[The, Commutator] Vol.1 Issue1

Games and puzzles

[The,ComThe Macsoc Magazine: FEBRUARY 2010

Kakuro

Kendoku

Fill in the boxes with the integers 1,2,...,9 so that each integer only occurs once in each string and the add up to the number adjoining that string

Try and fill each row and each column with the integers 1,2,3 and 4 exactly once so that the numbers in each nest can be combined under the designated product to equal the prescribed number.

Then try it with the integers 1,2,...,8:

FIRST EDITION

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

“Proof” 1

“Proof” 2

x= 0 ‘ x(x - 1)= 0 ‘ x- 1 = 0 ‘ x= 1 ‘ 1= 0

0 = 0 + 0 + 0 + 0f = (1 - 1)+ (1 - 1)+ (1 - 1)+ (1 - 1)+ f = 1 + (- 1 + 1)+ (- 1 + 1)+ (- 1 + 1)+ f = 1+ 0 + 0 + 0 + f = 1

Sodoku

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

Easier

Difficult

7 1 3 9 7

5 1

Sponsored by: The Department of Mathematics and the IMA

2

6 1 2 9 9 5 6 8 5 6 3 1 9 4 9 8 5 2

2 9 8 3 1 7

9 7 5 4 6 9 7 2 3 6 6 4

1 5 3

5 1 4

February 2010 [The , Commutator] 16

Contents News 2 Editorial News Features

4

Group theory

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

6

Editor, Designer, Graphics Carl Chaplin Assistant Editors Stuart Andrew Andrew Bestel Those who were also actively involved in the organisation David Halfpenny Emma Cummin Sandy Black Fiona Doherty Louise Ogden Paul McFadden Michael Stringer Neil Fullarton Jaspal Puri

Mathematics & People 10 Are men really better at maths than women?

11 Nazi Germany and Mathematics Discovery

12 Finding Neptune How Neptune was discovered to exist, its mass and orbit calculated before it was observed

Assorted Articles 13 Foundations of Mathematics 14 Why study Mathematics?

15 How to tile the sphere with differential equations Games 16 games and puzzles 2 [The , Commutator] February 2010

University Restructuring and You! The Faculty of Information & Mathematical Sciences (FIMS) is soon to be replaced by the School of Mathematics and Statistics under the new college structure that will supersede the old faculty structure from 1st August 2010.

The most notable affect for students will be the combining of the existing FIMS graduate school with that of the Faculty of Chemistry, Faculty of Engineering and the Faculty Physics and Astronomy. The existing faculty based graduate schools will form a single graduate school for the College of Science and Engineering.

Students were informed of the University wide changes by the Senior Vice Principal and Deputy Vice Chancellor Andrea Nolan, together with the promise of zero disruption while the changes are implemented.

A few words of thanks, a brief outline of what [The, Commutator] is about and then a definition of the Commutator.

Undergraduate students from August will be awarded their degree by

their college instead of their faculty but there will be little change in their day to day experiences.

The main changes will be in the University Financial structure which was previously comprised of ten budgets, corresponding to nine faculties and the University Services. The new structure will be five larger budgets, corresponding to four colleges and University Services. The University has risen to within the top 20 in the UK rankings and within the top 100 in the world rankings. To carry on this trend Glasgow needs to respond to the increase in demand for inter-disciplinary academic research and this restructuring of University finances is designed to facilitate this. Currently research over faculty boundaries requires multiple research grants and so there is unnecessary resistance and less coordination than can be expected from a single college research fund. Fewer budgets also allow for greater efficiency as there should be fewer transaction and co-ordination costs and delays. The university as a whole expects to be more agile in its re-

Thank You My first duty as editor is to thank all those involved in the project. I have sometimes been frustrated with the level of indifference among undergraduates and I have to say that it has been very refreshing to meet and work with such a lively and proactive team here in Glasgow. This enterprise has been built on their enthusiasm it is to them that I am the most grateful.

I would also like to thank the Glasgow University Department of Mathematics and the Maclaurin Society for their full support both in terms of time spent organising and advising as well as their financial support. It is their financial support that has allowed us to print the magazine in full colour with a professional finish.

The Macsoc itself and by association this magazine, is funded by the The Institute of Mathematics and its Applications (IMA) which is the UK's learned and professional society for mathematics and its applications. It promotes mathematics research, education and careers, and the use of mathematics in business, industry and commerce.

What’s it all about?

Our vision for the magazine is to have a publication that first and foremost communicates ideas which have sparked an interest in the writers. With this both the understanding and the enthusiasm benefits the reader. Our aim is to create something which is both a Vista for first and second year students who are yet to decide on which disciplines will make up their degree and also for honors students to broaden their knowledge using the lecture courses provided at Glasgow University as a foundation.

The publication also serves to promote discussion on all aspects of Mathematics be it sociological, psychological, philosophical or historical. Mathematics has affected everyone’s lives both within and outwith the department and so, although not central to mathematical research objectives, there are many areas of discussion should be of interest to us all.

For the writers themselves we hope to provides a platform for undergraduates to be involved in an extra curricular pursuit where they can develop skills outside of those that are offered as part of the undergraduate programme.

What is the commutator?

For an outline of group theory see the following feature article. The commutator of two elements a,b of a group G is denoted by [a,b] and is defined as follows: [a,b]:= a- 1 b- 1 ab. An important feature of a Group is commutativity, that is the extent to which ab = ba. Of course we take for granted that addition is always commutative and that many of the multiplicative groups we encounter day to day do not impose an order effect on our calculation. The savvy shopper would have to add to his whit the ability to position each item of his weekly shop on the conveyor belt at the supermarket check out in the cheapest possible order in a world without commutativity. As an undergraduate you will encounter matrix multiplication as a first example of a non-commutative group operation. However David Halfpenny provides an example of the rubik’s cube in his article on page 5. If you rotate some of the pieces about one axis and then perform a rotation about a different axis the rubik’s cube will end up in a different configuration to that achieved by performing the same two rotations in reverse order, because these two group elements which act

Chaos theory

An introduction to a theory which reveals the deterministic equations which govern the seemingly random and probabilistic events.

News

Word from the Editor

“The needs of our students are central to the activity of the University and the restructure has not, and will not, alter that”

sponse to external opportunities and threats, while larger budgets should make larger investments more feasible.

For those undergraduates who are planning a future in academic research, the benefits of the restructuring will be immediately apparent as heads of colleges will have larger budgets and more administrative support. The University’s standings in both the UK and the World rankings should reflect any increase in the volume or quality of research while the College of Science and Engineering should hold more weight than the individual faculties it replaces.

As such all undergraduates should feel an increase in the benefit that graduating from Glasgow University will bring.

Carl Chaplin

on the cube do not commute with each other in the Group.

Taking the set of all commutators of a group we find that each pair of elements that commute with each other are sent to the identity element by their commutator and we are left with the set of commutators which don’t commute with each other. Taking the subgroup which is generated by this set tells us something about the extent to which that group commutes. This is called the derived subgroup and it is an important tool in the understanding of the structure of a group.

It is with this that we go straight into our first feature which starts with some basic group theory and goes on to demonstrate the application of group theory to the understanding of the rubiks cube.

Carl Chaplin

February 2010 [The , Commutator] 3

Feature Group Theory

THE GROUP

Mathematics of the Rubik’s cube

PUORG EHT

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

What is a group?

Much of the mathematics you will have learnt at school and in the first few years of your university career will have involved manipulating formulars and algebraic expressions involving unknowns. Abstracting from this we formalise the idea of dealing with elements which are somehow compatible under some sort of operation. Vague so far, but all avenues to the world of the abstract are vague, deliberately so. Once you have a foothold you can begin to admire the power you gain when you liberate yourself from the specifics of any particular example.

We therefore start with a set of elements such as the integers and we define a binary operation which acts on a pair of elements of this set to generate another element: f(a,b)= c. However we choose to construct an operation it would be useful if putting two elements in gave us something which is again in the set of interest. For example if we use * to represent our chosen operation, 5 * 23 = Blue would be very unhelpful to us because blue isn’t an integer. It is however liberating to know that you could construct a binary operation that would act on a subset of the set of all colours, we only illustrate abstract algebra with numbers as this is a context which the reader will be familiar with. Let’s define our binary operation on the set S by the function: fS : # S " S and we denote f(s,t) by s * t.

