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In this issue… F 12 Optimal Pac-man 17 Interstellar travel: the mathematics of wormholes
26 Music playlists, Facebook and Twier 36 Re-inventing 2D
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I 4 In conversation with Dr Hannah Fry about the mathematics of love.
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31 A Fields Medal at UCL: Klaus Friedrich Roth
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Page Three model What’s hot and what’s not Dear Dirichlet Crossnumber The Symposium of the Muses: exploring the link between the arts and the sciences
48 Top 10: mathematicians’ haircuts
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Editorial Director Rafael Prieto Curiel Communications Director Anna Lambert Communications and social media Maia Miglioranza Pice Preeyakorn Huda Ramli Creative Director Samuel Brown Editorial and contents Sebastián Bahamonde Beltrán Mahew Scroggs Pietro Servini Mahew Wright Image and design Jessie Jing Website Manager Adam Townsend @chalkdustmag facebook.com/chalkdustmag chalkdustmagazine.com
Why a new magazine? The idea for this project emerged one evening, as we were solving maths problems whilst sharing biscuits and coffee (a relaxing evening for any self-respecting mathematician). Discussion turned to how it would be good to have a space to share our ideas, our amazing mathematical tricks and fascinating curiosities with more colleagues, not only in the present but also in generations to come (because we work, ultimately, with our legendary forefathers - Euler, Galois, Cauchy and more - whose presence lives on in their papers). Why chalkdust? Picking the name is always the hardest task! We wanted to name our project in such a way that people would associate it with maths, but that at the same time would let the audience know that we were not creating a journal but a magazine for those who enjoy maths. Along with our website, Facebook and Twier pages, we hope to provide you, the reader, with information about the latest news, events and research in the world of maths that we so love. We aim to ensure that you discover something interesting inside, regardless of whether you are somebody who has always thought that maths was too hard or a professor who has dedicated their whole life to this most beautiful of sciences. Rafael Prieto Curiel Editorial Director
Acknowledgements: Producing this magazine would not have been possible without the help of many different people. Particular thanks go to the academic staff at UCL, especially Professors Robb McDonald and Steven Bishop and Dr Luciano Rila, for their enthusiasm, encouragement and helpful suggestions; to Erica Tyson from the IMA and Fiona Nixon and Stephen Hugge from the LMS for their valuable support; to Mariana Ceccoi from UCLU for her time; and to the team at Free Hype, particularly Archy de Berker and Rain Soo, for their creative assistance.
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S′ = Π − βSZ − δS Z ′ = βSZ + ζR − αSZ R′ = δS + αSZ − ζR Source: flickr.com/joelf
This is a model of a zombie outbreak as proposed in When Zombies Aack! by Philip Munz, Ioan Hudea, Joe Imad and Robert J. Smith? (the question mark is not a typo; it is part of his name). The model features three populations: S, the susceptible or living humans; Z, the zombies; and R, the removed or dead humans. Humans can move between these three groups. The rates of movement are modelled by the following parameters: α β δ ζ Π
The rate at which humans can kill zombies. The rate at which live humans are infected and become zombies. The rate of non-zombie related death in the live humans. The rate at which the dead rise again as zombies. The human birth rate (assumed to be constant).
The terms in the model can be nicely shown in the following diagram:
Munz, Hudea, Imad and Smith? showed that unless a zombie outbreak could be ended quickly by a swi killing of zombies then the human population will eventually become entirely zombie. 3
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In Conversation with Dr Hannah Fry A L, U C L
“I genuinely struggle to find a topic where maths can’t offer you at least some insight.” Hannah Fry is a lecturer at the Centre for Advanced Spatial Analysis (CASA) at UCL. In addition to her research on the mathematics of social systems, Hannah also does a lot of public engagement - showing the general public some of the fascinating ways that Maths can be used in the real world. She’s given TED talks, spoken on TV and radio, made YouTube videos, and performed in science stand-up and stage shows. Most recently, she has wrien a book called The Mathematics of Love, and presented the BBC documentary Climate Change by Numbers.
