvin Minskys registermachine, which stores arbitrary largenumbers by pushing blocks intoempty space. A design for the registerswas constructed in Conways Game ofLife by Dean Hickinson in 1990 (link[3]). Later, it was used by Paul Chapman to implement a complete registermachine that demonstrates universalcapability in 2002 (link [4]).

For most cellular automata, thereare congurations (states) that are unreachable: no state will produce themby the application of the evolutionrules. These states are called Gardensof Eden, because they can only appearas initial states. As an example, consider a trivial set of rules that evolveevery cell into 0. The onedimensionalRule 0 exhibits this functionality(Fig.10). For this automaton, any state withnonzero cells is a Garden of Eden.

Considering the Life universe, Marn Heule, Christiaan Hartman, KeesKwekkeboom and Alain Noels fromthe DelftUniversity of Technologysystematically searched the entirespace of 10by10 patterns with fourfold rotational symmetry, nding aGarden of Eden with 92 specied cells


Figure 9. Link to the Glider gun and UTMdemonstrations:

http://thecommutator.com/?p=51

Figure 10. Rule 0 is the simplest possible Garden of Eden. Its evolution depicts the trivial set

of rules that evolve every cell into 0.

Conways Rules

Gardens of Eden


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

Numerous variants have been considered over the years but none has everhad the same success or has been accepted as broadly as Conways Life.The apparent, vast number of choicesof rules, 229 to be accurate, is rather illusory. The rules not permitting rotational and reective symmetry, or theones not providing a quiescent state (astate such that if all cells are in that

state at a given time t, will remain inthe same state at time t+1 as well) arenot acceptable. Given restrictions such

as totality, thousands of millions ofcandidates remain still many, but notentirely impossible to sort through.Some promising threedimensionalanalogues of Life have been found,and although possible, working withhigher dimensions is very slowly computable, clumsier and dicult tograsp, since the front surfaces of theobjects prevent everything that is happening in the background to be seenin its entirety. The regular twodimensional Life has been arguablyfound sucient to allow for the discovery of undreamed.

The surprising power of recursive rules is best illustrated witha few specic cases presenting different approaches that have beenexamined throughout the years.The simplest recursive rule allowing growth is: any cell touching anoccupied cell experiences a birth inthe next generation. Then even asingle occupied cell acts as a seedfor unlimited growth. All its neighbours become occupied in generation 1, forming a 3by3 square,

leading itself to a 5by5 square, thena 7by7 square, then a 9by9 square

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


Figure 11. The smallest known Garden ofEden in the Life universe, found at the TU

DelftFigure 12. Rule 30 is known to have many of

the properties desirable for practical cryptography

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

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

Multidimensional Life andOther Variations


Edward Fredkins universe

26 [The, Commutator] Sept 2012

and therefore not interesting enough.

Hungarianborn mathematicianStanislaw Ulam found ways to eliminate some of the wouldbe occupiedcells so that the pattern acquires richerstructure. It grows in complexity aswell as size, producing an evergrowing coral reef from a starting patternof a single occupied cell and may beeven played in three dimensions.

One other approach was developedin the early 1960s by Edward Fredkin.

His variation uses two states, on ando, just like Conways Life. Only thefour orthogonally connected neighbours count, i.e. Fredkins variationoperates over a von Neumann neighbourhood. A cell will be on in the nextgeneration if, and only if, one or three(an odd number) of its neighbours areon presently. Otherwise, if zero or two(an even number) of its neighbours areon, the cell will be o. A specic property of this variation is that no patternof live cells can ever die away as anyconguration whatsoever becomes

four copies of itself. The quadruplingtime is a power of 2 and varies withthe size and complexity of the initialpattern. Another important specicityof Fredkins game is that there is noway of working out which was theoriginal pattern from a given furtherstate of the same pattern. On the negative side, the type of selfreproduction is trivial because every patternreproduces. Therefore, it is not a special feature of a patterns organisationthat allows reproduction.