You will notice immediately that addition of the integers can be considered in this way: g:Z # Z " Z ,g (s,t) = s + t.

One inportant feature of an algebraic object such as a group is associativity. This is simply the requirement that for all a,b,c in our set: (a * b)* c = a * (b * c). This simply means that evaluating a * band then taking *c of this yields the same result as evaluating b * c and then taking a *. This seems like a technicality at first but actually it is a necessary axiom, without which we can’t prove the simplest of results, as you will see. The associative axiom will become your best friend as you reed on.

From the integers we also observe a very special element which we call the identity element.

Under addition this is zero but if we consider the set of rationals with zero excluded under the binary operation of multiplication the identity element is 1. You might ask: ‘what these two elements have in common? ‘ and by way of an answer they both play the same role in the context of their respective binary operations. Formally e is an identity element if for all s in S , e * s = s * e = s.

Finally we can think about the expression s * t as t doing something to s and we arrive at a new element, but there ought to be some mechanism for getting from (s * t)back to s and so we introduce the notion of inverses. That is for each -1

t in S there must exist t -1

which satisfies:

-1

t* t = t * t = e.

Together with our definition of e and associativity we get: -1

-1

(s * t)* t = s * (t* t ) = s * e = s.

If we furnish a set with a binary operation and

impose the axioms of associativity and inverses we get a group.

Some elementary group theory As a first step towards studying different types of groups, once we have a group we can take subsets of the original set which are themselves closed under the group operation and we call such a subset a subgroup. For example the even integers form a subgroup of the integers under addition since addition of even integers always yields another even integer.

Once we have more than one group we can consider maps which preserve the group operation. This is not as straight forward as it sounds as we may have a map between two groups which have two completely different operations such as addition in one group and multiplication in the other. We need functions between two groups where it doesn’t matter whether you perform the group operation on two elements in the first group and then take the image of the mapping or take the image of two elements and then take the product under the second group operation. When a function obeys this simple rule we call it a group homomorphism and an interesting example of a group homomorphism is the natural logarithm.

The natural logarithm takes as its domain the group of non-zero positive real number under multiplication (denoted by (R 2 0,# )) and its codomain is the group of all real numbers under addition (denoted by (R ,+ )). This homomorphism can be thought of in the conventional sense

figure 1 the natural logarithm using the graph in figure 1 but its significance can be appreciated by the fact that it is a group homomorphism. In fact this particular homomorphism helped put man on the moon. The point being that multiplication is labour intensive compared to addition and so before the advent of computers and pocket calculators, slide rules were used to reduce a multiplication down to a simple sum. Suppose we needed to find the product of two elements in (R 2 0,# ) , we would first use the logarithm function to find the two corresponding elements in (R ,+ ), we would then perform the operation in this group (i.e. add them) and then we would perform the inverse map to get back to an element in (R 2 0,# )which would correspond to the product of the two elements we started with.

Immediately we have a practical application of some elementary group theory, all be it that now this happens inside a computer where once we had to do it for ourselves. We can go further though, the map is injective, that is no too elements are mapped to the same point; further it is surjective, every real number has a corresponding real number greater than zero under the inverse of the natural logarithm. A homomorphism which

4 [The , Commutator] February 2010

is injective and surjective is called an isomorphism. A fancy word you might think but it is quite deserving of its own fancy title since it tells us that these two groups are structurally the same.

That is going between the two is as simple as renaming the elements and doesn’t require us to change anything structurally.

It is from these humble begins that one of the greatest mathematical achievements of the 20th century was made, that is to classify all finite simple groups. It was a collaborative effort which took up more than 10,000 journal pages to document.

What do groups do?

In nature we observe symmetry in all life, from the geometry of celestial bodies down to the quantum particle. As such the study of and ability to model symmetric systems is extremely important to physicists and chemists alike.

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

The simplest illustration of this is the Dihedral group on n elements, D 2n . For example D 6 can be represented as the symmetries of an equilateral triangle. Its elements are the two rotations (by integer multiples of 2r /3) and three reflections and its identity element is the element which doesn’t move the triangle at all. We take the binary operation of two elements to be the application of one element to the triangle followed by the other.

Clearly by geometry this group is closed under this operation since we always recover our triangle, it is just the corners which are permuted. If we restrict this set of isometries of the plane to just reflections or just rotations we observe two naturally occurring subgroups.

By way of example, quantum mechanics showed that the elementary systems that make up matter, such as electrons and protons, are truly identical, not just very similar, so that symmetry in their arrangement is exact, not approximate as in the macroscopic world. Systems were also seen to be described by functions of position that are subject to the usual symmetry operations of rotation and reflection, as well as to others not so easily described in concrete terms, such as the exchange of identical particles. Elementary particles were observed to reflect symmetry properties in more esoteric spaces. In all these cases, symmetry can be expressed by certain operations on the systems concerned, which have properties revealed by Group Theory.

Carl Chaplin

figure 2

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

Invented in 1981 by professor of architecture, Erno Rubik, the Rubik’s cube is the best selling puzzle game in history, selling over 350 million cubes since its invention. Nowadays there are many types of Rubik’s cube with all kinds of dimensions. We will, however, concentrate on the traditional 3 # 3 # 3 structure. This type of cube has the usual 6 faces, on each of which are 9 distinct squares, each of which have one of 6 prescribed colours. The cube is built in such a way that each 1 # 3 # 3 face has the ability to rotate in a clockwise or anticlockwise direction. The purpose of this puzzle is to perform a certain algorithm of these rotations to a randomly mixed up cube so that all the squares on each face of the cube are of the same colour.

Notation There are many texts available containing solutions to the Rubik’s cube. Perhaps the most popular of which is "Notes on Rubik’s magic cube" by David Singmaster. We shall use notation developed in this text.

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

f (front), b (back), l (left), r (right), u (up) and d (down).

Using these denotations, we can refer to the corners and edges of the cube. For example, fdr corresponds to the corner located on the front face at the bottom right hand corner. Similarly, blcorresponds to the edge on the left side of the back face. So the first letter specifies the face and the following letters give the position on the face.

Let M be a move on the Rubik’s cube. Then we can define the product of any two moves M 1,M 2 , to be M 1 * M 2 where the move M 1 is performed first and then M 2 . Moreover, if we denote G to be the set of all moves on a Rubik’s cube, G is in fact a group under the operation *, as follows: Let M 1,M 2,M 3 ! G . Since M 1 and M 2 are moves then M 1 * M 2 is clearly also a move so G is closed under the defined operation. * is associative M 1 * (M 2 * M 3) = (M 1 * M 2)* M 3 , since:

and the inverse of a move is simply the reverse of the original performed. The identity element of G is simply not performing a move. Thus G is a group.

The cube itself contains 20 permutable pieces, namely 8 corners and 12 edges. In fact, the set of moves that permute the corners of the cube is a subgroup of the group G . Similarly, the set of moves that permute the edges of the cube is also a subgroup of G . These subgroups will be denoted by C and E , respectively.

As stated earlier, the individual moves on the Rubik’s cube are the rotations of each face. From this, clearly our group of moves, G , is generated by these r /2 clockwise rotations of each face.

These generators are denoted by F,B,L,R,U ,D , each corresponding to the obviously appropriate face. Moreover, we can represent each generator using cycle notation. Consider a generator F . If we apply the generator F to a corner on the front face, fur, say, it is permuted such that the fur corner goes to where the fdr corner was originally situated. Similarly, under F , the fdr corner goes to where the fdl corner started, see figure 2.

It follows that we can represent the permutations of the corners of f, under the generator F , as the cycle (fur fdr fdl ful). Similarly, for the edges we have (fr fd fl fu). Moreover, since the corners and edges are the only permutable pieces is the cube, F can be represented by these to disjoint cycles, i.e.

F = (fur fdr fdl ful)(fr fd fl fu) 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 is true that we can regard every element of C as a permutation of S8 , and every element of E as a permutation of S12 . So, if we label each of the corners of the cube by integers in {1,g ,8} as in diagram 2. We are able to show that there exists a group homomorphism: r C :G $ S8 , defined by r C (M ) " v .i.e.

figure 3 and, U ,R ! G ,r C (U * R) = (14872) r C (U )$r C (R) = (1432)(3487).