M R W Would you like to tell our readers a bit about your mathematical background? I did my undergraduate degree in Maths here at UCL and I much preferred the applied side. I then did my PhD in fluid dynamics with Prof. Frank Smith, doing lots of lovely asymptotic analysis. My postdoc was a bit different. I think fluids is a really great place to train, but it’s hard to find a really good postdoc in fluids, and you can’t pick the subject that you want to work on. So this postdoc came up using mathematics to look at social systems - things like trade, migration and security. And I just thought it was an interesting topic and came over here to CASA, and I’ve been here ever since! Because there’s loads of mathematicians at CASA, aren’t there? chalkdustmagazine.com
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chalkdust Yes there’s a big group of us actually. There’s quite a few PhD students who are joint supervised by people in the maths department. And there’s also physicists and computer scientists. So you get a really nice mix. I think people oen don’t realise that mathematicians work in so many different areas of research, across lots of university departments. It’s genuinely useful for so many applications. Yes, I think that really links in with the point I was trying to make with my book, The Mathematics of Love. You can’t think of maths just as this abstract thing that exists only in isolation. I genuinely struggle to find a topic where maths can’t offer you at least some use or insight. I mean, I’ve published in archaeology journals! So I wanted to prove that even if you take love, the thing that is as far away from maths as possible, there is still some genuinely nice maths to be found. So which aspects of love have been modelled in your book then? It takes you on a lile tour of every possible aspect of dating life. It starts with somebody who is single and desperately hoping to meet somebody, exploring how to increase your chances of bumping into somebody you like at a party or meeting someone online. It then looks at strategies you can employ when dating, how to choose when to sele down, and even how to plan your wedding, with optimal seating plans and invitation lists. And finally there’s my favourite bit, which is a serious academic study - a collaboration between psychologists and mathematicians - looking at long term relationships through a mathematical lens. Has writing the book made you think differently about your own love life at all? Well I think the stuff at the end about long term relationships and arguing definitely has some relevance and is really interesting. The main point of it is that you should have lile positive arguments oen, rather than leing things build up. So now, if there’s a choice of leing something go or not, I choose not to let it go. My favourite bit was the chapter on how to decide when to sele down with a partner. People could make some quite big life decisions from that! I’ve had loads of people on twier say to me, “Thanks a lot for your book, but I’ve realised now it’s over!” 5
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chalkdust Another thing I like about the book is how similar models can be used to explore lots of different phenomena. For example, choosing a long term partner and deciding which interviewee to hire. Yeah, you see that’s what I find really exciting about maths, you find these connections between these things that don’t seem to be connected at all. So similar maths can be used to describe arms races and arguments between married couples. As soon as someone says this to you, you can see the analogy and it becomes really obvious. But it’s not really obvious until you transcribe it into mathematical language. Yes, sometimes writing down the model is the really useful part. Yes, I think the whole point is that you want mathematical models to start off being tangible and common sense. Once you’ve wrien them down in a concrete way, that’s when you can start finding the counter intuitive results, and there are plenty of them. I had a conversation the other day with a friend who thought that applying mathematics to love was going a bit too far. I think she thought that your book was trying to tell people the exact truth about how to conduct their love life. I completely agree that these models are a representation of reality, not reality itself. And I think that people who put too much faith in mathematical models are demonstrating as much of a misunderstanding of maths as people who don’t trust models at all. So the example that I always give to my students is Google Maps. If it says your journey is an hour and twenty minutes, nobody believes it will take exactly that long, but having that representation is much more useful than having nothing at all. Some people might say that some of the stuff in the book isn’t maths, it’s psychology or economics, what would you say to that? Well, I think the lines get really blurred, and the discipline boundaries are a bit old fashioned. It’s certainly true that a lot of economists are doing similar roles to mathematicians, likewise physicists, likewise some psychologists. But I think the difference comes down to what you think is important and interesting. In my job, the process is: you collect the data about the real world, you analyse it chalkdustmagazine.com
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chalkdust using statistics, you create the mathematical model and you use the mathematical model to give you insights. And the last bit is what I really love. But other people in this oice do exactly the same job as me but they focus on dierent stages. So the computer scientists like data mining and construction and other people find the statistics really interesting.
