[The, Commutator] Vol.1 Issue1

Fill in the boxeswith the inte-gers 1,2,...,9 sothat each integeronly occursonce in eachstring and theadd up to thenumber adjoin-ing that string

Try and fill each row andeach column with the integers1,2,3 and 4 exactly once sothat the numbers in each nestcan be combined under thedesignated product to equalthe prescribed number.

Then try it with the inte-gers 1,2,...,8:

[The,Com-

7 1 23 9 7 5

16 1 2 99 5 6

8 5 6 3 1 94 9

8 5 2

Difficult Easier

Games and puzzles

February 2010 [The , Commutator] 16

FIRST EDITION

The Macsoc Magazine: FEBRUARY 2010

Sponsored by: The Department of Mathematics and the IMA

Find the floor in the logic

( )

x

x x

x

x

0

1 0

1 0

1

1 0

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f

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= + - + + - + + - + +

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“Proof” 1 “Proof” 2

This month . Don’t believe me? Well here are two “proofs”:1 0=

Kendoku

Sodoku Fill in each row and each column with the integers 1,2,...,9 exactly once

9 72 9 5 48 1

3 6 91 5

7 2 37 5

3 6 1 46 4

Kakuro

What’s it all about?Our vision for the magazine is to have a

publication that first and foremost commu-nicates ideas which have sparked an inter-est in the writers. With this both theunderstanding and the enthusiasm benefitsthe reader. Our aim is to create somethingwhich is both a Vista for first and secondyear students who are yet to decide onwhich disciplines will make up their de-gree and also for honors students tobroaden their knowledge using the lecturecourses provided at Glasgow University asa foundation.

The publication also serves to promotediscussion on all aspects of Mathematics beit sociological, psychological, philosophicalor historical. Mathematics has affectedeveryone’s lives both within and outwiththe department and so, although not cen-tral to mathematical research objectives,there are many areas of discussion shouldbe of interest to us all.

For the writers themselves we hope toprovides a platform for undergraduates tobe involved in an extra curricular pursuitwhere they can develop skills outside ofthose that are offered as part of the under-graduate programme.

What is the commutator?For an outline of group theory see the

following feature article. The commutatorof two elements of a group is de-noted by and is defined as follows:

. An important feature of aGroup is commutativity, that is the extentto which . Of course we take forgranted that addition is always commuta-tive and that many of the multiplicativegroups we encounter day to day do not im-pose an order effect on our calculation. Thesavvy shopper would have to add to hiswhit the ability to position each item of hisweekly shop on the conveyor belt at the su-permarket check out in the cheapest possi-ble order in a world withoutcommutativity. As an undergraduate youwill encounter matrix multiplication as afirst example of a non-commutative groupoperation. However David Halfpenny pro-vides an example of the rubik’s cube in hisarticle on page 5. If you rotate some of thepieces about one axis and then perform arotation about a different axis the rubik’scube will end up in a different configura-tion to that achieved by performing thesame two rotations in reverse order, be-cause these two group elements which act

on the cube do not commute with eachother in the Group.

Taking the set of all commutators of agroup we find that each pair of elementsthat commute with each other are sent tothe identity element by their commutatorand we are left with the set of commutatorswhich don’t commute with each other. Tak-ing the subgroup which is generated bythis set tells us something about the extentto which that group commutes. This iscalled the derived subgroup and it is an im-portant tool in the understanding of thestructure of a group.

It is with this that we go straight intoour first feature which starts with somebasic group theory and goes on to demon-strate the application of group theory to theunderstanding of the rubiks cube.

Carl Chaplin

[ , ]:a b a b ab1 1= - -

ab ba=

[ , ]a b,a b G

Contents

2 [The , Commutator] February 2010 February 2010 [The , Commutator] 3

News

News2 Editorial

NewsFeatures

4 Group theory

12 Finding NeptuneDiscovery

An introduction to abstract algebra and a look atthe mathematics of the rubik’s cube.

How Neptune was discovered to exist, its massand orbit calculated before it was observed

Word fromthe Editor

6 Chaos theoryAn introduction to a theory which reveals the de-terministic equations which govern the seeminglyrandom and probabilistic events.

Mathematics & People10 Are men really better at mathsthan women?11 Nazi Germany and Mathematics

Assorted Articles13 Foundations of Mathematics14 Why study Mathematics?15 How to tile the sphere with dif-ferential equationsGames16 games and puzzles

Thank YouMy first duty as editor is to thank all

those involved in the project. I have some-times been frustrated with the level of in-difference among undergraduates and Ihave to say that it has been very refreshingto meet and work with such a lively andproactive team here in Glasgow. This en-terprise has been built on their enthusiasmit is to them that I am the most grateful.

I would also like to thank the GlasgowUniversity Department of Mathematicsand the Maclaurin Society for their fullsupport both in terms of time spent organ-ising and advising as well as their financialsupport. It is their financial support thathas allowed us to print the magazine infull colour with a professional finish.

The Macsoc itself and by associationthis magazine, is funded by the The Insti-tute of Mathematics and its Applications(IMA) which is the UK's learned and pro-fessional society for mathematics and itsapplications. It promotes mathematics re-search, education and careers, and the useof mathematics in business, industry andcommerce.

A few words of thanks, a briefoutline of what [The, Commutator]is about and then a definition of theCommutator.

Editor, Designer, GraphicsCarl Chaplin

Assistant EditorsStuart AndrewAndrew Bestel

Those who were also activelyinvolved in the organisation

David HalfpennyEmma CumminSandy BlackFiona DohertyLouise OgdenPaul McFaddenMichael StringerNeil FullartonJaspal Puri

The Faculty of Information & Mathe-matical Sciences (FIMS) is soon to be re-placed by the School of Mathematics andStatistics under the new college struc-ture that will supersede the old facultystructure from 1st August 2010.

The most notable affect for studentswill be the combining of the existingFIMS graduate school with that of theFaculty of Chemistry, Faculty of Engineer-ing and the Faculty Physics and Astron-omy. The existing faculty basedgraduate schools will form a singlegraduate school for the College of Scienceand Engineering.

Students were informed of the Uni-versity wide changes by the Senior VicePrincipal and Deputy Vice ChancellorAndrea Nolan, together with the prom-ise of zero disruption while the changesare implemented.

“The needs of our students are central tothe activity of the University and the re-structure has not, and will not, alter that”

Undergraduate students from Au-gust will be awarded their degree by

their college instead of their faculty butthere will be little change in their day today experiences.

The main changes will be in the Uni-versity Financial structure which waspreviously comprised of ten budgets,corresponding to nine faculties and theUniversity Services. The new structurewill be five larger budgets, correspon-ding to four colleges and UniversityServices. The University has risen towithin the top 20 in the UK rankingsand within the top 100 in the worldrankings. To carry on this trend Glas-gow needs to respond to the increase indemand for inter-disciplinary academicresearch and this restructuring of Uni-versity finances is designed to facilitatethis. Currently research over facultyboundaries requires multiple researchgrants and so there is unnecessary re-sistance and less coordination than canbe expected from a single college re-search fund. Fewer budgets also allowfor greater efficiency as there should befewer transaction and co-ordinationcosts and delays. The university as awhole expects to be more agile in its re-

sponse to external opportunities andthreats, while larger budgets shouldmake larger investments more feasible.

For those undergraduates who areplanning a future in academic research,the benefits of the restructuring will beimmediately apparent as heads of col-leges will have larger budgets and moreadministrative support. The Univer-sity’s standings in both the UK and theWorld rankings should reflect any in-crease in the volume or quality of re-search while the College of Science andEngineering should hold more weightthan the individual faculties it replaces.As such all undergraduates should feelan increase in the benefit that graduat-ing from Glasgow University will bring.

Carl Chaplin

University Restructuring and You!

Feature Group Theory


Invented in 1981 by professor of architecture,Erno Rubik, the Rubik’s cube is the best sellingpuzzle game in history, selling over 350 millioncubes since its invention. Nowadays there aremany types of Rubik’s cube with all kinds of di-mensions. We will, however, concentrate on thetraditional structure. This type of cubehas the usual 6 faces, on each of which are 9 dis-tinct squares, each of which have one of 6 pre-scribed colours. The cube is built in such a waythat each face has the ability to rotatein a clockwise or anticlockwise direction. The pur-pose of this puzzle is to perform a certain algo-rithm of these rotations to a randomly mixed upcube so that all the squares on each face of thecube are of the same colour.

Notation There are many texts available containing so-

lutions to the Rubik’s cube. Perhaps the most pop-ular of which is "Notes on Rubik’s magic cube" byDavid Singmaster. We shall use notation devel-oped in this text.

Now, imagine holding a Rubik’s cube in frontof you so that you have just a single face facingyou. We label each face as follows:

(front), (back), (left),(right), (up) and (down).