Notice that (1432) and (3487) are not disjoint, so upon putting this into disjoint form, we have: (1432)(3487) = (14872).

Thus r c is a group homomorphism. This logic is also true for r E:G $ S12 . Note also that elements of the group, can be considered as permutations of S20 with a corresponding group G homomorphism defined similarly as with C and E. The commutator of moves M 1,M 2 in G , [M 1,M 2]can be used to permute a small number of pieces on the cube whilst leaving the majority unaltered. This is an important sequence as it allows us to change unsolved parts of the cube without disturbing other pieces that have already been solved. The commutator is very effective in solving corners and edges of the cube as they usually lie in the intersection of two faces and are therefore open to permutation under the commutator of two generating moves, provided that these generators correspond to faces adjacent to each other.

As one can see, the connection between the Rubik’s cube and concepts in group theory is very strong. For more information on the connection between puzzles such as the Rubik’s cube and group theory, see texts such as "Adventures in group theory" by David Joyner and David Singmaster’s "Notes on Rubik’s magic cube".

David Halfpenny

r C (M 1 * M 2) = r C (M 1)r C (M 2) Note that if this is true for any two generators of G , it holds for all elements of G . So for

February 2010 [The , Commutator] 5

Feature Chaos Theory in this area must be the roulette wheel.

The roulette wheel is nowhere near as “random” as large scale casinos would have you believe. In fact, a bouncing roulette ball obeys exactly the same laws of physics as a bouncing tennis ball. So why is it so difficult, neigh on impossible, to predict where the roulette ball will land? Predicting the motion of a tennis ball seems simple enough and generally speaking a minute inaccuracy in the measurement of the initial conditions (such as the height dropped, the angle, air resistance… etc.) will result in a very small inaccuracy of prediction.

The ball may land only a few millimetres from where it was expected to land. A roulette ball, however, is involved in a chaotic system. This means

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

Chaos theory is a radical, topical scientific discovery. It is not only a prolific field in mathematics but has proven to be applicable in a staggering number of areas and provides us with new tools in which we may explain and interact with the world round about us.

What is Chaos Theory?

Popular perceptions of Chaos theory most notably include ‘the butterfly effect.’ The idea is that a tiny influence such a single butterfly flapping its wings can cause a seemingly disproportionate outcome, that is, tornado at the other side of the world. This indeed is the idea at the very heart of Chaos Theory; that minuscule differences in ‘initial conditions’ can have drastic consequences on a final result (Chorafas, 1994).

Chaos theory can be applied to describing the behaviour of certain dynamical systems which are functions of time. Uniquely in a chaotic system, although behaviour appears random and unpredictable the systems are actually deterministic. To understand that even small, seemingly unimportant alterations in the initial conditions change the entire outcome of a system is to understand the basic principle of Chaos theory.

The Discovery of Chaos Theory Chaos theory was discovered in the 1960’s by a collaboration of scientists who, against the grain, considered these unexpected results and ideas. However the discovery of the theory is popularly attributed to Dr. Edward Lorenz who coined the phrase the “butterfly effect” and is commonly known as the “Father of Chaos” (Lorenz, 1989). Lorenz had made a weather stimulating programme on his computer where he worked at the Massachusetts Institute of Technology. This very complex program had many complicated equations determining the probability of certain weather forecasts. He would input some conditions (e.g. the temperature, wind speed etc.) and it would generate a graph of the predicted weather. The program was considered very highly and incredibly seemed as though it would never repeat a previous sequence. It was believed that if the exact initial weather conditions were put in to this program, it could infallibly mimic the real varying weather conditions outside. On one occasion as the program was running on the initial conditions that Lorenz entered he decided that he wanted to take a better look at the final outcome of the

terminate at an equilibrium point

approach a limit cycle

The roulette wheel is nowhere near as “random” as large scale casinos would have you believe.

return to the original point giving a closed path

figure 1 The three possible phase paths of a two dimensional system weather. Instead of allowing the program to run he cheated, he read off the values from the earlier stimulation and input these as the initial conditions. Now although the computer would calculate numbers to six decimal places, when Lorenz inputted the values half way through

Conditions for Chaos If we consider Chaotic behaviour in nonlinear systems there is one key condition that must be met for this behaviour to occur. This condition is that the phase space is at least three dimensional. The reason for this is the Poincare-Bendixon Theorem that states, if we are dealing with two dimensional autonomous systems, for example:

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

figure 2 two very different outcomes, with very similar initial conditions the stimulation he rounded them to three decimal places. It transpired this minute change in initial conditions completely altered the outcome of the predicted weather.

The above graph is an example of the kind of divergence that Lorenz would have been presented with. It clearly demonstrates that the small difference in initial conditions makes an increasingly large difference as the time value increases. It is for this reason that weather forecasts are more inaccurate the further into the future one attempts to predict their behaviour.

Struck by the disproportionate effect that initial conditions had on the final outcome of this system Edward Lorenz certainly invented the theory’s most enduring image in 1979, in an address to the American Association for the Advancement of Science entitled, “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” (Lorenz, 1993)

6 [The , Commutator] February 2010

Then the phase path must eventually do one of three things: terminate at an equilibrium point; approach a limit cycle; return to the original point giving a closed path, as illustrated by figure 1.

This means we can know what will happen to these systems, we can in theory predict their behaviour. They are not therefore ‘random’ oscillations (Acheson, 1997).

Lorenz Equations Lorenz 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 simplified model of thermal convection in a layer of fluid.

They are as follows:

xo= 10 (y - x) o= rx - y - zx y o= - 8 z + xy z 3 Where originally: • x is proportional to the speed of motion of the air due to convection.

The

Roulette

Gambling is certainly one area where being able to accurately predict the future results would of course lead to phenomenal successes. Although such a truly accurate system would be any gamblers’ Holy Grail, if it was to be universally available it would destroy the very nature of the practice itself. Gambling was conventionally thought to be predominantly based on • y is a measure of the temperature difference between the warm, rising air and the cool, falling air.

• z is a measure of the vertical temperature difference as you move through the system from top to bottom.

• rthe measure of the difference in the temperature If we find the equilibrium points for these three equations it is evident: x = 0,y = 0,z = 0, is an equilibrium point for all values of r. It turns out that this point is only stable for values of r 1 1. If we now increase r beyond this point we find two new equilibrium points and these turn out only to be stable for:1 1 r 1 24.74. These are the only equilibrium points that exist.

Lorenz, although recognizing the regularity in these systems, was also very aware that these where no ordinary systems. Their sensi-

Wheel

sciences of probability and perhaps even luck. Although there are many tricks top level gamblers employ to increase the likelihood of winning, it seemed hard to imagine that any gambling system was deterministic in any way. That is until Chaos Theory entered the picture. We may ask what kind of systems exhibit these chaotic examples and the most famous one tivity put huge restraints on the traditional ways one may make use of a system. As for the weather the implications of the findings where mixed, although it was now theoretically possible to predicate any forecast given the conditions, it was now even more than before seeming practically impossible to provide the data required to do so. As he wrote in his 1963 paper: “When our results…. are applied to the atmosphere… they indicate that prediction of the sufficiently distant future is impossible by any method, unless the present conditions are known exactly. In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-longranging forecasting would seem to be non-existent.”

(Lorenz, 1963: 130–141) These Chaotic systems, unconventional in themselves, would have to be handled in different more unconventional ways it seems if

that if there is even a minute inaccuracy in the measurements of how high the ball was dropped from, how fast the table was spinning or indeed the dimensions of the table, then the predicted resting place of the roulette ball would be massively inaccurate. Instead of landing just a few millimetres from where it was predicted to land, like the tennis ball, the roulette ball may well end up on the other side of the table. This is a prime example of a chaotic system.

Emma Cummin

they where to aid the project of scientific enquiry.

Even if the success of the theory when applied to a weather model seemed to be curtailed by limitations in other practices (of weather observation for instance), the theory was still groundbreaking. One of the reasons for this was its incredible potential, it wasn’t long before systems exhibiting chaotic dynamics where being recognized across a large number of fields, from mathematical models to biological patterns.