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M P Why do you think that maths has such a big PR problem? For instance, physics is equally nerdy, but they’ve got space and the large Hadron Collider and so on. What theoretical physicists actually do is basically maths, and somehow physics ends up with all the glamour whereas maths has very lile. Well actually I think that there’s loads of glamour in maths. For example, using Hilbert spaces for image recognition, the emergence of fractals in city growth, or predictive crime. However the only example that mathematicians ever bother to bring up is codebreaking and prime numbers! I don’t think mathematicians value making their subject exciting in the same way that physicists do. Do you think that there’s anything that mathematicians who aren’t interested in the applications of maths can do to engage with the public? So, there’s a guy called Cedric Villani, who’s won a Fields Medal and he’s just wrien a book called Birth of A Theorem about the mathematical process. I think that’s a really captivating story. And then there’s the Fermat’s Last Theorem documentary about Andrew Wiles that Simon Singh made. Both of those are really exciting, but again they’re not really about the maths. You’re taking maths and you’re anchoring it to a human story that people can latch on to. It’s very difficult for people to appreciate the abstract. It’s not enough to say, “by the way, this maths is really beautiful.” For a lot of people, their main experience of maths is of finding it boring and dry at school. Is there anything we can do to change that? Well I don’t think that you can change the curriculum. Also, teachers know so much about geing pupils to understand maths that I would never try to pretend to know the answer to that. But what I think you can do is go into schools and do exciting things. I do stage shows about maths, and there was one in particular in Palace Theatre on Charing Cross Road, with 3000 school children. It was basically 3 maths lectures, which I know sounds really boring, but we had Ma Parker, a comedian who’s incredibly funny. All the kids were laughing and screaming and loving all these lile bits of maths that they can anchor to things they know about already. That really lets you show them how far maths can go.
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W M As we are both aware, there is a huge gender imbalance in maths, particularly at more senior levels. At UCL at least, nearly 50% of undergraduate students are female, but for PhD students, it’s more like 15%. Why do you think this is? Well, first things first, it’s definitely not because of lack of ability. Let’s just get that out there. I also don’t think that it’s because people are sexist. I think the reason is that everything used to be really male focused, and the support structures that are in place haven’t ever been reassessed for more gender specific support. If you look at qualitative overviews of the difference in gender in mathematics, you can see a difference between how men and women view their own mathematical ability. Boys are like, this maths is hard; whereas the girls are like, I find this maths hard.
Ada Lovelace
I’m making a documentary for the BBC about Ada Lovelace, a female mathematician in the 19th Century, who worked closely with Charles Babbage who created the analytical engine, which was essentially the first computer. I’ve been doing a lot of research about her recently and you see the same story with her as you see with women in mathematics now. The amount of foresight that she had about the analytical engine was huge. Babbage himself didn’t realise its potential, but she did. She was the first person to work out that you could possibly use computers to do things other than maths. And that’s really incredible, but she really lacked confidence in her ability.
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chalkdust That’s so interesting! For example, Babbage was trying to get money for this analytical engine, which Ada wrote the first program for. He went to Robert Peel, the prime minister at the time, and ended up having this shouting match with him and achieving nothing. Ada wrote Babbage this 2000 word leer begging him to let her do the PR for the analytical engine because she was so much calmer and she completely understood it’s potential. Although Babbage did respect Ada, he said no, and because she lacked confidence, she then didn’t push it. She thought that he knew best and she didn’t have faith in her own ability. And if she had, who knows what would have happened. The analytical engine could have been built and then the first computer could have been created a hundred years before it actually was. And there’s lots of parallels between that and the way that women are now. I think it’s very difficult to imagine yourself doing something if you’ve never seen anyone like you doing it. So I personally think the visibility of women in maths and the availability of role models are massively important. Yes, I agree. So oen when I do school events, I get girls come up to me and ask “Is it really hard because they’re all boys?” And I’m like “no”, because particularly at undergraduate level that isn’t even the case. Actually I feel glad we’re living in a time when people are so keen to try and understand what’s going wrong and how to fix it. You can follow Hannah on Twier @FryRSquared. The documentary Climate Change by Numbers is currently on BBC iPlayer, and The Mathematics of Love is now available in bookshops. Anna Lambert is a PhD student at UCL working on mathematical models of bioreactors. You can contact her on Twier @anna_lambert
Who would you like to see interviewed in our next issue? Let us know your ideas on Twier @chalkdustmag, on facebook.com/chalkdustmag or at contact@chalkdustmagazine.com.
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Blackboards The mathematical equivalent of a Lile Black Dress. Blackboards will never go out of style.