Using these denotations, we can refer to thecorners and edges of the cube. For example,corresponds to the corner located on the front faceat the bottom right hand corner. Similarly, cor-responds to the edge on the left side of the backface. So the first letter specifies the face and the fol-lowing letters give the position on the face.

Let be a move on the Rubik’s cube. Thenwe can define the product of any two moves

, to be where the move isperformed first and then . Moreover, if we de-note to be the set of all moves on a Rubik’scube, is in fact a group under the operation ,as follows:

Let . Since and are

moves then is clearly also a move so isclosed under the defined operation. is associative

since: ,

and the inverse of a move is simply the reverse of theoriginal performed. The identity element of is simplynot performing a move. Thus is a group.

The cube itself contains 20 permutable pieces,namely 8 corners and 12 edges. In fact, the set ofmoves that permute the corners of the cube is asubgroup of the group . Similarly, the set ofmoves that permute the edges of the cube is alsoa subgroup of . These subgroups will be de-noted by and , respectively.

As stated earlier, the individual moves on theRubik’s cube are the rotations of each face. Fromthis, clearly our group of moves, , is generatedby these clockwise rotations of each face.These generators are denoted by ,each corresponding to the obviously appropriateface. Moreover, we can represent each generatorusing cycle notation. Consider a generator . Ifwe apply the generator to a corner on the frontface, , say, it is permuted such that the fur cor-ner goes to where the corner was originallysituated. Similarly, under , the corner goesto where the corner started, see figure 2.

It follows that we can represent the permuta-tions of the corners of , under the generator ,as the cycle . Similarly, for theedges we have . Moreover, since thecorners and edges are the only permutable piecesis the cube, can be represented by these to dis-joint cycles, i.e.

This logic is obviously true for each face.From the above, with the additional fact that

there are 8 corners and 12 edges on the cube, it istrue that we can regard every element of as apermutation of , and every element of as apermutation of . So, if we label each of the cor-ners of the cube by integers in as in di-agram 2. We are able to show that there exists agroup homomorphism: , defined by

.i.e.

Note that if this is true for any two generatorsof , it holds for all elements of . So for

and,.

Notice that and are not dis-joint, so upon putting this into disjoint form, wehave: .

Thus is a group homomorphism. Thislogic is also true for . Note also thatelements of the group, can be considered as per-mutations of with a corresponding grouphomomorphism defined similarly as with and

.The commutator of moves in ,

can be used to permute a small numberof pieces on the cube whilst leaving the majorityunaltered. This is an important sequence as it al-lows us to change unsolved parts of the cube with-out disturbing other pieces that have already beensolved. The commutator is very effective in solv-ing corners and edges of the cube as they usuallylie in the intersection of two faces and are there-fore open to permutation under the commutatorof two generating moves, provided that these gen-erators correspond to faces adjacent to each other.

As one can see, the connection between theRubik’s cube and concepts in group theory is verystrong. For more information on the connectionbetween puzzles such as the Rubik’s cube andgroup theory, see texts such as "Adventures ingroup theory" by David Joyner and DavidSingmaster’s "Notes on Rubik’s magic cube".

David Halfpenny

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What is a group?Much of the mathematics you will have learnt

at school and in the first few years of your univer-sity career will have involved manipulating for-mulars and algebraic expressions involvingunknowns. Abstracting from this we formalise theidea of dealing with elements which are somehowcompatible under some sort of operation. Vagueso far, but all avenues to the world of the abstractare vague, deliberately so. Once you have afoothold you can begin to admire the power yougain when you liberate yourself from the specificsof any particular example.

We therefore start with a set of elements suchas the integers and we define a binary operationwhich acts on a pair of elements of this set to gen-erate another element: . However wechoose to construct an operation it would be use-ful if putting two elements in gave us somethingwhich is again in the set of interest. For exampleif we use to represent our chosen operation,

would be very unhelpful to us be-cause blue isn’t an integer. It is however liberatingto know that you could construct a binary opera-tion that would act on a subset of the set of allcolours, we only illustrate abstract algebra withnumbers as this is a context which the reader willbe familiar with. Let’s define our binary operationon the set by the function:

and we denote by .You will notice immediately that addition of theintegers can be considered in this way:

.One inportant feature of an algebraic object

such as a group is associativity. This is simply therequirement that for all in ourset: . This simply meansthat evaluating and then taking of thisyields the same result as evaluating and thentaking . This seems like a technicality at firstbut actually it is a necessary axiom, without whichwe can’t prove the simplest of results, as you willsee. The associative axiom will become your bestfriend as you reed on.

From the integers we also observe a very spe-cial element which we call the identity element.Under addition this is zero but if we consider theset of rationals with zero excluded under the bi-nary operation of multiplication the identity ele-ment is 1. You might ask: ‘what these twoelements have in common? ‘ and by way of an an-swer they both play the same role in the contextof their respective binary operations. Formallyis an identity element if for all in ,

.Finally we can think about the expression

as doing something to and we arrive at anew element, but there ought to be some mecha-nism for getting from back to and so weintroduce the notion of inverses. That is for each

in there must exist which satisfies:

.Together with our definition of and associa-

tivity we get:

.If we furnish a set with a binary operation and

impose the axioms of associativity and inverseswe get a group.

Some elementary group theoryAs a first step towards studying different

types of groups, once we have a group we can takesubsets of the original set which are themselvesclosed under the group operation and we call sucha subset a subgroup. For example the even inte-gers form a subgroup of the integers under addi-tion since addition of even integers always yieldsanother even integer.

Once we have more than one group we canconsider maps which preserve the group opera-tion. This is not as straight forward as it sounds aswe may have a map between two groups whichhave two completely different operations such asaddition in one group and multiplication in theother. We need functions between two groupswhere it doesn’t matter whether you perform thegroup operation on two elements in the firstgroup and then take the image of the mapping ortake the image of two elements and then take theproduct under the second group operation. Whena function obeys this simple rule we call it a grouphomomorphism and an interesting example of agroup homomorphism is the natural logarithm.

The natural logarithm takes as its domain thegroup of non-zero positive real number undermultiplication (denoted by ) and itscodomain is the group of all real numbers underaddition (denoted by ). This homomor-phism can be thought of in the conventional sense

using the graph in figure 1 but its significance canbe appreciated by the fact that it is a group homo-morphism. In fact this particular homomorphismhelped put man on the moon. The point being thatmultiplication is labour intensive compared to ad-dition and so before the advent of computers andpocket calculators, slide rules were used to reducea multiplication down to a simple sum. Supposewe needed to find the product of two elements in

, we would first use the logarithm func-tion to find the two corresponding elements in

, we would then perform the operation inthis group (i.e. add them) and then we would per-form the inverse map to get back to an element in

which would correspond to the productof the two elements we started with.

Immediately we have a practical applicationof some elementary group theory, all be it thatnow this happens inside a computer where oncewe had to do it for ourselves. We can go furtherthough, the map is injective, that is no too ele-ments are mapped to the same point; further it issurjective, every real number has a correspondingreal number greater than zero under the inverseof the natural logarithm. A homomorphism which

is injective and surjective is called an isomor-phism. A fancy word you might think but it isquite deserving of its own fancy title since it tellsus that these two groups are structurally the same.That is going between the two is as simple as re-naming the elements and doesn’t require us tochange anything structurally.

It is from these humble begins that one of thegreatest mathematical achievements of the 20thcentury was made, that is to classify all finite sim-ple groups. It was a collaborative effort whichtook up more than 10,000 journal pages to docu-ment.

What do groups do?In nature we observe symmetry in all life,

from the geometry of celestial bodies down to thequantum particle. As such the study of and abilityto model symmetric systems is extremely impor-tant to physicists and chemists alike.

It is with this in mind that, having said whata group is, I answer the question of what a groupdoes: a group models symmetry.

The simplest illustration of this is the Dihedralgroup on n elements, . For example can berepresented as the symmetries of an equilateraltriangle. Its elements are the two rotations (by in-teger multiples of ) and three reflections andits identity element is the element which doesn’tmove the triangle at all. We take the binary oper-ation of two elements to be the application of oneelement to the triangle followed by the other.Clearly by geometry this group is closed underthis operation since we always recover our trian-gle, it is just the corners which are permuted. If werestrict this set of isometries of the plane to just re-flections or just rotations we observe two naturallyoccurring subgroups.

By way of example, quantum mechanicsshowed that the elementary systems that make upmatter, such as electrons and protons, are trulyidentical, not just very similar, so that symmetryin their arrangement is exact, not approximate asin the macroscopic world. Systems were also seento be described by functions of position that aresubject to the usual symmetry operations of rota-tion and reflection, as well as to others not so eas-ily described in concrete terms, such as theexchange of identical particles. Elementary parti-cles were observed to reflect symmetry propertiesin more esoteric spaces. In all these cases, symme-try can be expressed by certain operations on thesystems concerned, which have properties re-vealed by Group Theory.