Chris Andrews

February 2010 [The , Commutator] 7

Feature Chaos Theory

The Tent Map

Lorenz system of equations in detail The weather is a complex and unpredictable series of events that occurs everyday and is extremely difficult to model. One important aspect in meteorology, and fluid mechanics, is the problem of convection: the process by which hot air rises and cold air sinks. In our atmosphere we know that the air at the bottom is generally higher than it is on the top. However, the convective process is not completely stable and in some cases there is no convection or complex turbulent motion can occur.

Let T be the temperature of the atmosphere then the temperature difference between the two altitudes can then help us describe the onset of convection and turbulent flows. If we consider a temperature difference to be dt, then if this occurs below a critical value, c then there is no convection, above the critical value steady convection occurs. The onset of convection has been researched and a dimensionless number called the Rayleigh’s number, R a , as described by Lord Rayleigh which s used to determine where the onset of convection occurs. However if the temperature difference is large enough then complex turbulent flows appear and from this poses a difficult question: for what value of temperature T do turbulent flows occur?

This phenomenon was investigated by Edward Lorenz, to which he discovered the Lorenz system of equations which are described on page 6 but will be outlined here in more detail: dx = v x(y - x) dt dy (1) = rx - y - xz dt dz = - bz + xy dt The parameter r in the model is Rayleigh’s number which is denoted as, (2) r = R a /R c where R C is the critical Rayleigh number for which convection occurs, and so in the investigation to where turbulence in the atmosphere occurs it is useful to consider different values of the parameterr .

Analysis of the Lorenz system Onset of Convection The Lorenz system in equation (1) is non-linear due to the xz and xy terms, therefore it is difficult to get an analytical solution and numerical methods must be used. However we can locate the critical points of the system by setting dx/dt= dy/dt= dz/dt= 0.

We can then choose x = y from the first equation in (1) and we obtain, x(r - 1 - z) = 0 2

x - bz = 0

(3)

From the first equation in (3) we can choose either: x = 0 to give x = 0,y = 0

and z = 0 as critical point CP1 , or, z = r - 1 to give x = ! b (r - 1), y = ! b (r - 1) and z = r - 1 as critical point CP2 . We note that the only critical point for r < 1 is CP1 = (0,0,0)since the second critical point gives complex values. Then for r > 1 the critical points CP1 and CP2 are valid. Thus we can see that r = 1 gives a bifurication point and so denotes the onset of convection in our model. This makes physical sense since the value r is taken to be dimensionalised where r = R a /R c where R c is taken to be the critical Rayleigh number.

Chaos 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-dimensional Logistic Map and the one-dimensional Tent Map that shall now be presented in more detail.

figure 1 The values of S are shown in the above diagrams. The first one shows a stable system when r = 14 and the second shows and unstable system when r = 28. A Initial conditions are x = 1,y = 1,z = 1.

Methods of analysis The Lorenz system of equations can be analysed in different pictorial fashions such as time plots, phase-space plots and spectral diagrams. These are useful in examining the different effects of r on the system which may provide some understand into its behaviour.

Time plots Time plots show the way in which a particular value of x,y and z and vary with time t. This generally is a decent method to show when a system develops chaotic behaviour. Usually a time plot consists of a harmonic wave which oscillates with time when the system is stable, however when the system becomes unstable the harmonic wave becomes inconsistent and may ’jump’ as shown in figure . This however is not a decent method to evaluate different values of r since it may require hundreds of plots to see when r changes from stable to unstable.

Phase-space portraits Phase-space portraits show the variations of x and y for example, or in the case of a three dimensional plot show how x,y and z vary after a specific time t. However, to evaluate different values of r it is not very effective since numerous plots are required to examine specifically when a system becomes unstable. For a stable system the plot oscillates towards a point and remains fixed there for all time as shown in the first diagram of figure. Interestingly with the lorenz system, in the phase-space portrait, if a system become unstable it develops strange attractors. The trajectories orbit around these attractors and in some cases produce limit cycles, whereby the trajectory orbits the same path for a certain amount of time. Usually when r is large enough the trajectory will orbit one strange attractor then suddenly jump to jump to the other and then back again without any real pattern, a sign of chaotic behaviour as seen in the figure for the xy-plane.

A brief summary of the Lorenz system of equations has been discussed. The values of r are useful to examine in the real world as the system (1) can be used to model convec-

8 [The , Commutator] February 2010

t and y() t are shown in the above diagrams. The first one shows a stable system figure 2 The values of x() when r = 14 and the second shows and unstable system when r = 28. The two vacant spaces that appear in the second diagram are called strange attractor which the trajectories orbit. A Initial conditions are x = 1,y = 1,z = 1. tion in the atmosphere. Interestingly, chaotic behaviour appear for a critical values of r which is usually around r = 25 in most literature and by running numerical calculations. These types of behaviour can be examined in forms of time plots and phase-

-space portraits to give a greater understanding of the systems behaviour. Further research has been to examine the system in terms of spectral plots in terms of discrete systems, that is choosing a select number of data points in the system, examining this against a Nyquist frequency and plotting against its amplitude. Over 100 diagrams in figure 3 have been plotted together for values between r = 0 to r = 30 and the ’noise’ that appear is approximately around r = 25 as well as slight chaotic behaviour around r = 14. However further investigations are needed into the practicality of this diagram, except it gives a decent pictorial description for different values of r which is ideal for examining the onset of chaotic behaviour.

Sandy Black

Economic C h a o s

figure 3 spectral diagram

Like in gambling (perhaps not too far off the practise altogether) one of the most forefront examples of a system in which prediction is everything

In the preceding articles we have discussed the way in which chaos theory is related to complicated systems such as the roulette wheel and how a slight change to an initial condition can significantly change the outcome of the event or even make the event outcome impossible to predict.

Properties of the tent map The tent map is an iterated function with its graph likened to the shape of a tent (figure 4). It demonstrates a range of dynamical behaviour ranging from predictable to chaotic The tent function is defined for all Figure 4 The Tent Map values on the unit interval [0,1]. This closed interval contains the image of this function so that when we input real values from this domain, the function outputs real values in, the interval [0,1].

Mathematically we write this mapping as f:[0,1] " [0,1]. Since the image of this function is contained in the domain we can begin with a number, say x, in the interval [0,1] and repeatedly apply or rather, iterate, the tent map function f: x,f(x),f(f(x)),f(f(f(x))).... The tent map is defined by the difference equation :

X n+ 1 = 1 - 2 X n -

where the tent function intersects the line ( y = xequivalent) X n+ 1 = X n 2 (1 - X n) = X n {since 0.5 # X n # 1.0}, which yields the other fixed pointX n = 2/3 .

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

0.659 " 0.682 " 0.636 " 0.728…

We can see that the value of 0.659 gradually moves away from the fixed point 2/3 but is still reasonably close to it which we expect since the change to the initial value, 2/3, is very insignificant. Any point close enough to either of the fixed points behave this way (unstable) and so we are able to predict the direction that the value goes under the tent map function, that is, away from the fixed point but still reasonably near to it.

Chaotic behaviour Now suppose that we choose two random non fixed values that are again very close to each other, say 0.258 and 0.259.

Applying the tent map, f, 10 times to each value gives the graph : We can see how 0.258 and 0.259 initially follow a similar pattern during the process of iteration which we expect to happen as the difference between the initial values isn’t at all significant. After the 8th iteration we start to see a hint of variance

1 2

This equation may be rewritten as:

X n+ 1 = 2X n ,0 # X n 1 0.5; X n+ 1 = 2 (1 - X n),0.5 # X n 1 1.0.

Predictable behaviour For certain points in the interval [0,1], when we plug them into the function we will not see any change to the value after x amount of iterations since they do not map to any other value but themselves. These points are called fixed points. We say that a is a fixed point of a function if and only if f(a) = a. In other words the point (a,f(a)) lies on the line y = x, so that the graph of the tent function has a common point with the line y = x . As we can see (figure 1) 0 is a fixed point of the tent map since f(0) = 0 i.e. the point (0,0) lies on the line y = x .

Are there any other fixed points for this map?