E-Learning Systems In the immortal words of the Black Eyed Peas, interactive whiteboards are so two thousand and late.
Pi Approximation Day Because everyone knows that 22 7 is a beer approximation to π than 3.14.
Pi Day The American date format is totally illogical. Plus, pi day is too mainstream now.
Iccanobif Numbers x0 = 1, x1 = 1, xn+1 = reverse( xn ) + reverse( xn−1 ). I’ll let you work out its amazing properties for yourself.
Fibonacci Numbers Rabbits, golden ratio, spirals, sunflowers, blah blah blah. We’ve heard it all before! Agree? Disagree? Let us know on Twier @chalkdustmag, at facebook.com/chalkdustmag or at contact@chalkdustmagazine.com. 11
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Optimal Pac-Man M S, U C L
In the classic arcade game Pac-Man, the player moves the title character through a maze. The aim of the game is to eat all of the pac-dots that are spread throughout the maze while avoiding the ghosts that prowl it. While playing Pac-Man recently, my concentration dried from the pac-dots and I began to think about the best route I could take to complete the level.
S B K In the 1700s, Swiss mathematician Leonhard Euler studied a related problem. The city of Königsberg had seven bridges, which the residents would try to cross while walking around the town. However, they were unable to find a route crossing every bridge without repeating one of them. In fact, the city dwellers could not find such a route because it is impossible to do chalkdustmagazine.com
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Diagram showing the bridges in Königsberg. If you have not seen this puzzle before, you may like to try to find a route crossing them all exactly once before reading on.
so, as Euler proved in 1735. He first simplified the map of the city, by making the islands into vertices (or nodes) and the bridges into edges.
A graph of the seven bridges problem.
This type of diagram has (slightly confusingly) become known as a graph, the study of which is called graph theory. Euler represented Königsberg in this way as he realised that the shape of the islands is irrelevant to the problem: representing the problem as a graph gets rid of this useless information while keeping the important details of how the islands are connected. Euler next noticed that if a route crossing all the bridges exactly once was possible then whenever the walker took a bridge onto an island, they must take another bridge off the island. In this way, the ends of the bridges at each island can be paired off. The only bridge ends that do not need a pair are those at the start and end of the circuit. This means that all of the vertices of the graph except two (the first and last in the route) must have an even number of edges connected to them; otherwise there is no route around the graph travelling along each edge exactly once. In Königsberg, each island is connected to an odd number of bridges. Therefore the route that the residents were looking for did not exist (a route now exists due to two of the 13
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chalkdust bridges being destroyed during World War II). This same idea can be applied to Pac-Man. By ignoring the parts of the maze without pac-dots the pac-graph can be created, with the paths and the junctions forming the edges and vertices respectively. Once this is done there will be twentyfour vertices, twenty of which will be connected to an odd number of edges, and so it is impossible to eat all of the pac-dots without repeating some edges or travelling along parts of the maze with no pac-dots.
The Pac-graph. The odd nodes are shown in red.
This is a start, but it does not give us the shortest route we can take to eat all of the pac-dots: in order to do this, we are going to have to look at the odd vertices in more detail.
T C P P The task of finding the shortest route covering all the edges of a graph has become known as the Chinese postman problem as it is faced by postmen—they need to walk along each street to post leers and want to minimise the time spent walking along roads twice—and it was first studied by Chinese mathematician Kwan MeiKo. As the seven bridges of Königsberg problem demonstrated, when trying to find a route, Pac-Man will get stuck at the odd vertices. To prevent this from happening, all the vertices can be made into even vertices by adding edges to the graph. Adding an edge to the graph corresponds to choosing an edge, or sequence of edges, for Pac-man to repeat or including a part of the maze without pac-dots. In chalkdustmagazine.com
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chalkdust order to complete the level with the shortest distance travelled, Pac-man wants to add the shortest total length of edges to the graph. Therefore, in order to find the best route, Pac-Man must look at different ways to pair off the odd vertices and choose the pairing which will add the least total distance to the graph. The Chinese postman problem and the Pac-Man problem are slightly different: it is usually assumed that the postman wants to finish where he started so he can return home. Pac-Man however can finish the level wherever he likes but his starting point is fixed. Pac-Man may therefore leave one odd node unpaired but must add an edge to make the starting node odd. One way to find the required route is to look at all possible ways to pair up the odd vertices. With a low number of odd vertices this method works fine, but as the number of odd vertices increases, the method quickly becomes slower. With four odd vertices, there are three possible pairings. For the Pac-Man problem there will be over 13 billion (1.37 × 1010 ) pairings to check. These pairings can be checked by a laptop running overnight, but for not too many more vertices this method quickly becomes unfeasible. With 46 odd nodes there will be more than one pairing per atom in the human body (2.53 × 1028 ). By 110 odd vertices there will be more pairings (3.47 × 1088 ) than there are estimated to be atoms in the universe. Even the greatest supercomputer will be unable to work its way through all these combinations. Beer algorithms are known for this problem that reduce the amount of work on larger graphs. The number of pairings to check in the method above increases like the factorial of the number of vertices. Algorithms are known for which the amount of work to be done increases like a polynomial in the number of vertices. These algorithms will become unfeasible at a much slower rate but will still be unable to deal with very large graphs.