Carl Chaplin

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figure 1 the natural logarithm

Whether you are new to abstract algebra or are in need of a little revision I will attempt to furnish you with all thetools you require to enjoy David Halfpenny’s piece which follows.

THE GROUP THEGROUPMathematics of the Rubik’s cube

Be it a lamentable problem or a long term goal, the Rubik’s cube capturesuniversal fascination. Of course, there is clearly a technical property to thispuzzling device, but how mathematical is it? Here, we will take a brief glimpseat the connection between the Rubik’s cube and elements of group theory.

figure 2

figure 3

• is a measure of the temperaturedifference between the warm, rising air and thecool, falling air.

• is a measure of the vertical temperaturedifference as you move through the systemfrom top to bottom.

• the measure of the difference in the tem-perature

If we find the equilibrium points for thesethree equations it is evident:

, is an equilibrium pointfor all values of . It turns out that this point isonly stable for values of . If we now in-crease beyond this point we find two newequilibrium points and these turn out only tobe stable for: . These are theonly equilibrium points that exist.

Lorenz, although recognizing the regular-ity in these systems, was also very aware thatthese where no ordinary systems. Their sensi-

tivity put huge restraints on the traditionalways one may make use of a system. As for theweather the implications of the findings wheremixed, although it was now theoretically pos-sible to predicate any forecast given the condi-tions, it was now even more than beforeseeming practically impossible to provide thedata required to do so. As he wrote in his 1963paper:

“When our results…. are applied to the atmos-phere… they indicate that prediction of the suffi-ciently distant future is impossible by any method,unless the present conditions are known exactly. Inview of the inevitable inaccuracy and incomplete-ness of weather observations, precise very-long-ranging forecasting would seem to be non-existent.”(Lorenz, 1963: 130–141)

These Chaotic systems, unconventional inthemselves, would have to be handled in dif-ferent more unconventional ways it seems if

they where to aid the project of scientific en-quiry.

Even if the success of the theory when ap-plied to a weather model seemed to be cur-tailed by limitations in other practices (ofweather observation for instance), the theorywas still groundbreaking. One of the reasonsfor this was its incredible potential, it wasn’tlong before systems exhibiting chaotic dynam-ics where being recognized across a large num-ber of fields, from mathematical models tobiological patterns.

Chris Andrews

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Chaos theory is a radical, topical scientificdiscovery. It is not only a prolific field in math-ematics but has proven to be applicable in astaggering number of areas and provides uswith new tools in which we may explain andinteract with the world round about us.

What is Chaos Theory?Popular perceptions of Chaos theory most

notably include ‘the butterfly effect.’ The ideais that a tiny influence such a single butterflyflapping its wings can cause a seemingly dis-proportionate outcome, that is, tornado at theother side of the world. This indeed is the ideaat the very heart of Chaos Theory; that minus-cule differences in ‘initial conditions’ can havedrastic consequences on a final result(Chorafas, 1994).

Chaos theory can be applied to describingthe behaviour of certain dynamical systemswhich are functions of time. Uniquely in achaotic system, although behaviour appearsrandom and unpredictable the systems are ac-tually deterministic. To understand that evensmall, seemingly unimportant alterations in theinitial conditions change the entire outcome ofa system is to understand the basic principle ofChaos theory.

The Discovery of Chaos TheoryChaos theory was discovered in the 1960’s

by a collaboration of scientists who, against thegrain, considered these unexpected results andideas. However the discovery of the theory ispopularly attributed to Dr. Edward Lorenzwho coined the phrase the “butterfly effect”and is commonly known as the “Father ofChaos” (Lorenz, 1989). Lorenz had made aweather stimulating programme on his com-puter where he worked at the MassachusettsInstitute of Technology. This very complex pro-gram had many complicated equations deter-mining the probability of certain weatherforecasts. He would input some conditions(e.g. the temperature, wind speed etc.) and itwould generate a graph of the predictedweather. The program was considered veryhighly and incredibly seemed as though itwould never repeat a previous sequence. It wasbelieved that if the exact initial weather condi-tions were put in to this program, it could in-fallibly mimic the real varying weatherconditions outside. On one occasion as the pro-gram was running on the initial conditions thatLorenz entered he decided that he wanted totake a better look at the final outcome of the

weather. Instead of allowing the program torun he cheated, he read off the values from theearlier stimulation and input these as the initialconditions. Now although the computer wouldcalculate numbers to six decimal places, whenLorenz inputted the values half way through

the stimulation he rounded them to three dec-imal places. It transpired this minute changein initial conditions completely altered the out-come of the predicted weather.

The above graph is an example of the kindof divergence that Lorenz would have beenpresented with. It clearly demonstrates that thesmall difference in initial conditions makes anincreasingly large difference as the time valueincreases. It is for this reason that weather fore-casts are more inaccurate the further into thefuture one attempts to predict their behaviour.Struck by the disproportionate effect that initialconditions had on the final outcome of this sys-tem Edward Lorenz certainly invented the the-ory’s most enduring image in 1979, in anaddress to the American Association for theAdvancement of Science entitled, “Does theflap of a butterfly’s wings in Brazil set off a tor-nado in Texas?” (Lorenz, 1993)

Conditions for ChaosIf we consider Chaotic behaviour in nonlin-

ear systems there is one key condition thatmust be met for this behaviour to occur. Thiscondition is that the phase space is at leastthree dimensional. The reason for this is thePoincare-Bendixon Theorem that states, if weare dealing with two dimensional autonomoussystems, for example:

Then the phase path must eventually do one ofthree things: terminate at an equilibrium point;approach a limit cycle; return to the originalpoint giving a closed path, as illustrated by fig-ure 1.

This means we can know what will happento these systems, we can in theory predict theirbehaviour. They are not therefore ‘random’ os-cillations (Acheson, 1997).

Lorenz EquationsLorenz used 12 equations when predicting

the weather; however, “the Lorenz Equations”refers to the three main formulas he used(Lorenz, 1989). These arose from a simplifiedmodel of thermal convection in a layer of fluid.They are as follows:

Where originally:• is proportional to the speed of motion

of the air due to convection.

( , )x f x y=o( , )y g x y=o

( )x y x10= -oy rx y zx= - -o

z z xy38

= - +o

x

Gambling is certainly one area wherebeing able to accurately predict the futureresults would of course lead to phenome-nal successes. Although such a truly accu-rate system would be any gamblers’ HolyGrail, if it was to be universally availableit would destroy the very nature of thepractice itself. Gambling was convention-ally thought to be predominantly based on

sciences of probability and perhaps evenluck. Although there are many tricks toplevel gamblers employ to increase the like-lihood of winning, it seemed hard to imag-ine that any gambling system wasdeterministic in any way. That is untilChaos Theory entered the picture. We mayask what kind of systems exhibit thesechaotic examples and the most famous one

in this area must be the roulette wheel.The roulette wheel is nowhere near

as “random” as large scale casinoswould have you believe. In fact, abouncing roulette ball obeys exactlythe same laws of physics as a bounc-ing tennis ball. So why is it so diffi-cult, neigh on impossible, to predictwhere the roulette ball will land? Pre-dicting the motion of a tennis ballseems simple enough and generallyspeaking a minute inaccuracy in themeasurement of the initial conditions(such as the height dropped, the angle,air resistance… etc.) will result in avery small inaccuracy of prediction.The ball may land only a few millime-tres from where it was expected toland. A roulette ball, however, is in-volved in a chaotic system. This means

that if there is even a minute inaccu-racy in the measurements of how highthe ball was dropped from, how fastthe table was spinning or indeed thedimensions of the table, then the pre-dicted resting place of the roulette ballwould be massively inaccurate. In-stead of landing just a few millimetresfrom where it was predicted to land,like the tennis ball, the roulette ballmay well end up on the other side ofthe table. This is a prime example of achaotic system.

Emma Cummin

The Roulette Wheel

The roulette wheel isnowhere near as “random”as large scale casinos wouldhave you believe.

Feature Chaos Theory

A brief Introduction This article aims to outline the discovery of ‘chaos’ and explain some of the basic ideas of the theory in terms ofthe Lorenz Equations. It shall then look in more detail at examples of where we may find ‘chaos’ .


approach a limit cycle return to the originalpoint giving a closed path

terminate at an equilib-rium point

figure 1 The three possible phase paths of a two dimensional system

figure 2 two very different outcomes, withvery similar initial conditions

-space portraits to give a greater under-standing of the systems behaviour. Furtherresearch has been to examine the system interms of spectral plots in terms of discretesystems, that is choosing a select number ofdata points in the system, examining thisagainst a Nyquist frequency and plottingagainst its amplitude. Over 100 diagrams infigure 3 have been plotted together for val-ues between to and the ’noise’that appear is approximately aroundas well as slight chaotic behaviour around

. However further investigations areneeded into the practicality of this diagram,except it gives a decent pictorial descriptionfor different values of which is ideal for ex-amining the onset of chaotic behaviour.