Having a look at figure 4 it is clear that there exists another fixed point at point A. The fixed point at A can be attained algebraically by solving is in economics; specifically, the stock exchange. Here there is a huge demand for any theory able to successfully predict the totally random seeming fluctuations in share prices for instance. Indeed Wall Street is one of the few places in the modern world where an as-

between the values but by the end of the 10th iteration we see a clear divergence between them.

Again we see the telltale features of a chaotic system - the slightest of changes to the initial conditions of the system can result in a seemingly unpredictable the outcome. We can see this same pattern emerge after a sufficiently large number of iterations (Simiu, 2002)

Sean Chan

trologer can make a good living and be taking seriously by a highly educated audience (Chorafas, 1994). Therefore the application of Chaos theory to the stock exchange and other areas of economics has been attempted enthusiastically by researchers but to limited

success. It seems the spirit is strong but the body of evidence weak; there is as yet no conclusive finding of the existence of a chaotic system in economic and financial data (Brock et

al, 1991).

Fiona Doherty

February 2010 [The , Commutator] 9

Mathematics and People Are Men Really Better at Maths than Women?

For centuries, the notion that women are innately less capable of studying mathematics then men has persisted even in the most educated of minds. In 2006, the proportion of students applying to study Mathematical and Computer Sciences at an undergraduate level in the UK who were female was just 21.2% . Many of us just accept this, citing that maths and science are traditionally more masculine subjects, but there has been a surprising amount of research on gender and mathematics in the past few decades. Is it fair to assume that mathematics is just better suited to men, and that because of a woman’s genetics she is always destined to be outshone by her male counterparts?

In short, the answer is no, that is to say it doesn’t fully explain the under representation of women in the field. Certainly, it has been shown that boys and girls learn differently due to genetics, but there is no evidence to suggest that girls are less able to learn and understand mathematics because of gender differences. This then suggests that environmental factors could possibly explain the differences, perhaps due to social influences or current teaching methods.

In the UK, mathematics is taught as a core subject, and all students must study it throughout their primary and the majority of their secondary education. This can make it challenging to determine a student’s attitudes towards the subject until it no longer becomes obligatory. In Scotland, many students have the choice to opt out of mathematics before studying for their Highers. In 2007, the proportion of students who studied Higher Mathematics that were female was 48.7%, which does not seem particularly low, but if we then consider the same statistic but in Advanced Higher Mathematics the proportion drops to 38.2% .

Equivalently, in England and Wales, the proportion of students studying an A Level in Mathematics in 2006 who were female was just 38.5% .

Previous data shows a similar trend going back for decades so it is clear that there are fewer girls then boys who choose to study mathematics once the subject becomes elective. Despite this, the percentage of girls achieving a passing grade was higher then for boys in both Mathematics Higher and Mathematics Advanced Higher (70.5% compared to 69.6% and 70.9% compared to 62.5% respectively). So in the 2007 data, the gender differences disappear when one accounts for the number of students who participate in courses.

This is a common result among many different studies across the world, and is known as the Theory of Differential Participation (Fennema & Sherman 1977). Although this theory is not fully accepted by all researchers in mathematics education, there is no dispute that fewer girls participate in the more advanced levels of the subject.

One measure, which has been employed by a number of countries across the world, has been the introduction of intervention programmes in schools. In the USA, one such program run by Fennema et al. in 1981, involved the broadcasting of videos designed to change attitudes about gen-

der-related differences in maths by giving information about careers, the educational relevance of maths and suggestions for activities to effect change. The videos were focused on specific target groups including students, teachers and parents.

The research was based on two assumptions, firstly, that if the girls’ knowledge on gender differences increased and their attitudes towards mathematics were improved then their participation in more advanced courses would also increase. And secondly, that in order to change the attitudes of the girls, then the expectations of their parents and teachers would also need to be altered assuming that the students are influenced by their social environment. Fennema and her colleagues did indeed find that the girls’ participation in advanced maths courses increased. They also showed an increased understanding of gender differences, saw mathematics as more useful for their future lives and were less likely to blame their failures on lack of ability. However, it should be noted that boys were also included in the study and their attitudes to mathematics improved in a similar way to the girls’. The success of this programme and others, however, shows how effective such interventions can be in improving the attitudes of female (and male) students towards mathematics. Before a child’s progression into secondary education, there is little difference in how children value the subject (Eccles et al. 1983) but once in secondary school girls begin to rank mathematics as their lowest valued subject compared to boys who have it ranked as one of the highest (Eccles et al. 1993). It has been suggested that this is due to a child’s increasing awareness of genderroles as they grow older. In other words, girl’s become influenced by their social environment in such a way that they begin to value mathematics less due to the expectations placed upon them from society.

There are a number of government initiatives in this country as well, targeted at improving not only the gender differences in mathematics but also the overall attitude of students to the subject.

However, they have had limited success, which suggests that if a student’s perception of mathematics is to be altered, then there should be as much emphasis on improving the attitude’s of teachers and parents. Many studies have shown that boys tend to believe that mathematics will be of greater value to them then girls do, and they also have a higher confidence in their abilities to learn the subject. Both boys and girls also tend to believe that mathematics is more suited to male learning (Sherman & Fennema 1978).

Equality in the classroom can be a controversial issue. All students in the UK are now entitled to a fair and equal education, and until now, that has meant that teachers endeavour to teach all their students in the same way. But there has been some evidence to show that the current teaching style adopted by many schools is not suited to the way that girls’ in particular learn. Boys respond better to an education focussed around competitive groups and individual work, whereas girls learn more effectively in cooperative groups (Fennema & Peterson 1986). However, much of a child’s education is focussed around the competitive model, partly because it seems boys are much more responsive to this style of teaching.

This raises the issue of equality, i.e. perhaps rather than equality in the classroom what we are really looking for is equality in the outcomes. This could mean that for some subjects, not just mathematics,

10 [The , Commutator] February 2010

children should be separated into gender groups, and the teaching organised around what suits each group respectively. Teachers inevitably treat boys and girls differently; a person’s gender is an unavoidable part of one’s personality. However, if the way in which mathematics is taught is having a detrimental effect on many girls’ capabilities in the subject, something does need to change.

Many would argue that education cannot possibly be equal if students are treated differently, but the alternative seems to be that for some children, their education suffers.

The complexity of this issue is overwhelming.

Although current research does show that gender differences in mathematics are decreasing, they do still exist especially in students’ beliefs about the subject and in career choices that involve mathematics. This is not to say that there are no high achieving women in the field, but when we look at the data from the classroom, there is no denying the smaller proportion of women then men who progress into the advanced levels of mathematics.

However, it is unclear whether an effective solution exists to this problem, or even if the gender imbalance in mathematics is a problem at all. Increased awareness of gender differences by students, teachers and parents, does appear to increase participation in the subject, but it could be argued that if we place such a high importance on an advanced mathematics education those who choose a different path, be them male or female, would seem inferior; an idea that can surely never lead to true equality.

Louise Ogden

Nazi Germany and Mathematics Open any history book about Nazi Germany and you'll find page after page of information about their propaganda machine. In the aftermath of the First World War, a growing number of the German Public, disillusioned with mainstream politics and crippled, both physically and psychologically, by economic poverty, turned to the Nazis in the polls. After seizing power and dismantling the Weimar Republic's democracy in 1933, Adolf Hitler's National Socialist German Worker's Party set about trying to change the hearts and minds of the German public, in accordance with their extremist ideology, one rooted in racism and nationalist pride. They played on already existing tensions within German society and used them to achieve their own needs. Their abhorrent views on the superiority of the ''Aryan Race'' over the Jewish community, the mentally ill, homosexuals and other minorities are well known to most and throughout their reign they committed some of humanities worst atrocities.

Their propaganda machine, under the leadership of Joseph Goebbels, was particularly concerned with changing education to fit their views. After all, if children were raised as Nazis then, at least theoretically, the party would be more secure. It is easy to see how sciences like biology and anatomy would be altered to provide an idealistic view of the ''superior'' race. Equally, social sciences like history were rewritten to cast a more optimistic view on the nation's history, to say the least. However, a non political subject such as mathematics has no clear link to any kind of ideology, at least not since Pythagoreans and other cults, and few history books recount in any great detail the Nazi views towards mathematics, and the effect their rule had on the subject's teaching. This article will provide a brief and hopefully interesting overview of this, and we will see how their views range from reasonable to bizarre.