S PM P For the Pac-Man problem, the shortest pairing of the odd vertices requires the edges marked in red to be repeated. Any route which repeats these edges will be optimal. For example, the route in green will be optimal. One important element of the Pac-Man gameplay that I have neglected are the 15
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The edges to be repeated and an example of an optimal route.
ghosts (Blinky, Pinky, Inky and Clyde), which Pac-Man must avoid. There is a high chance that the ghosts will at some point block the route shown above and ruin Pac-Man’s optimality. However, any route repeating the red edges will be optimal: at many junctions Pac-man will have a choice of edges he could continue along. It may be possible for a quick thinking player to utilise this freedom to avoid the ghosts and complete an optimal game. Additionally, the skilled player may choose when to take the edges that include the power pellets, which allow Pac-Man to reverse the roles and eat the ghosts. Again cleverly timing these may allow the player to complete an optimal route. Unfortunately, as soon as the optimal route is completed, Pac-Man moves to the next level and the player has to do it all over again ad infinitum. Mahew Scroggs is a PhD student at UCL working on finite and boundary element methods. You can contact him on Twier @mscroggs. His website, mscroggs.co.uk, is full of maths. He hopes his website will soon feature a video of him completing Pac-Man optimally.
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Interstellar Travel: the Mathematics of Wormholes M W, U C L
Wormholes have been fascinating science fiction writers for decades, allowing protagonists to travel instantaneously to remote parts of the cosmos, the distant future or even entirely different universes. Last year Christopher Nolan’s blockbuster film Interstellar featured a wormhole as its key plot device, and recently won an Oscar for its visual depiction of them. The film centres around a crew of astronauts travelling through a rip in space and time in the hope of finding a future home for humanity. Although the idea of a wormhole might sound like one of the more farfetched ideas coming from the minds of science fiction writers, in fact there is a considerable amount of active research into the science behind them. The original screenplay for Interstellar was actually developed by a physicist at Caltech named Kip Thorne, who has been studying the mathematical properties of wormholes for nearly thirty years. Thorne collaborated with Double Negative Ltd, a special effects company based in Great Portland Street in central London, to ensure that the wormholes displayed in the film obeyed the correct laws of physics. And amazingly the cuing-edge super-accurate visualization soware available to the 17
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chalkdust Hollywood special-effects team actually enabled physicists to see new phenomena that they hadn’t anticipated. So what are wormholes, and how can we describe them mathematically? The answer to this question requires us to know a lile bit about physicists’ currently favoured description of gravity, the general theory of relativity. Kip Thorne has been one of the big names in this field of general relativity for over half a century, and so before we explore the mathematics of wormholes we will take a short detour through Einstein’s crowning achievement.