Sandy Black

r 0= r 30=r 25=

r 14=

r

The weather is a complex and unpre-dictable series of events that occurs every-day and is extremely difficult to model. Oneimportant aspect in meteorology, and fluidmechanics, is the problem of convection: theprocess by which hot air rises and cold airsinks. In our atmosphere we know that theair at the bottom is generally higher than itis on the top. However, the convectiveprocess is not completely stable and in somecases there is no convection or complex tur-bulent motion can occur.

Let be the temperature of the atmos-phere then the temperature difference be-tween the two altitudes can then help usdescribe the onset of convection and turbu-lent flows. If we consider a temperature dif-ference to be , then if this occurs below acritical value, then there is no convection,above the critical value steady convectionoccurs. The onset of convection has been re-searched and a dimensionless number calledthe Rayleigh’s number, , as described byLord Rayleigh which s used to determinewhere the onset of convection occurs. How-ever if the temperature difference is largeenough then complex turbulent flows ap-pear and from this poses a difficult question:for what value of temperature do turbu-lent flows occur?

This phenomenon was investigated byEdward Lorenz, to which he discovered theLorenz system of equations which are de-scribed on page 6 but will be outlined herein more detail:

(1)

The parameter r in the model is Rayleigh’snumber which is denoted as,

(2)where is the critical Rayleigh number

for which convection occurs, and so in theinvestigation to where turbulence in the at-mosphere occurs it is useful to consider dif-ferent values of the parameter .

Analysis of the Lorenz system

Onset of ConvectionThe Lorenz system in equation (1) is

non-linear due to the and terms, there-fore it is difficult to get an analytical solutionand numerical methods must be used. How-ever we can locate the critical points of thesystem by setting .We can then choose from the firstequation in (1) and we obtain,

(3)

From the first equation in (3) we canchoose either: to give

and as critical point , or,

to give ,and as critical point . We notethat the only critical point for is

since the second critical pointgives complex values. Then for the crit-ical points and are valid. Thus wecan see that gives a bifurication pointand so denotes the onset of convection inour model. This makes physical sense sincethe value is taken to be dimensionalisedwhere where is taken to be thecritical Rayleigh number.

Methods of analysisThe Lorenz system of equations can be

analysed in different pictorial fashions suchas time plots, phase-space plots and spectraldiagrams. These are useful in examining thedifferent effects of r on the system whichmay provide some understand into its be-haviour.

Time plotsTime plots show the way in which a par-

ticular value of and and vary withtime . This generally is a decent method toshow when a system develops chaotic be-haviour. Usually a time plot consists of aharmonic wave which oscillates with timewhen the system is stable, however whenthe system becomes unstable the harmonicwave becomes inconsistent and may ’jump’as shown in figure . This however is not adecent method to evaluate different valuesof r since it may require hundreds of plots tosee when r changes from stable to unstable.

Phase-space portraitsPhase-space portraits show the varia-

tions of and for example, or in the caseof a three dimensional plot show howand vary after a specific time . However,to evaluate different values of it is not veryeffective since numerous plots are requiredto examine specifically when a system be-comes unstable. For a stable system the plotoscillates towards a point and remains fixedthere for all time as shown in the first dia-gram of figure. Interestingly with the lorenzsystem, in the phase-space portrait, if a sys-tem become unstable it develops strange at-tractors. The trajectories orbit around theseattractors and in some cases produce limitcycles, whereby the trajectory orbits thesame path for a certain amount of time. Usu-ally when r is large enough the trajectorywill orbit one strange attractor then sud-denly jump to jump to the other and thenback again without any real pattern, a signof chaotic behaviour as seen in the figure forthe -plane.A brief summary of the Lorenz system ofequations has been discussed. The values of

are useful to examine in the real world asthe system (1) can be used to model convec-

tion in the atmosphere. Interestingly, chaoticbehaviour appear for a critical values ofwhich is usually around in most lit-erature and by running numerical calcula-tions. These types of behaviour can beexamined in forms of time plots and phase-

Ra

( )dtdx

x y xv= -

dt

dyrx y xz= - -

dtdz

bz xy= - +

/r R Ra c=RC

x y=

( )x r z1 0- - =

x bz 02

- =

x 0= ,x y0 0= =

z 0= CP1 z r 1= -

( )x b r 1!= - ( )y b r 1!= -z r 1= - CP2

r 1<( , , )CP 0 0 01 =

r 1>CP1 CP2

r 1=

r/r R Ra c= Rc

,x y zt

T

tdc

T

r

xz xy

/ / /dx dt dy dt dz dt 0= = =

x y,x y

z tr

xy

r

rr 25=

Feature Chaos Theory


figure 1 The values of are shown in the above diagrams. The first one shows a stable system whenand the second shows and unstable system when . A Initial conditions are

.

Sr 14= r 28=

, ,x y z1 1 1= = =

Lorenz system of equations in detail

figure 2 The values of and are shown in the above diagrams. The first one shows a stable systemwhen and the second shows and unstable system when . The two vacant spaces that appearin the second diagram are called strange attractor which the trajectories orbit. A Initial conditions are

.

()x t ()y tr 14= r 28=

, ,x y z1 1 1= = =

In the preceding articles we have discussedthe way in which chaos theory is related to com-plicated systems such as the roulette wheel andhow a slight change to an initial condition can sig-nificantly change the outcome of the event or evenmake the event outcome impossible to predict.

Properties of the tent mapThe tent map is

an iterated functionwith its graphlikened to the shapeof a tent (figure 4). Itdemonstrates arange of dynamicalbehaviour rangingfrom predictable tochaotic

The tent func-tion is defined for allvalues on the unit interval . This closed in-terval contains the image of this function so thatwhen we input real values from this domain, thefunction outputs real values in, the interval [0,1].Mathematically we write this mapping as

. Since the image of this functionis contained in the domain we can begin with anumber, say , in the interval and repeat-edly apply or rather, iterate, the tent map function

: . The tent mapis defined by the difference equation :

This equation may be rewritten as:

Predictable behaviourFor certain points in the interval , when

we plug them into the function we will not see anychange to the value after amount of iterationssince they do not map to any other value but them-selves. These points are called fixed points. We saythat is a fixed point of a function if and only if

. In other words the point lieson the line , so that the graph of the tentfunction has a common point with the line . As we can see (figure 1) 0 is a fixed point of thetent map since i.e. the point lieson the line .Are there any other fixed points for this map?

Having a look at figure 4 it is clear that thereexists another fixed point at point A. The fixedpoint at A can be attained algebraically by solving

where the tent function intersects the line ( equivalent)

{since },which yields the other fixed point .

Suppose we choose a value close to either ofthe fixed points, lets say 0.659 (extremely close tothe fixed point 2/3 ). After a few iterations underthe tent map function:

We can see that the value of 0.659 gradually movesaway from the fixed point 2/3 but is still reason-ably close to it which we expect since the changeto the initial value, 2/3, is very insignificant. Anypoint close enough to either of the fixed points be-have this way (unstable) and so we are able to pre-dict the direction that the value goes under the tentmap function, that is, away from the fixed pointbut still reasonably near to it.

Chaotic behaviour Now suppose that wechoose two random non fixed values that areagain very close to each other, say 0.258 and 0.259.Applying the tent map, , 10 times to each valuegives the graph :We can see how 0.258 and 0.259 initially follow asimilar pattern during the process of iterationwhich we expect to happen as the difference be-tween the initial values isn’t at all significant. Afterthe 8th iteration we start to see a hint of variance

between the values but by the end of the 10th iter-ation we see a clear divergence between them.Again we see the telltale features of a chaotic sys-tem - the slightest of changes to the initial condi-tions of the system can result in a seeminglyunpredictable the outcome. We can see this samepattern emerge after a sufficiently large number ofiterations (Simiu, 2002)

Sean Chan

:[ , ] [ , ]f 0 1 0 1"

x [ , ]0 1

f , ( ), ( ( )), ( ( ( )))...x f x f f x f f f x

X X1 221

n n1 = - -+

, . ;

( ), . . .