As with most subjects, the Nazis tilted the teaching of their subjects to not-so-subtly impose their views on pupils. Below, a few genuine maths questions from a Nazi textbook are provided: 1) The construction of a lunatic asylum costs 6,000,000 RM. How many houses, at 11,000 RM, could be built for that amount?

2) To keep a mentally ill person costs 4 RM per day, a cripple 5.5 RM per day and a criminal 3.5 RM per day. Many civil servants receive only 4 RM per day, white collar workers barely 3.5 RM and unskilled workers not even 2 RM per day for their families.

According to conservative estimates that there are 300,000 mentally ill, epileptics etc in care.

How much do these people cost to keep in total, at a cost of 4 RM per day?

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

After school level, Nazi influence was also clear in mathematics at universities. After reforms made by the ruling Nazis in 1933, each lecturer had to perform a traditional nazi salute at the beginning and end of each class.

Also, in 1933, ''Dozentenschaft'', an organization which brought all academics under the Nazi banner, provided character profiles of all lecturers to make sure they were ''suitable'' for their jobs. However, rather than examine their scientific work, these ''report cards'' made sure that the academic community were politically coherent with Nazi ideology, at least on the face of things. Any expression of sympathy for oppressed minorities, or worse still communist movements, could lead to anything from losing their job to a spell in a forced labour camp. The repressive nature of the Nazi state placed paranoia in the halls of universities, and mathematics was no exception, despite its non political roots.

Intrusion into mathematics by Nazis didn't stop there. E.A. Weiss, a mathematician with Nazi sympathies at Bonn University, is known for creating a series of Mathematical ''Camps'' in 1933. Inspired by the Hitler Youth camps which aimed to unite and train people from a young age, Weiss tried to combine this with mathematical study. In his 1933 pamphlet, ''Mathematics: For What?'' he tried to align mathematics with fascist ideology. In the work, he dismisses the claim that Mathematics can be studied simply for the sake of studying mathematics. He argues that students must have a purpose to study mathematics and that maths ''for the sake of maths'' is hard to justify- not a particularly unusual claim, if flawed. Nazi overtones soon surface as he asks if it is ''really German'' to study

things for this reason. He claims that mathematics is a character building process, promoting (amongst other things) clarity of speech and writing, courage, manners and comradeship between student and professor.

He saw maths as a personality moulding process and, with this in mind, he organized a Mathematics Camp in Kronenburg with 10 other students (8 male, 2 female: a direct result of the Nazis alienating women students) at a ruined castle. After unfurling the swastika, they took part in the usual Nazi camp activities of hiking and military style games, but combining them with mathematical studies. A typical timetable contained 5 hours of mathematical studies throughout the day. The camps ran until 1938, when the demands of war curtailed any future plans.

Although the Nazi intrusion into mathematics was not as bold as in other sciences, the very fact they encroached on a completely non political subject is a stark reminder on how far they were willing to go. Mathematical papers began to slant towards ideology.

For example, in Tietjen's work, ''Space or Number?'' he compared Germans to space and the Jews to numbers. In the paper, although numbers are not dismissed (indeed, they are necessary), he believes the study of space though observation and logic, will lead to numbers, and this analogy is his effort to emphasize superiority. Although ridiculous to the reasonable minded reader, these ideas were indeed part of the Nazi drive towards a perfect ''Aryan'' state, and mathematics unfortunately wasn't spared in their quest.

Paul McFadden

February 2010 [The , Commutator] 11

Discovery

Assorted Articles Foundations of Mathematics

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 earlier Le Verrier had discovered the planet "with the point of his pen".

For thousands of years man has been aware of planets beyond our own: Jupiter, Saturn, Venus, Mars and Mercury, each taking the names of deities from the Greek Pantheon though known of when that very construct was an inkling in the distant future. All could be seen from Earth, observed from Rome to Xianyang with the naked eye, and perceived even in that time of more primitive science as having a decisive effect or influence on our lives. But this was not the movement of the spheres immortalised in Shakespearean literature, nor was it to be found in the astrological mumblings of mystics. Rather it lay in the quest to complete an ordered picture of the solar system where the laws of physics ruled, and to do so took mathematical artistry and determination the likes of which few possessed.

The road to their unveiling began in 1690 when the English astronomer John Flamsteed spotted a point of light which he duly classified as a star. Over the coming decades other astronomers made the same observations, and the same mistake. It took the polymath Sir William Herschel to show them the error of their ways when in 1781 he saw the same distant object, believing it to be a comet. He soon realised that what he observed might not have been a comet after all; taking up his pen he jotted down some calculations to confirm his suspicions and realised that he had indeed discovered a planet, older than any of the sons of Adam but a new and ground-breaking discovery. Uranus had now been unveiled and telescopes across the known world peered upwards, what had been to our eyes a bit part of far remote scenery in the cosmic play was now exposed as a key player.

From a mathematical standpoint the discovery of Uranus was of limited importance, the methods used by Herschel had already been explored with the known planets of the solar system, similarly the characteristics which led him to reevaluate his initial designation of Uranus as a comet was a result of the pre-existing corpus of knowledge. However there was a new challenge posed by the discovery, as Joseph Jerome Le Francais de Lalande found in 1782 when he noted Uranus following a significantly different orbit to that predicted. The best mathematical models available could not account for the variance, increasing the data set to include earlier observations of Uranus did little to improve it. They were faced with a stark choice between appealing to an undiscovered trans-uranian planet and rewriting the Law of Gravity. The problem remained unre-

solved for decades until the 1820s and early 1830s when improved measurements, predictions, and Uranus’ continued contempt for them, began to give way to a belief that the problem might lie not in the mathematical methods used, or indeed in the accuracy of observations on the known planets, but in an incomplete data set which omitted a key player from the cast list, the hunt for the latest planet was on.

Astronomers the world over began to search the night skies in earnest, the degree to which Uranus’ orbit was altered gave a starting point suggesting a celestial body not exceeding 12th magnitude. Many came close to discovering the hypothetical planet, some, including Lalande himself, even observed it but attributed any noted positional variation to errors in their observations or as some other species of celestial inhabitant.

On this occasion it was up to theoreticians to find the planet which practical astronomers were unable to pin down. Perturbation theory, a method which seeks to determine the discernible effect of known bodies on other known bodies, had been in use for some time. The basic principle begins with the solution to a known problem which can then be modified through an iterative process, giving a Power Series which quantifies the difference between the true value and the known problem and hence providing the answer to the problem they are trying to solve. Effectively it is an application of differential equations, but reversing the method to find details about an unknown body proved far more difficult, leading some to believe that the problems may be insurmountable.

The nature of the problems were twofold, firstly in order to find the missing planet they had to deduce a method of inverting Perturbation theory, taking what they did know about Uranus (the effect the known planets had on it and Uranus’ effect on them) and then using that to find out about this hidden planet. A comparable problem in nature, though certainly not in difficulty, might be reduced to having the result of an integral and attempting to find the integral itself, but without known limits how to deal with the constant? Even then before finding the solution to their ‘known unknown’ of a planet they would have to take away the iterative ‘add ons’ of the power series to obtain the initial value to their planet’s position.

In practical terms there were two problems, the first, a limited data series resulting from its fairly recent discovery, was eroding by the early 1840s as Uranus neared the completion of its first orbit

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since the discovery by Herschel the second, finding a mathematician with the necessary insight, ability and tenacity to tackle the problem remained, until like buses two emerged in quick succession.

The method employed by both was similar; using a formula for the perturbations of mean longitude it was possible to construct an equation based on three differential equations belonging to the unknown planet. An assumption is used to obtain the fundamental values which are then applied to the formula, after which by changing the form of the equation trigonometric relationships can be applied reducing the problem to a matter of eliminating the unknown values from the equations. Having reduced the equations it is then possible to use approximations to obtain a value for i , which can be improved by repeated approximations, with values for the longitude, eccentricity and mass of the planet can be obtained.