Edward Fredkin became wealthy running a computercompany that specialises in imageprocessing. It wasperhaps there that he learnt how to view reality as bitsof information. At the age of thirtyfour, without evenhaving a bachelors degree, Fredkin became a full professor at the Massachusetts Institute of Technology.More recently, he has been a Distinguished Career Professor at Carnegie Mellon University, at Boston University and a Visiting Professor at MIT.

There is a notion that the laws of physics are discreteand the universe evolves as the result of deterministiccomputation, such as a cellular automaton, and it datesback to pioneering German computer engineer KonradZuse. His proposal, referred to as Zuses thesis, has ledto the development of the eld of digital physics. Oneinteresting consequence of Zuses thesis is that entropydoes not increase. Zuse is also credited for developingand building the rst fully functional programcontrolled digital computer, the Z3, in 1941 (in Berlin). Thethesis was elaborated upon by Edward Fredkin, whoproposed the related notion of nite Nature which isthe assumption that, at some scale, space and time arediscrete and the number of possible states of every nitevolume of spacetime is nite.

Fredkin and others have suggested that spacetime itself is granular composed of discrete units and thatour entire universe is a cellular automaton run by anenormous computer, and what we call motion is onlysimulated motion. According to his idea, a given cellularautomaton can imitate life and it will depict reality moreand more precisely when applied to smaller bits of matter. A moving particle at the ultimate microlevel may beessentially the same as the one of Conways Life gliders,appearing to move on the macrolevel, whereas actuallythere is only an alteration of states of basic spacetimecells in obedience to transition rules that have yet to bediscovered. An electron is nothing more than a patternof information, and an electron in a path is a pattern that

is moving. At the most fundamental level the automatonwill describe the physical worlds forms of movementwith perfect precision because at that level the universeis a cellular automaton in three dimensions. Fredkin be

The Los Alamos National Laboratory was established in 1943 and one of the pioneers there, John vonNeumann recruited Ulam from Princeton. The rstcomputers at the laboratory were enormously huge,unlike their todays personal desktop and mobilecousins. Ulam posed a very interesting problem forLos Alamoss computers, which at the time were usedmainly for mathematical computations:

Think of any positive whole number. If it is even, divideit by two. If it is odd, triple it and add one. Keep applyingthis same rule over and over. What happens to the number?

The histories of the initial values are surprisinglyunpredictable. Say you choose 10. Ten is even, so youhalve it and get 5. Five is odd, so you triple it and addone to get 16. Sixteen is halved to 8, then 4, then 2, then1. One is again odd, so it is tripled and added to oneto get 4. Then number enters the endless loop 421421421

Formally, in modular arithmetic notation, we denethe function f as follows:

We now form a sequence by performing this operationrepeatedly, beginning with any positive integer, andtaking the result at each step as the input at the next:

( )/2 0 (mod2)

3 1 1 (mod2)f n

n if n

n if n.

/

/=

+'

0

) 0.a

n for i

f(a ifor1i

i 2=

=

-

'

Stanislaw Ulam and the 3N+1 Problem


lieves that this giant lattice, which we all refer to simplyas the world, consists of logical units composed of fundamental grains, each computing at a completely locallevel its state in the next picosecond as a function of the

states of its neighbouring cells. The informationprocessed at this basic level is the factory for our reality.Yet if the universe is indeed a computer, and the physicalnatural laws are its software, then we will always be cuto from knowing what the hardware is like. Turing certainly showed that computers are equivalent to universalmachines, so that a program does not in any essentialsense depend on being realised by a single specic kindof hardware. Since in Fredkins universe we ourselveswould already be virtual metaentities in such a cosmicprogram, we could never know anything about the primary machine. We could hope that it never crashed, forwe would go down with it.

The Fredkin Prize

In 1980, Carnegie Mellon University announced theestablishment of a $100,000 prize (by Edward Fredkinhimself), called Fredkin Prize, for the rst computer prorgram to become World Chess Champion and the beginning of annual computer versus human competition. Theprize was threetiered:1) The rst award of $5,000 was given to Ken Thompsonand Joe Condon from Bell Laboratories, who in 1981 developed the rst chess machine to achieve master status.