X X X

X X X

2 0 0 5

2 1 0 5 10

n n n

n n n

1

1

1

1

#

#

=

= -

+

+

[ , ]0 1

x

a( )f a a= ( , ( ))a f a

y x=y x=

( )f 0 0= ( , )0 0y x=

X Xn n1 =+ y x=( )X X2 1 n n- = . .X0 5 10n# #

/X 2 3n =

. . . . …0 659 0 682 0 636 0 728" " "

f

[ , ]0 1

E c o n o m i cC h a o sLike in gambling (perhaps nottoo far off the practise alto-gether) one of the most fore-front examples of a system inwhich prediction is everything

is in economics; specifically, thestock exchange. Here there is ahuge demand for any theoryable to successfully predict thetotally random seeming fluctu-ations in share prices for in-stance. Indeed Wall Street isone of the few places in themodern world where an as-

trologer can make a good livingand be taking seriously by ahighly educated audience(Chorafas, 1994). Therefore theapplication of Chaos theory tothe stock exchange and otherareas of economics has been at-tempted enthusiastically by re-searchers but to limited

success. It seems the spirit isstrong but the body of evidenceweak; there is as yet no conclu-sive finding of the existence ofa chaotic system in economicand financial data (Brock etal, 1991).Fiona Doherty

Figure 4 The Tent Map

The Tent MapChaos theory is also related to many mathematical models where there is a growing interest in

the area. Examples include the two-dimensional Small Horseshoe Map and the one-dimensionalLogistic Map and the one-dimensional Tent Map that shall now be presented in more detail.

figure 3 spectral diagram

For centuries, the notion that women are in-nately less capable of studying mathematics thenmen has persisted even in the most educated ofminds. In 2006, the proportion of students apply-ing to study Mathematical and Computer Sciencesat an undergraduate level in the UK who were fe-male was just 21.2% . Many of us just accept this,citing that maths and science are traditionallymore masculine subjects, but there has been a sur-prising amount of research on gender and mathe-matics in the past few decades. Is it fair to assumethat mathematics is just better suited to men, andthat because of a woman’s genetics she is alwaysdestined to be outshone by her male counterparts?In short, the answer is no, that is to say it doesn’tfully explain the under representation of womenin the field. Certainly, it has been shown that boysand girls learn differently due to genetics, butthere is no evidence to suggest that girls are lessable to learn and understand mathematics be-cause of gender differences. This then suggeststhat environmental factors could possibly explainthe differences, perhaps due to social influencesor current teaching methods.

In the UK, mathematics is taught as a coresubject, and all students must study it throughouttheir primary and the majority of their secondaryeducation. This can make it challenging to deter-mine a student’s attitudes towards the subjectuntil it no longer becomes obligatory. In Scotland,many students have the choice to opt out of math-ematics before studying for their Highers. In 2007,the proportion of students who studied HigherMathematics that were female was 48.7%, whichdoes not seem particularly low, but if we then con-sider the same statistic but in Advanced HigherMathematics the proportion drops to 38.2% .Equivalently, in England and Wales, the propor-tion of students studying an A Level in Mathemat-ics in 2006 who were female was just 38.5% .Previous data shows a similar trend going backfor decades so it is clear that there are fewer girlsthen boys who choose to study mathematics oncethe subject becomes elective. Despite this, the per-centage of girls achieving a passing grade washigher then for boys in both Mathematics Higherand Mathematics Advanced Higher (70.5% com-pared to 69.6% and 70.9% compared to 62.5% re-spectively). So in the 2007 data, the genderdifferences disappear when one accounts for thenumber of students who participate in courses.This is a common result among many differentstudies across the world, and is known as the The-ory of Differential Participation (Fennema & Sher-man 1977). Although this theory is not fullyaccepted by all researchers in mathematics educa-tion, there is no dispute that fewer girls participatein the more advanced levels of the subject.

One measure, which has been employed by anumber of countries across the world, has beenthe introduction of intervention programmes inschools. In the USA, one such program run byFennema et al. in 1981, involved the broadcastingof videos designed to change attitudes about gen-

der-related differences in maths by giving infor-mation about careers, the educational relevance ofmaths and suggestions for activities to effectchange. The videos were focused on specific targetgroups including students, teachers and parents.The research was based on two assumptions,firstly, that if the girls’ knowledge on gender dif-ferences increased and their attitudes towardsmathematics were improved then their participa-tion in more advanced courses would also in-crease. And secondly, that in order to change theattitudes of the girls, then the expectations of theirparents and teachers would also need to be al-tered assuming that the students are influencedby their social environment. Fennema and her col-leagues did indeed find that the girls’ participa-tion in advanced maths courses increased. Theyalso showed an increased understanding of gen-der differences, saw mathematics as more usefulfor their future lives and were less likely to blametheir failures on lack of ability. However, it shouldbe noted that boys were also included in the studyand their attitudes to mathematics improved in asimilar way to the girls’. The success of this pro-gramme and others, however, shows how effec-tive such interventions can be in improving theattitudes of female (and male) students towardsmathematics. Before a child’s progression into sec-ondary education, there is little difference in howchildren value the subject (Eccles et al. 1983) butonce in secondary school girls begin to rank math-ematics as their lowest valued subject comparedto boys who have it ranked as one of the highest(Eccles et al. 1993). It has been suggested that thisis due to a child’s increasing awareness of gender-roles as they grow older. In other words, girl’s be-come influenced by their social environment insuch a way that they begin to value mathematicsless due to the expectations placed upon themfrom society.

There are a number of government initiativesin this country as well, targeted at improving notonly the gender differences in mathematics butalso the overall attitude of students to the subject.However, they have had limited success, whichsuggests that if a student’s perception of mathe-matics is to be altered, then there should be asmuch emphasis on improving the attitude’s ofteachers and parents. Many studies have shownthat boys tend to believe that mathematics will beof greater value to them then girls do, and theyalso have a higher confidence in their abilities tolearn the subject. Both boys and girls also tend tobelieve that mathematics is more suited to malelearning (Sherman & Fennema 1978).

Equality in the classroom can be a controver-sial issue. All students in the UK are now entitledto a fair and equal education, and until now, thathas meant that teachers endeavour to teach alltheir students in the same way. But there has beensome evidence to show that the current teachingstyle adopted by many schools is not suited to theway that girls’ in particular learn. Boys respondbetter to an education focussed around competi-tive groups and individual work, whereas girlslearn more effectively in cooperative groups (Fen-nema & Peterson 1986). However, much of achild’s education is focussed around the compet-itive model, partly because it seems boys aremuch more responsive to this style of teaching.This raises the issue of equality, i.e. perhaps ratherthan equality in the classroom what we are reallylooking for is equality in the outcomes. This couldmean that for some subjects, not just mathematics,

children should be separated into gender groups,and the teaching organised around what suitseach group respectively. Teachers inevitably treatboys and girls differently; a person’s gender is anunavoidable part of one’s personality. However,if the way in which mathematics is taught is hav-ing a detrimental effect on many girls’ capabilitiesin the subject, something does need to change.Many would argue that education cannot possiblybe equal if students are treated differently, but thealternative seems to be that for some children,their education suffers.

The complexity of this issue is overwhelming.Although current research does show that genderdifferences in mathematics are decreasing, they dostill exist especially in students’ beliefs about thesubject and in career choices that involve mathe-matics. This is not to say that there are no highachieving women in the field, but when we lookat the data from the classroom, there is no denyingthe smaller proportion of women then men whoprogress into the advanced levels of mathematics.However, it is unclear whether an effective solu-tion exists to this problem, or even if the genderimbalance in mathematics is a problem at all. In-creased awareness of gender differences by stu-dents, teachers and parents, does appear toincrease participation in the subject, but it couldbe argued that if we place such a high importanceon an advanced mathematics education thosewho choose a different path, be them male or fe-male, would seem inferior; an idea that can surelynever lead to true equality.

Louise Ogden

Are Men Really Better atMaths than Women?

Open any history book about Nazi Ger-many and you'll find page after page of infor-mation about their propaganda machine. Inthe aftermath of the First World War, a grow-ing number of the German Public, disillu-sioned with mainstream politics andcrippled, both physically and psychologi-cally, by economic poverty, turned to theNazis in the polls. After seizing power anddismantling the Weimar Republic's democ-racy in 1933, Adolf Hitler's National SocialistGerman Worker's Party set about trying tochange the hearts and minds of the Germanpublic, in accordance with their extremist ide-ology, one rooted in racism and nationalistpride. They played on already existing ten-sions within German society and used themto achieve their own needs. Their abhorrentviews on the superiority of the ''Aryan Race''over the Jewish community, the mentally ill,homosexuals and other minorities are wellknown to most and throughout their reignthey committed some of humanities worstatrocities.

Their propaganda machine, under theleadership of Joseph Goebbels, was particu-larly concerned with changing education tofit their views. After all, if children wereraised as Nazis then, at least theoretically, theparty would be more secure. It is easy to seehow sciences like biology and anatomywould be altered to provide an idealistic viewof the ''superior'' race. Equally, social scienceslike history were rewritten to cast a more op-timistic view on the nation's history, to saythe least. However, a non political subjectsuch as mathematics has no clear link to anykind of ideology, at least not since Pythagore-ans and other cults, and few history books re-count in any great detail the Nazi viewstowards mathematics, and the effect their rulehad on the subject's teaching. This article willprovide a brief and hopefully interestingoverview of this, and we will see how theirviews range from reasonable to bizarre.

As with most subjects, the Nazis tilted theteaching of their subjects to not-so-subtly im-pose their views on pupils. Below, a few gen-uine maths questions from a Nazi textbookare provided:

1) The construction of a lunatic asylum costs6,000,000 RM. How many houses, at 11,000 RM,could be built for that amount?