A V B I G R Albert Einstein first formulated his general theory of relativity to describe the force of gravity one hundred years ago this year (1915) and the theory has remarkably remained unchanged ever since. Einstein rejected the classical Newtonian view that gravity is just a force between massive objects. Instead, the force of gravity manifests itself by curving both space and time. What does this mean precisely? You may remember Newton’s first law, that if no force acts on an object it will either remain stationary or travel with a constant speed along a straight line. However as soon as you apply a force to an object, it will begin to accelerate and will no longer necessarily travel along a straight path. Einstein’s insight was that instead of treating gravity as a force in this regard, he could modify what it means for a line to be straight. In spaces with curvature, particles don’t move along classical Euclidean straight lines, they travel along what are called geodesics. Think of the surface of a sphere, like the earth. The quickest way to get between two points on this surface are the arcs of great circles, like the lines of longitude. These are the geodesics in this space. In Einstein’s general relativity, chalkdustmagazine.com
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A great circle on the surface of a sphere. Source: Wikimedia Commons
chalkdust space and time are curved, and freely falling particles under the influence of gravity travel along the geodesics (“straight lines”) of the spacetime. So how is this curved spacetime described mathematically? We start with a special type of vector space, called a manifold, with four dimensions: one dimension for time and three dimensions for space. On this manifold, we add some additional structure in the form of a metric gab that tells us how to calculate distances and angles. The space we are all used to is traditional Euclidean flat space in Cartesian coordinates. The metric gab in this case is just the identity matrix, which tells us that the line element is ds2 = dx2 + dy2 + dz2 . This line element allows us to calculate the length of curves in space. Einstein’s great insight was that maer distorts and curves this flat metric, and this relationship is described by the famous Einstein’s field equations, 1 8πG R ab − Rgab = 4 Tab . 2 c On the le hand side of the equation we have terms involving the Ricci tensor R ab and its trace, the Ricci scalar R. These tensors are functions of the metric which measure the amount of curvature present at each point of our spacetime. On the right hand side we have G, Newton’s gravitational constant; c, the speed of light; and Tab which is called the energy momentum tensor. This describes the amount, type and distribution of energy and maer in space. Given an energy momentum tensor describing a maer distribution, the aim is to then solve Einstein’s equation for the metric g. In general this is very difficult to do since the equations are highly non-linear. This is because the matter is telling spacetime how to curve, but this curvature then effects how the maer in our spacetime is moving, so we get lots of complicated feedback effects. In all but highly idealised symmetric situations finding exact solutions to Einstein’s equations is almost imposTwo masses curving spacetime. sible. 19
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T M W
Model of a two dimensional wormhole.
So how do we describe a wormhole using general relativity? Essentially a wormhole is a topological feature of our manifold connecting two completely separate regions of space with a tunnel. Topologists would call a wormhole “a handle” of a multiply-connected space. To visualise this take a flat sheet of paper and fold it in half smoothly (without creasing the fold!), cut a small hole in each half, and then connect a tube or cylinder between the two separate holes. You now have a wormhole in two dimensional space. It’s as simple as that!
Wormholes were first studied mathematically in relativity as early as 1921 by the German mathematician Herman Weyl. However it wasn’t until Thorne began studying them in the 1980s that they started to be taken seriously by relativists. Thorne’s friend Carl Sagan, one of the great popularisers of science of the twentieth century, was working on a science fiction novel Contact, which was later turned into a Hollywood movie. Sagan wanted to feature wormholes in the novel, so asked his friend Kip Thorne to look at wormhole solutions to Einstein’s equations. Shortly aer, Thorne proposed the idea of a traversable wormhole that would allow an explorer to travel through a wormhole in both directions from one part of the universe to another very quickly. A whole new area of physics research was spawned. Recall that if we are given a distribution of maer, Einstein’s equation allows us to solve for the metric of our spacetime, although we’ve already seen that this is typically a very difficult procedure. In Thorne’s 1988 paper, Wormholes in spacetime and their use for interstellar travel, he turned this procedure on its head. Instead of inpuing maer into the equation and solving for the metric, he inserted the metric describing a wormhole into the le hand side of Einstein’s equation, and then derived conditions for the maer that would be required to create this wormhole. This is a much simpler procedure since we no longer need to solve a complicated set of coupled partial differential equations. So what does the metric of a wormhole look like? Thorne and his graduate student chalkdustmagazine.com
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chalkdust Mike Morris proposed what is now called the Morris-Thorne metric, which is given in spherical polar coordinates by ds2 = −eΦ(r) dt2 +
1 1−
b (r ) r
dr2 + r2 (dθ 2 + sin2 θdϕ2 ).