John Couch Adams first encountered the problem in 1841 after reading a paper on the subject of Uranus’ anomalous orbit by George Airy and resolved to begin an investigation, “as soon as possible after attaining my degree”. Having graduated in 1843 decorated with the assorted laurels of Cambridge, including the Smith prize won by Lord Kelvin two years later, he was finally free to begin his work and set about making the necessary calculations. By September of 1845 having found repeated iterations were making little change, he communicated the result to Professor Challis, in October he door stepped the Astronomer Royal, Airy, twice, missing him on both occasions and leaving a written account of his findings. After the discovery of Neptune he was able to further refine his work resulting in the version published in the eventual, though belated, paper “On the perturbations of Uranus” as an Appendix to the Nautical Almanac.

However Adams was not the only one to have

have found the problem a tantalising prospect and conflicting claims were soon to lead to a less than scientific squabble. Urbain Le Verrier completed a mathematics degree before beginning his studies and a career in chemistry. A failed job application later and he found himself as a tutor at l'École Polytechnique specialising in celestial mechanics. Initially he focused on the motion of Mercury, but by 1845 he had turned his attention to Uranus. Initially he carried out an extensive series of calculations, concluding in a paper of November 1845 that there was no way of accounting for its disturbed orbit by the known planets. By June of 1846 in another paper he had concluded that another planet must exist which accounted for the residual effect on Uranus’ orbit, giving an approximation of its distance from the sun and a calculation of its longitude. Finally, in August he was able to publish a paper with a predicted orbit and mass for the new planet, sent his result to Galle at the Berlin Observatory who received it on 23rd September and that very night sighted the predicted planet, within 1 degree of the theoretical prediction. As Arago put it, Le Verrier had discovered a planet "with the point of his pen", Neptune was revealed and from then until 1930, and again from 2006 onwards, the solar system’s planetary company was complete.

Though there was no question of who had ultimately triggered the discovery the English scientific establishment was eager to claim credit for the discovery (a second planet in under a century would have been quite a feat). Indeed Le Verrier’s June paper had triggered a last minute push, coordinated by Airy, to beat him to the punch, but the method used was time consuming and Challis’ outdated star-map rendered the search futile. Only after hearing of the discovery did they realise their search had twice spotted the planet but not recognised the significance. In spite of the acrimony between some of the characters involved, Adams himself acknowledged the priority claim of Le Verrier, providing ample demonstration of the saying “publish or die” as a planet might just hit you.

Michael Stringer

It is often said that maths and absolute knowledge go hand in hand; indeed, it is often claimed that the theorems of mathematics are the only things whose truth we can be undoubtedly sure of, with all other ‘facts’ being temporary or ambiguous. From the ancient days of Plato and Pythagoras, to the revolutionary times of the Enlightenment, maths has always been admired for its truth and its beauty. But how certain can we actually be of the claims of the absolute truth of maths? We must look to its foundations, and see that they are built solidly, consistently, and without errors; otherwise, the whole of maths could come crashing down around us.

This was a task faced by mathematicians at the beginning of the 20th century. An attempt at formalising the foundations of maths, purely in terms of sets, had been made by Georg Cantor at the end of the 1800s, however this was shown to be inconsistent; it had not been formulated rigorously enough. Cantor defined a set to be any collection of objects of our imagination, and allowed new sets to be created out of old in various ways, such as taking unions and intersections.

However, in 1901, Bertrand Russell showed that this model for the foundations of mathematics was not consistent. Russell devised a set, X , consisting of all sets which are not members of themselves. A contradiction arises, as it is easy to show that X both is and is not a member of itself.

Clearly, a different approach was needed.

Throughout the early years of the 20th century, what is now known as axiomatic set theory was developed. An axiom is essentially just a rule which is assumed to be true, or self-evident. Axioms are commonplace in mathematics: from Euclid’s axiomatisation of geometry in 300 BCE, to the axioms required in the definition of a vector space. Axiomatisation is a necessary procedure, as when trying to prove something, we cannot just continually break down what we are doing into smaller and simpler steps; eventually, we must stop, and accept that (or decide if) the step we are trying to do is true.

By 1922, after much debate, a list of nine axioms was produced; these were the axioms upon which the entirety of mathematics would be based. They are known as the ZFC axioms (as they were primarily developed by Zermelo and Fraenkel, and include something known as the axiom of choice), and are still accepted as the foundation of mathematics today: any proof you do can be broken down into a (probably very long) chain of applications of these nine axioms.

Having been let down by set theories before, much scrutiny was made by logicians over the consistency of the axioms. An axiomatic set theory is consistent if no contradictions can be derived using its axioms. Unfortunately, in 1931, Kurt Gödel showed that it is not possible to prove whether or not the ZF axioms (that is, the ZFC axioms but with the axiom of choice removed) are consistent or not. It has, however, been proved that if the ZF axioms are consistent, then so are the ZFC axioms (that is, adding the axiom of choice to our list does not introduce any contradictions).

It is, however, widely believed that ZFC is consistent; if it were not, it is thought that a contradiction would have been discovered by now.

Another problem we have to deal with is how to be sure that we have picked the ‘right’ axioms to be part of our theory. What criteria are we using to decide if an axiom is ‘self-evident’? For instance, it is fairly clear that if two sets, A and B, contain the same members as each other, we should want it to be true that A=B. And, indeed, an axiom saying just this is on our list of nine. But there 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 the ZF axioms was hotly debated for decades. Objections were made, since many found the truth of the axiom not to be intuitively obvious, and, in fact, if accepted in the theory, the axiom of choice can be use to prove true statements which some mathematicians call ‘undesirable’. An example of such a result is that it is possible to decompose a rigid, geometric sphere into a finite number of pieces, then rearrange them so you get two spheres of the same size as the original. This is known as the Banach-Tarski paradox. That being said, it is not really correct to claim that one of ZF and ZFC is ‘right’ and the other is ‘wrong’ – they are simply different theories, starting from different assumptions. Mathematically, it is possible to work with any consistent set of axioms, and see what we can prove from them. Strictly, every theorem stated should begin with “If the list of axioms I am working with is assumed to be true, then…”, however it is usually assumed we are working in ZFC, and this is thankfully omitted!

There is a another, more crippling problem when trying to formulate a rigorous foundation for maths, as was discovered by in the 1930s, by Gödel. The results, known as Gödel’s incompleteness theorems, say that all consistent axiomatic systems are incomplete, meaning that there are certain true statements whose truths are unprovable using the rules of the system. The outlook is perhaps a little bleak for us, then: either ZFC is inconsistent (and so, in fact, every proposition can be proved true, such as 1+1=3), or ZFC is consistent, but incomplete, and so some true statements are undecidable by the axioms. This is not a flaw inherent to the ZFC theory itself, but one which afflicts any sufficiently powerful axiomatic system. We can either prove too much, or too little (in the inconsistent and incomplete cases, respectively), and so any attempt at formulating a complete, consistent description of mathematics is doomed to failure.

As much of a blow as this is to mathematics, it is a fact that is intimately tied to the use of an axiomatic system. All hope is not lost, however; we still have (what is believed to be) a consistent list of axioms, from which every result you have ever been taught, and ever will be taught, has been proved. It is simply an immutable part of mathematics that there are certain statements whose truth cannot be decided upon. There are, in fact, already known examples of statements which cannot be proved in the ZFC theory: interested readers should investigate the continuum hypothesis and inaccessible cardinals. Despite these setbacks, thanks to the work of mathematicians which started over a century ago, we are left today with a theory of mathematics which is as rigorous, consistent and complete as it can be – and at the end of the day, what more can be asked for?

Neil Fullarton

February 2010 [The , Commutator] 13

Assorted Articles Tiling the sphere using differential equations

Why study mathematics?

A common question for undergraduates of mathematics to encounter is why do they study mathematics? Depending on how this is asked the question could mean what are the benefits of mathematical results and their applications, or alternatively, what is it about the subject that the student enjoys. This article is an undergraduates offering of an (incomplete) answer to both interpretations of the question. mechanics of piping gas from Russia to the UK to keep everyone warm with no leaks or explosions along the way. Similar examples can be found for subjects like Newtonian Mechanics or Mathematical Biology. So instead of giving a detailed example of an application of applied mathematics this article will show that techniques of applied mathematics can be used in the study of pure mathematics, which in turn, as shown above, can be used to study applied problems.