2) To keep a mentally ill person costs 4 RMper day, a cripple 5.5 RM per day and a criminal3.5 RM per day. Many civil servants receive only4 RM per day, white collar workers barely 3.5 RMand unskilled workers not even 2 RM per day fortheir families.

According to conservative estimates that thereare 300,000 mentally ill, epileptics etc in care.How much do these people cost to keep in total, ata cost of 4 RM per day?

3) The Jews are aliens in Germany. In 1933there were 66,060,000 inhabitants in the GermanReich, 499,682 of whom were Jews, What is thepercent of aliens?

After school level, Nazi influence was alsoclear in mathematics at universities. After re-forms made by the ruling Nazis in 1933, eachlecturer had to perform a traditional nazisalute at the beginning and end of each class.Also, in 1933, ''Dozentenschaft'', an organiza-tion which brought all academics under theNazi banner, provided character profiles ofall lecturers to make sure they were ''suitable''for their jobs. However, rather than examinetheir scientific work, these ''report cards''made sure that the academic communitywere politically coherent with Nazi ideology,at least on the face of things. Any expressionof sympathy for oppressed minorities, orworse still communist movements, could leadto anything from losing their job to a spell ina forced labour camp. The repressive natureof the Nazi state placed paranoia in the hallsof universities, and mathematics was no ex-ception, despite its non political roots.

Intrusion into mathematics by Nazis did-n't stop there. E.A. Weiss, a mathematicianwith Nazi sympathies at Bonn University, isknown for creating a series of Mathematical''Camps'' in 1933. Inspired by the Hitler Youthcamps which aimed to unite and train peoplefrom a young age, Weiss tried to combine thiswith mathematical study. In his 1933 pam-phlet, ''Mathematics: For What?'' he tried toalign mathematics with fascist ideology. Inthe work, he dismisses the claim that Mathe-matics can be studied simply for the sake ofstudying mathematics. He argues that stu-dents must have a purpose to study mathe-matics and that maths ''for the sake of maths''is hard to justify- not a particularly unusualclaim, if flawed. Nazi overtones soon surfaceas he asks if it is ''really German'' to study

things for this reason. He claims that mathe-matics is a character building process, pro-moting (amongst other things) clarity ofspeech and writing, courage, manners andcomradeship between student and professor.He saw maths as a personality mouldingprocess and, with this in mind, he organizeda Mathematics Camp in Kronenburg with 10other students (8 male, 2 female: a direct re-sult of the Nazis alienating women students)at a ruined castle. After unfurling theswastika, they took part in the usual Nazicamp activities of hiking and military stylegames, but combining them with mathemat-ical studies. A typical timetable contained 5hours of mathematical studies throughout theday. The camps ran until 1938, when the de-mands of war curtailed any future plans.

Although the Nazi intrusion into mathe-matics was not as bold as in other sciences,the very fact they encroached on a completelynon political subject is a stark reminder onhow far they were willing to go. Mathemati-cal papers began to slant towards ideology.For example, in Tietjen's work, ''Space orNumber?'' he compared Germans to spaceand the Jews to numbers. In the paper, al-though numbers are not dismissed (indeed,they are necessary), he believes the study ofspace though observation and logic, will leadto numbers, and this analogy is his effort toemphasize superiority. Although ridiculousto the reasonable minded reader, these ideaswere indeed part of the Nazi drive towards aperfect ''Aryan'' state, and mathematics unfor-tunately wasn't spared in their quest.

Paul McFadden

Nazi Germany and Mathematics

Mathematics and People


have found the problem a tantalising prospectand conflicting claims were soon to lead to a lessthan scientific squabble. Urbain Le Verrier com-pleted a mathematics degree before beginninghis studies and a career in chemistry. A failedjob application later and he found himself as atutor at l'École Polytechnique specialising in ce-lestial mechanics. Initially he focused on the mo-tion of Mercury, but by 1845 he had turned hisattention to Uranus. Initially he carried out anextensive series of calculations, concluding in apaper of November 1845 that there was no wayof accounting for its disturbed orbit by theknown planets. By June of 1846 in anotherpaper he had concluded that another planetmust exist which accounted for the residual ef-fect on Uranus’ orbit, giving an approximationof its distance from the sun and a calculation ofits longitude. Finally, in August he was able topublish a paper with a predicted orbit and massfor the new planet, sent his result to Galle at theBerlin Observatory who received it on 23rd Sep-tember and that very night sighted the predictedplanet, within 1 degree of the theoretical predic-tion. As Arago put it, Le Verrier had discovereda planet "with the point of his pen", Neptunewas revealed and from then until 1930, andagain from 2006 onwards, the solar system’splanetary company was complete.

Though there was no question of who hadultimately triggered the discovery the Englishscientific establishment was eager to claim creditfor the discovery (a second planet in under acentury would have been quite a feat). IndeedLe Verrier’s June paper had triggered a lastminute push, coordinated by Airy, to beat himto the punch, but the method used was timeconsuming and Challis’ outdated star-map ren-dered the search futile. Only after hearing of thediscovery did they realise their search had twicespotted the planet but not recognised the signifi-cance. In spite of the acrimony between some ofthe characters involved, Adams himself ac-knowledged the priority claim of Le Verrier,providing ample demonstration of the saying“publish or die” as a planet might just hit you.

Michael Stringer

For thousands of years man has been aware ofplanets beyond our own: Jupiter, Saturn, Venus,Mars and Mercury, each taking the names ofdeities from the Greek Pantheon though known ofwhen that very construct was an inkling in the dis-tant future. All could be seen from Earth, observedfrom Rome to Xianyang with the naked eye, andperceived even in that time of more primitive sci-ence as having a decisive effect or influence on ourlives. But this was not the movement of thespheres immortalised in Shakespearean literature,nor was it to be found in the astrological mum-blings of mystics. Rather it lay in the quest to com-plete an ordered picture of the solar system wherethe laws of physics ruled, and to do so took math-ematical artistry and determination the likes ofwhich few possessed.

The road to their unveiling began in 1690when the English astronomer John Flamsteedspotted a point of light which he duly classified asa star. Over the coming decades other as-tronomers made the same observations, and thesame mistake. It took the polymath Sir WilliamHerschel to show them the error of their wayswhen in 1781 he saw the same distant object, be-lieving it to be a comet. He soon realised that whathe observed might not have been a comet after all;taking up his pen he jotted down some calcula-tions to confirm his suspicions and realised thathe had indeed discovered a planet, older than anyof the sons of Adam but a new and ground-break-ing discovery. Uranus had now been unveiled andtelescopes across the known world peered up-wards, what had been to our eyes a bit part of farremote scenery in the cosmic play was now ex-posed as a key player.

From a mathematical standpoint the discov-ery of Uranus was of limited importance, themethods used by Herschel had already been ex-plored with the known planets of the solar system,similarly the characteristics which led him to re-evaluate his initial designation of Uranus as acomet was a result of the pre-existing corpus ofknowledge. However there was a new challengeposed by the discovery, as Joseph Jerome Le Fran-cais de Lalande found in 1782 when he notedUranus following a significantly different orbit tothat predicted. The best mathematical modelsavailable could not account for the variance, in-creasing the data set to include earlier observa-tions of Uranus did little to improve it. They werefaced with a stark choice between appealing to anundiscovered trans-uranian planet and rewritingthe Law of Gravity. The problem remained unre-

solved for decades until the 1820s and early 1830swhen improved measurements, predictions, andUranus’ continued contempt for them, began togive way to a belief that the problem might lie notin the mathematical methods used, or indeed inthe accuracy of observations on the known plan-ets, but in an incomplete data set which omitted akey player from the cast list, the hunt for the latestplanet was on.

Astronomers the world over began to searchthe night skies in earnest, the degree to whichUranus’ orbit was altered gave a starting pointsuggesting a celestial body not exceeding 12thmagnitude. Many came close to discovering thehypothetical planet, some, including Lalande him-self, even observed it but attributed any noted po-sitional variation to errors in their observations oras some other species of celestial inhabitant.

On this occasion it was up to theoreticians tofind the planet which practical astronomers wereunable to pin down. Perturbation theory, amethod which seeks to determine the discernibleeffect of known bodies on other known bodies,had been in use for some time. The basic principlebegins with the solution to a known problemwhich can then be modified through an iterativeprocess, giving a Power Series which quantifiesthe difference between the true value and theknown problem and hence providing the answerto the problem they are trying to solve. Effectivelyit is an application of differential equations, butreversing the method to find details about an un-known body proved far more difficult, leadingsome to believe that the problems may be insur-mountable.