2) Seven years later, the intermediate prize of $10,000 forthe rst chess machine to reach international master status was awarded in 1989 to ve Carnegie Mellon graduate students who built the chessplaying computer DeepThought, the precursor to Deep Blue, at the university.

3) The $100,000 third tier of the prize was awarded to theIBM team, who built the rst computer chess machinethat beat a world chess champion at the Fourteenth Na-tional Conference on Articial Intelligence (AAAI97, heldfrom 27th July to 31st July 1997, at the convention centrein Providence, Rhode Island).

Sept 2012 [The, Commutator] 27

Of all simple modications of Lifesrules, the most widely played is a version called 3-4 Life, whose name comesfrom the MIT Life group. Perhaps 34Life was one of the variations Conwayhimself was experimenting with before settling on the version of Life thatwe know today.

As for 34 Lifes rules, they are evensimpler than the conventional Lifesones: a cell will be on in the next timestep if, and only if, it has three or fourMoore neighbours that are on now,

without considering its own presentstate.

Most of what is known about theuniverse of 34 Life has been discovered by computer hacking, just as withConways Life. In this article, as it usedto be some 30 or 40 years ago, the social group which is referred to as hackers is the one that collectivelydescribes those computer programmers who regard crafting beautifulcode as the most important thing inthe world, and use their wizardry tosolve dicult and fascinating prob

lems in, usually but not only restrictedto, maths and logic. Unfortunately,nowadays the term hacker is immediately associated with an antisocialcomputer geek, often using his extraordinary programming powers towrite malicious programs and hackentire networks, websites or computers.

Back to our 34 Life variation, whatturns out to be a crucial dierence isthat it is more progrowth, and thus itfails to match the vast richness of con

The smallest i such that ai = 1 is called the total stoppingtime of n. The conjecture asserts that every n has awelldened total stopping time. For instance, thestopping time of 5 is 5 because:

5 16 8 4 2 1

Ulam found that every number he tested eventuallyentered the 421 cycle. Of course, some numbers takelonger. For instance, twentyseven takes 109 steps, atone point reaching a maximum value of 9232. The interesting fact is that no one has ever found a numberthat does not enter the 421 cycle. By the same token,no one has ever been able to prove that all numbersindeed enter this very same cycle. The problem, initially posed in the 1930s, is known today as Collatzproblem, Collatz conjecture, Kakutanis problem, Ulamsproblem, Syracuse problem, Thwaites conjecture and the3N+1 problem and remains unsettled. If there are anynumbers that do not enter the loop, they must be verylarge. All numbers up to 1016 have been tested andfound to enter the loop.

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

In May 2011, Gerhard Opfer of the Hamburg University released a paper claiming that the Collatz conjecture is true (link [1]). Eventually some aws in theproof were found and he corrected his statement.



ventional Life. Very small patterns are more likely to survive under Conways rules than under those of 34 Life.In the latter, any big random pattern seems to grow forever, whereas in regular Life unlimited growth is possiblebut only with ingenious constructions such as the Gun,Stabilised switch engine, or Breeder.

Cellular automata have applications in many diversebranches of science, such as biology, chemistry, physicsand astronomy.

Cellular automation models can be used for modellingof trac ow, where the road is discretised into cells, eacheither being empty or containing a car moving with agiven speed. As the cars interact, collective phenomenasuch as trac jams can be modelled. The Nagel-Schreck-enberg model, Biham-Middleton-Levine model and Rule 184are examples of such models.

Another onedimensional cellular automaton of specialinterest is Rule 30 because it is chaotic, with central column 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 generatorfor large integers in the Mathematica software(developedby Wolfram Research). Moreover, Rule 30 is known tohave many of the properties desirable for practical cryptography. It d

Date post:	28-Mar-2016
Category:	Documents
Upload:	the-commutator
View:	241 times
Download:	11 times

[The, Commutator] vol.3 issue 1

Documents