For this example the reader must be familiar with the idea of a 3-manifold. A manifold is a space that looks similar to a Euclidean space on a small enough scale. So, for example, the earth is a 2-manifold as locally (on a small scale) it looks like a flat 2-dimensional Euclidean surface but globally is a 3-dimensional sphere, where the rules of geometry are different. 3-manifolds are defined analogously, as an example the reader may know that locally the universe seems like a 3-dimensional Euclidean space but globally it may be something different.

It is known that there are eight classifications of 3-manifolds. The distinction between classifications can made by studying the nonEuclidean global structure: specifically, by studying paths of shortest distance in the space (here the space is 3-manifold). If the space is defined in an appropriate way it is possible to differentiate along a path between two points to give the velocity and differentiate again to get the acceleration along the path. The study then begins resemble a mechanics problem and in fact with only a little further manipulation tools from Newtonian, Hamiltonian and Lagrangian mechanics can all be used to study these abstract spaces.

Although the above illustration of applied techniques in pure mathematics is not as explicit as the graph example, it is hope the reader has still gained a sense of the interconnected nature of mathematical techniques.

And to say it explicitly: The implication is that studying one area of mathematics can have applications to subjects far from the original area of study.

This article now takes an inevitably more personal stance as it addresses the second interpretation: why do students enjoy mathematics?

Mathematics can be very hard. Students are given problems to which they can apply tools they have learnt in lectures or from books and they can compare similar questions to try to solve the problem. Most problems are attempted knowing that there is a solution; the student just has to find it. And this can get frustrating. Hours can pass on one question with only dead ends reached. Problems like these are hard. So where does the satisfaction come in? Personally it is from

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To illustrate the benefits of mathematics two examples will be used. The first will show how pure mathematics can be used in an applied setting with obvious benefits and the second will come from applied mathematics with effects on pure mathematics.

The first example comes from graph theory. A graph is a graphical model that shows relationships between collections of objects. So as in figure 1 each circle represents an object and two objects are related to one another if a line joins their respective circles. The problem linked to a graph such as the one in figure 1, by asking it is possible to colour the graph with two colours in such a way that no two connected circles have the same colour. If this cannot be done with two colours what about three or four and so on. The first attempt to colour this with two colours is bound to fail: the circles labeled A, B and H are connected in a triangle so if we pick A to be red and H to be green, then C cannot be either red or green. A third colour must be used, say blue. Now the graph can be coloured as shown in figure two. So how does this related to a non-mathematical area? One of the classic examples is that of exam timetabling. Table 1 shows the choice seven students, labeled 1 - 7, make out of a selection of 8 exams, labeled A through to H. Some students are sitting three exams and some are sitting two. The problem is to minimize the number of times an exam hall is booked whist making sure there are no exam conflicts, i.e. no student has two exams booked at the same time. For this, conveniently enough, the graph of figure 2 works as a representation of the problem. Each circle represents an exam as given by its letter, and the connecting lines show if exams cannot be set at the same time. For example, student two is taking exams B and D so their circles are linked and student seven is taking exams B and C so these are linked. Now the colouring takes on a meaning: each set of coloured circles represents a non conflicting booking, and this set of bookings is known to be the most efficient since the fewest number of colours were used.

This example was chosen to give a quick illustration of how graphs are applied; there are many more applications ranging from mapping internet sites to tracking the spread diseases to understanding molecules at an atomic level. It must be noted that pure mathematics is a huge subject; graph theory is one area and this was one application.

For the applied mathematics student benefits of study can usually be derived from examples given in courses and sometimes directly from course titles. For example, everyone readily sees the importance of studying fluid mechanics; it is important to understand the

c in the following way m = 1- c n = c- a - b o = a - b.

It is now possible to define triangle maps. These are conTo do this we consider an equation know as the hypergeo- structed as the ratios of the solutions at each singular point i.e. metric equation: w (1)(z) 2 sk (z)= (k0) d w dw w z) z(z- 1) 2 + [c - (a + b + 1)z] - abw = 0. k ( dz dz with k taking one of the values 0,1 or 3 . In the Here the independent variable z is considered case that the angular parameters m , n and v to be a complex number that we also allow to have absolute value between 0 and 1 (that be infinite. This equation has 3 singular is they lie within unit distance of the oripoints at 0,1 and 3 . These can be thought gin in the complex plane) the sk (z) of as point where the coefficients of the maps the upper half complex plane to equation become infinite. This is easy a triangle on the surface of a sphere to see if we divide through by the with the edges being circular arcs. coefficient of the second derivative.

The points 0,1 and 3 are mapped to the vertices of this triangle. Its The solutions to this about these singular points are given in angles are rm , rn andro . terms of Gauss’s hypergeometric In the special case that we can series: find integers p,q,r such that the (a)n (b)n n angular parameters can be writF (a,b,c |z)= / n3= 0 z ten as: (c)n n! where m = 1 ,n = 1 ,o = 1 p q r (a)n = a(a + 1)(a + 2)+ n - 1) is Then this triangle will tile the sphere called the Pochhammer symbol. The as in figure 3. values a,b and c are complex conThe tiling is achieved, like any stants called parameters. other, by taking copies of the triangle in In order to produce a sphere tiling we figure 3 a single tile on a sphere the above image and reproducing it all need to construct the solutions about the over the surface. Hence it is possible to tile a singular points. They are: sphere with a differential equation. w (00)(z)= F (a,b,c |z)

The purpose of this article is to see how ideas from differential equations can be used to generate a sphere tiling.

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w (01)(z)= z(1- c)F (a - c + 1,b - c + 1,2 - c |z) at the point 0, w (10)(z)= F (a,b,1 + a + b - c |1 - z) w (11)(z)= (1 - z)(c- a- b)F (c - a,c - b,1 + c - a - b |1 - z) at the point 1 and w (30)(z)= z- a F (a,a - c + 1,1 + a - b |z- 1) w (31)(z)= z- b F (b,b - c + 1,1 + b - a |z- 1) at the point 3 . It is also helpful to introduce the new parameters, called angular parameter, that we can define from a,b and A problem that has had hours spent on it to no avail months before can, if the solution was fully understood, be conquered in minutes. Gaining a good understanding of something that once seems really hard is very satisfying.

figure 2 two points. The first is seeing the solution, even if it is explained by someone else, and the second comes when revising the problem at a later stage.

The next reason is concerned with exactly what mathematics is. It is clearly a science, but it is different from other sciences in that it has a fascinating range of areas. Mathematics has models that explain the motions of stars, the world of finance, how the many types of numbers interact, there are even mathematical models that can predict parts of your probable future and some that can represent distances of infinity in a circle of radius 1! Maybe this can be summarized by this fun argument: 1.

“God ever geometrises.”- Plato 2. “Geometrical properties are characterized by their invariance under a group of transformations” – Felix Klein 3. “If Plato and Klein are correct, then God must be a group theorist.” – Stewart and Golubitsky: Fearful Symmetry To study mathematics is to study almost anything.

Stuart Andrew

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

During the Italian Renaissance, Leonardo Da Vinci wanted to create paintings that looked as real as possible. He is quoted as saying “The most praiseworthy form of painting is the one that most resembles what it imitates.” (One might think of mathematical models). To get his paintings as life like as possible Da Vinci developed a mathematical system known as linear perspective and the notion of a vanishing point that allowed him to create the illusion of depth on a flat 2-dimensional canvas. To show how important mathematics was to Da Vinci, consider his comment: “Let no man who is not a Mathematician read the elements of my work.” Da Vinci obviously had a passion for mathematics, but not all painters have needed this. In fact if the work of Escher is considered then it is clear mathematical training is not always needed to draw even advanced mathematical ideas. It appears that the Dutch artist had formal no mathematical training, yet his prints caused quite a stir in mathematical com-

munities as they represented hyperbolic tessellations, as in the Circle Limit series, or Riemannian surfaces, as in print gallery. Through this work Escher ended up friends with the famous geometer HSM Coxeter. And as the arts are discussed it must also be noted mathematics has been used extensively in literature. For examples see Lewis Carrols “Alice in wonderland” and ”Through the Looking Glass” or Yevgeny Zamyatins “We.” Mathematics can be studied to gain a better understand of problems in the world, it can be studied by those who enjoy creating sound logic, or who enjoy solving problems, it can also help in work outside of maths such as the arts.

Andrew Bestel

February 2010 [The , Commutator] 15