The nature of the problems were twofold,firstly in order to find the missing planet they hadto deduce a method of inverting Perturbation the-ory, taking what they did know about Uranus (theeffect the known planets had on it and Uranus’ ef-fect on them) and then using that to find out aboutthis hidden planet. A comparable problem in na-ture, though certainly not in difficulty, might bereduced to having the result of an integral and at-tempting to find the integral itself, but withoutknown limits how to deal with the constant? Eventhen before finding the solution to their ‘knownunknown’ of a planet they would have to takeaway the iterative ‘add ons’ of the power series toobtain the initial value to their planet’s position.In practical terms there were two problems, thefirst, a limited data series resulting from its fairlyrecent discovery, was eroding by the early 1840sas Uranus neared the completion of its first orbit

since the discovery by Herschel the second, find-ing a mathematician with the necessary insight,ability and tenacity to tackle the problem re-mained, until like buses two emerged in quicksuccession.

The method employed by both was similar;using a formula for the perturbations of mean lon-gitude it was possible to construct an equationbased on three differential equations belonging tothe unknown planet. An assumption is used to ob-tain the fundamental values which are then ap-plied to the formula, after which by changing theform of the equation trigonometric relationshipscan be applied reducing the problem to a matterof eliminating the unknown values from the equa-tions. Having reduced the equations it is then pos-sible to use approximations to obtain a value for

, which can be improved by repeated approxi-mations, with values for the longitude, eccentric-ity and mass of the planet can be obtained.

John Couch Adams first encountered theproblem in 1841 after reading a paper on the sub-ject of Uranus’ anomalous orbit by George Airyand resolved to begin an investigation, “as soonas possible after attaining my degree”. Havinggraduated in 1843 decorated with the assortedlaurels of Cambridge, including the Smith prizewon by Lord Kelvin two years later, he was finallyfree to begin his work and set about making thenecessary calculations. By September of 1845 hav-ing found repeated iterations were making littlechange, he communicated the result to ProfessorChallis, in October he door stepped the As-tronomer Royal, Airy, twice, missing him on bothoccasions and leaving a written account of hisfindings. After the discovery of Neptune he wasable to further refine his work resulting in the ver-sion published in the eventual, though belated,paper “On the perturbations of Uranus” as an Ap-pendix to the Nautical Almanac.

However Adams was not the only one to have

i

Discovery


Finding Neptune

In 1989 Voyager 2 was 12 years into it’s mission as it passed Neptune,transmitting this startling image back to Earth. However 143 years earlierLe Verrier had discovered the planet "with the point of his pen".

It is often said that maths and absolute knowl-edge go hand in hand; indeed, it is often claimedthat the theorems of mathematics are the onlythings whose truth we can be undoubtedly sureof, with all other ‘facts’ being temporary or am-biguous. From the ancient days of Plato andPythagoras, to the revolutionary times of the En-lightenment, maths has always been admired forits truth and its beauty. But how certain can weactually be of the claims of the absolute truth ofmaths? We must look to its foundations, and seethat they are built solidly, consistently, and with-out errors; otherwise, the whole of maths couldcome crashing down around us.

This was a task faced by mathematicians atthe beginning of the 20th century. An attempt atformalising the foundations of maths, purely interms of sets, had been made by Georg Cantor atthe end of the 1800s, however this was shown tobe inconsistent; it had not been formulated rigor-ously enough. Cantor defined a set to be any col-lection of objects of our imagination, and allowednew sets to be created out of old in various ways,such as taking unions and intersections.

However, in 1901, Bertrand Russell showedthat this model for the foundations of mathemat-ics was not consistent. Russell devised a set, X ,consisting of all sets which are not members ofthemselves. A contradiction arises, as it is easy toshow that X both is and is not a member of itself.Clearly, a different approach was needed.

Throughout the early years of the 20th cen-tury, what is now known as axiomatic set theorywas developed. An axiom is essentially just a rulewhich is assumed to be true, or self-evident. Ax-ioms are commonplace in mathematics: from Eu-clid’s axiomatisation of geometry in 300 BCE, tothe axioms required in the definition of a vectorspace. Axiomatisation is a necessary procedure, aswhen trying to prove something, we cannot justcontinually break down what we are doing intosmaller and simpler steps; eventually, we muststop, and accept that (or decide if) the step we aretrying to do is true.

By 1922, after much debate, a list of nine ax-ioms was produced; these were the axioms uponwhich the entirety of mathematics would bebased. They are known as the ZFC axioms (as theywere primarily developed by Zermelo andFraenkel, and include something known as theaxiom of choice), and are still accepted as thefoundation of mathematics today: any proof youdo can be broken down into a (probably verylong) chain of applications of these nine axioms.

Having been let down by set theories before,much scrutiny was made by logicians over theconsistency of the axioms. An axiomatic set theoryis consistent if no contradictions can be derivedusing its axioms. Unfortunately, in 1931, KurtGödel showed that it is not possible to provewhether or not the ZF axioms (that is, the ZFC ax-ioms but with the axiom of choice removed) areconsistent or not. It has, however, been provedthat if the ZF axioms are consistent, then so are theZFC axioms (that is, adding the axiom of choiceto our list does not introduce any contradictions).It is, however, widely believed that ZFC is consis-tent; if it were not, it is thought that a contradic-tion would have been discovered by now.

Another problem we have to deal with is howto be sure that we have picked the ‘right’ axiomsto be part of our theory. What criteria are we usingto decide if an axiom is ‘self-evident’? For in-stance, it is fairly clear that if two sets, A and B,contain the same members as each other, weshould want it to be true that A=B. And, indeed,an axiom saying just this is on our list of nine. Butthere are other axioms, whose truths are not so

self-evident, which also form part of our list.One such axiom is the previously mentioned

axiom of choice, whose inclusion along with theZF axioms was hotly debated for decades. Objec-tions were made, since many found the truth ofthe axiom not to be intuitively obvious, and, infact, if accepted in the theory, the axiom of choicecan be use to prove true statements which somemathematicians call ‘undesirable’. An example ofsuch a result is that it is possible to decompose arigid, geometric sphere into a finite number ofpieces, then rearrange them so you get twospheres of the same size as the original. This isknown as the Banach-Tarski paradox. That beingsaid, it is not really correct to claim that one of ZFand ZFC is ‘right’ and the other is ‘wrong’ – theyare simply different theories, starting from differ-ent assumptions. Mathematically, it is possible towork with any consistent set of axioms, and seewhat we can prove from them. Strictly, every the-orem stated should begin with “If the list of ax-ioms I am working with is assumed to be true,then…”, however it is usually assumed we areworking in ZFC, and this is thankfully omitted!

There is a another, more crippling problemwhen trying to formulate a rigorous foundationfor maths, as was discovered by in the 1930s, byGödel. The results, known as Gödel’s incomplete-ness theorems, say that all consistent axiomaticsystems are incomplete, meaning that there arecertain true statements whose truths are unprov-able using the rules of the system. The outlook isperhaps a little bleak for us, then: either ZFC is in-consistent (and so, in fact, every proposition canbe proved true, such as 1+1=3), or ZFC is consis-tent, but incomplete, and so some true statementsare undecidable by the axioms. This is not a flawinherent to the ZFC theory itself, but one whichafflicts any sufficiently powerful axiomatic sys-tem. We can either prove too much, or too little (inthe inconsistent and incomplete cases, respec-tively), and so any attempt at formulating a com-plete, consistent description of mathematics isdoomed to failure.

As much of a blow as this is to mathematics,it is a fact that is intimately tied to the use of anaxiomatic system. All hope is not lost, however;we still have (what is believed to be) a consistentlist of axioms, from which every result you haveever been taught, and ever will be taught, hasbeen proved. It is simply an immutable part ofmathematics that there are certain statementswhose truth cannot be decided upon. There are,in fact, already known examples of statementswhich cannot be proved in the ZFC theory: inter-ested readers should investigate the continuumhypothesis and inaccessible cardinals. Despitethese setbacks, thanks to the work of mathemati-cians which started over a century ago, we are lefttoday with a theory of mathematics which is asrigorous, consistent and complete as it can be –and at the end of the day, what more can be askedfor?

Neil Fullarton

Foundations of Mathematics

Assorted Articles

The purpose of this article is to see how ideas fromdifferential equations can be used to generate a spheretiling.

To do this we consider an equation know as the hypergeo-metric equation:

Here the independent variable z is consideredto be a complex number that we also allow tobe infinite. This equation has 3 singularpoints at 0,1 and . These can be thoughtof as point where the coefficients of theequation become infinite. This is easyto see if we divide through by thecoefficient of the second derivative.

The solutions to this aboutthese singular points are given interms of Gauss’s hypergeometricseries:

whereis

called the Pochhammer symbol. Thevalues and are complex con-stants called parameters.

In order to produce a sphere tiling weneed to construct the solutions about thesingular points. They are:

at the point 0,

at the point 1 and

at the point . It is also helpful to introduce the new param-eters, called angular parameter, that we can define from and

in the following way

It is now possible to define triangle maps. These are con-structed as the ratios of the solutions at each singular point i.e.

with taking one of the values 0,1 or . In thecase that the angular parameters , and

have absolute value between 0 and 1 (thatis they lie within unit distance of the ori-

gin in the complex plane) the maps the upper half complex plane to

a triangle on the surface of a spherewith the edges being circular arcs.The points 0,1 and are mappedto the vertices of this triangle. Itsangles are , and .

In the special case that we canfind integers such that theangular parameters can be writ-ten as:

Then this triangle will tile the sphereas in figure 3.

The tiling is achieved, like anyother, by taking copies of the triangle in

the above image and reproducing it allover the surface. Hence it is possible to tile a

sphere with a differential equation.

Stuart Andrew

( 1) [ ( 1) ] 0.z zdzd w c a b z dz

dw abw2

2

- + - + + - =

, ,p q r1 1 1m n o= = =

, ,p q r

rornrm

3

()s zk

vnmk

( ) ( 1)( 2) 1)a a a a nn = + + + -

,a b c

()()()

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( )

( )

kk

k0

1

=

.a bo = -c a bn = - -1 cm = -

() ( , 1,1 | )w z z F b b c b a z( ) b1 1= - + + -3- -

() ( , ,1 |1 )w z F a b a b c z( )10 = + + - -

() ( 1, 1,2 | )w z z F a c b c c z( ) ( )c01 1= - + - + --

() ( , , | )w z F a b c z( )00 =

() (1 ) ( , ,1 |1 )w z z F c a c b c a b z( ) ( )c a b11 = - - - + - - -- -

() ( , 1,1 | )w z z F a a c a b z( ) a0 1= - + + -3- -

( , , | ) ( ) !( ) ( )

F a b c z c na b

zn

n n

n

n

0= 3

=/

3

3,a b

c

3

To illustrate the benefits of mathematicstwo examples will be used. The first will showhow pure mathematics can be used in an ap-plied setting with obvious benefits and the sec-ond will come from applied mathematics witheffects on pure mathematics.

The first example comes from graph the-ory. A graph is a graphical model that showsrelationships between collections of objects. Soas in figure 1 each circle represents an objectand two objects are related to one another if aline joins their respective circles. The problemlinked to a graph such as the one in figure 1, byasking it is possible to colour the graph withtwo colours in such a way that no two con-nected circles have the same colour. If this can-not be done with two colours what about threeor four and so on. The first attempt to colourthis with two colours is bound to fail: the circleslabeled A, B and H are connected in a triangleso if we pick A to be red and H to be green,then C cannot be either red or green. A thirdcolour must be used, say blue. Now the graphcan be coloured as shown in figure two. Sohow does this related to a non-mathematicalarea? One of the classic examples is that ofexam timetabling. Table 1 shows the choiceseven students, labeled 1 - 7, make out of a se-lection of 8 exams, labeled A through to H.Some students are sitting three exams andsome are sitting two. The problem is to mini-mize the number of times an exam hall isbooked whist making sure there are no examconflicts, i.e. no student has two exams bookedat the same time. For this, convenientlyenough, the graph of figure 2 works as a repre-sentation of the problem. Each circle repre-sents an exam as given by its letter, and theconnecting lines show if exams cannot be set atthe same time. For example, student two is tak-ing exams B and D so their circles are linkedand student seven is taking exams B and C sothese are linked. Now the colouring takes on ameaning: each set of coloured circles representsa non conflicting booking, and this set of book-ings is known to be the most efficient since thefewest number of colours were used.

This example was chosen to give a quick il-lustration of how graphs are applied; there aremany more applications ranging from map-ping internet sites to tracking the spread dis-eases to understanding molecules at an atomiclevel. It must be noted that pure mathematicsis a huge subject; graph theory is one area andthis was one application.

For the applied mathematics student bene-fits of study can usually be derived from exam-ples given in courses and sometimes directlyfrom course titles. For example, everyone read-ily sees the importance of studying fluid me-chanics; it is important to understand the

mechanics of piping gas from Russia to the UKto keep everyone warm with no leaks or explo-sions along the way. Similar examples can befound for subjects like Newtonian Mechanicsor Mathematical Biology. So instead of givinga detailed example of an application of appliedmathematics this article will show that tech-niques of applied mathematics can be used inthe study of pure mathematics, which in turn,as shown above, can be used to study appliedproblems.

For this example the reader must be famil-iar with the idea of a 3-manifold. A manifoldis a space that looks similar to a Euclideanspace on a small enough scale. So, for example,the earth is a 2-manifold as locally (on a smallscale) it looks like a flat 2-dimensional Euclid-ean surface but globally is a 3-dimensionalsphere, where the rules of geometry are differ-ent. 3-manifolds are defined analogously, as anexample the reader may know that locally theuniverse seems like a 3-dimensional Euclideanspace but globally it may be something differ-ent.

It is known that there are eight classifica-tions of 3-manifolds. The distinction betweenclassifications can made by studying the non-Euclidean global structure: specifically, bystudying paths of shortest distance in the space(here the space is 3-manifold). If the space isdefined in an appropriate way it is possible todifferentiate along a path between two pointsto give the velocity and differentiate again toget the acceleration along the path. The studythen begins resemble a mechanics problem andin fact with only a little further manipulationtools from Newtonian, Hamiltonian and La-grangian mechanics can all be used to studythese abstract spaces.

Although the above illustration of appliedtechniques in pure mathematics is not as ex-plicit as the graph example, it is hope thereader has still gained a sense of the intercon-nected nature of mathematical techniques.And to say it explicitly: The implication is thatstudying one area of mathematics can have ap-plications to subjects far from the original areaof study.

This article now takes an inevitably morepersonal stance as it addresses the second in-terpretation: why do students enjoy mathemat-ics?

Mathematics can be very hard. Studentsare given problems to which they can applytools they have learnt in lectures or from booksand they can compare similar questions to try tosolve the problem. Most problems are attemptedknowing that there is a solution; the student justhas to find it. And this can get frustrating. Hourscan pass on one question with only dead endsreached. Problems like these are hard. So wheredoes the satisfaction come in? Personally it is from

two points. The first is seeing the solution, evenif it is explained by someone else, and the secondcomes when revising the problem at a later stage.

Assorted Articles


Tiling the sphere using differential equationsWhy study mathematics?

figure 3 a single tile on a sphere

A problem that has had hours spent on it to noavail months before can, if the solution was fullyunderstood, be conquered in minutes. Gaining agood understanding of something that once seemsreally hard is very satisfying.

The next reason is concerned with exactlywhat mathematics is. It is clearly a science, butit is different from other sciences in that it hasa fascinating range of areas. Mathematics hasmodels that explain the motions of stars, theworld of finance, how the many types of num-bers interact, there are even mathematical mod-els that can predict parts of your probablefuture and some that can represent distances ofinfinity in a circle of radius 1! Maybe this canbe summarized by this fun argument:

1. “God ever geometrises.”- Plato2. “Geometrical properties are character-

ized by their invariance under a group of trans-formations” – Felix Klein

3. “If Plato and Klein are correct, then Godmust be a group theorist.” – Stewart and Golu-bitsky: Fearful Symmetry

To study mathematics is to study almost anything.

This article finishes with a look at one finalapplication of mathematics: to the arts.

During the Italian Renaissance, LeonardoDa Vinci wanted to create paintings that lookedas real as possible. He is quoted as saying “Themost praiseworthy form of painting is the onethat most resembles what it imitates.” (Onemight think of mathematical models). To gethis paintings as life like as possible Da Vinci de-veloped a mathematical system known as lin-ear perspective and the notion of a vanishingpoint that allowed him to create the illusion ofdepth on a flat 2-dimensional canvas. To showhow important mathematics was to Da Vinci,consider his comment: “Let no man who is nota Mathematician read the elements of mywork.” Da Vinci obviously had a passion formathematics, but not all painters have neededthis. In fact if the work of Escher is consideredthen it is clear mathematical training is not al-ways needed to draw even advanced mathe-matical ideas. It appears that the Dutch artisthad formal no mathematical training, yet hisprints caused quite a stir in mathematical com-

munities as they represented hyperbolic tessel-lations, as in the Circle Limit series, or Rie-mannian surfaces, as in print gallery. Throughthis work Escher ended up friends with the fa-mous geometer HSM Coxeter. And as the artsare discussed it must also be noted mathemat-ics has been used extensively in literature. Forexamples see Lewis Carrols “Alice in wonder-land” and ”Through the Looking Glass” orYevgeny Zamyatins “We.”

Mathematics can be studied to gain a betterunderstand of problems in the world, it can bestudied by those who enjoy creating soundlogic, or who enjoy solving problems, it canalso help in work outside of maths such as thearts.

Andrew Bestel

A common question for undergraduates of mathematics to encounter is why do they study mathematics? De-pending on how this is asked the question could mean what are the benefits of mathematical results and their ap-plications, or alternatively, what is it about the subject that the student enjoys. This article is an undergraduatesoffering of an (incomplete) answer to both interpretations of the question.

Exam PaperA B C D E F G H

students

1 X X X2 X X3 X X4 X X X5 X X6 X X7 X X

table 1

figure 1

figure 2

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