This metric describes two separate regions of universe, connected by a tube. These different regions can either be interpreted as two separate universes, or the same universe at a different place in time and space. The function b(r ) describes the shape of the wormhole, and the throat of the wormhole, where the two different parts of the universe are connected, is located at r0 where b(r0 ) = r0 . This function has to obey a condition called the flare out condition, b′ (r ) < 1, which ensures that the throat remains open.
A Morris-Thorne wormhole. Source: Wikimedia Commons
E M Relativists usually assume based on physical grounds that all maer in our spacetime obeys certain energy conditions. The weakest of these energy conditions is called the “null energy condition”. In the context of a fluid, this simply states that the pressure p added to the energy density ρ of the fluid at each point is greater than zero, ρ + p ≥ 0. This is a very sensible and physical thing to assume, classically both energy and pressure are positive for all known types of maer currently known to physicists in the universe. However, when inserting the above Morris-Thorne metric into Einstein’s equations a problem arises. It is found that the energy required to keep the throat of the wormhole open long enough for an observer to pass through, without exceeding the speed of light, would require a violation of the null energy condition. 21
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chalkdust This is because in order to keep the throat open, a repulsive gravitational force is required whereas typically gravity behaves as an aractive force. So does that mean wormholes have to remain the fantasies of science fiction writers? Well, not necessarily! Physicists call maer that violates the null energy condition “exotic maer”. In fact, it is entirely mathematically consistent that particles with negative mass could exist, and these would violate the null energy condition. And physicists are far from knowing all types of maer in the universe. Dark energy and dark maer make up about 96% of the energy content of the universe, and we are still prey clueless as to what this maer and energy is made of. Dark energy, which is responsible for the universe’s expansion rate accelerating, exhibits a negative pressure and in some cosmologists’ models violates the null energy condition. This repulsive pressure could be utilised to keep a wormhole open. Another possible saviour for wormholes comes from modified theories of gravity. It is strongly believed by many physicists that general relativity is not part of the final theory of everything that they have been searching for for centuries. The two fundamental theories of physics, general relativity and quantum field theory, are fundamentally incompatible, and many believe that general relativity will have to be modified on small scales to take account of quantum mechanics. Many also believe genArtistic impression from an oberal relativity needs to be modified on large server’s perspective as they cross cosmological scales as well: current observa- the event horizon of a wormhole. tions show that the universe is accelerating in Source: Wikimedia Commons its expansion, which cannot be explained in a fully satisfactory way without modifying relativity. Physicists have proposed many alternative theories to general relativity, and in many of these theories it is possible to get wormhole solutions to the governing equations without invoking any exotic maer. It’s not currently believed that wormholes can exist naturally in the universe, except possibly at quantum scales, but it may be possible that a highly advanced chalkdustmagazine.com
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chalkdust civilisation with enough technology could build one. This would have some interesting consequences. For example, this may allow the possibility of time travel, giving rise to such paradoxes as the famous grandfather paradox: if you go back in time and kill your grandfather, how were you alive in order to go back in time and kill your grandfather? Such paradoxes might be unpalatable, and we may need new physical principles to rule out their existence. Thorne has suggested one resolution to this paradox by hypothesising that we may only be able to build wormholes that travel into the future and not the past, but it is still unclear whether this avoids all paradoxes.
Wormholes may still be incredibly speculative objects, and the technology required to build one is far beyond anything human civilisation would be capable of for many millennia. But studying them has been a very fruitful area of research for physicists exploring the mathematical properties of general relativity. Pushing a theory to its limit and looking at extreme examples allows us to explore where a theory might break down. Usually the ideas of science fiction are inspired by cuing-edge physics; this is a fascinating area where science fiction has inspired a whole new exciting field of cuing edge research. Mahew Wright is a PhD student at UCL, working in the fields of general relativity and cosmology.
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Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the Profâ&#x20AC;&#x2122;s help? Send your problems to deardirichlet@chalkdustmagazine.com.
Dear Dirichlet, that log(x) < x-1 for all x This week's problem sheet asks me to show quick sketch backs it up as > 0. To me this seems obviously true -- a ies or something but I'm not well. I think I'm supposed to use Taylor ser sure where to start. Can you help? DIRICHLET SAYS: Starting is always very difficult, but if you don't make a move now, you may later regret it. You are operating on the impression that you ought to use Taylor series, but this is not a certainty. Perhaps enquire about other approaches and see whether the problem opens up to you. Of the other examples you encounter, do you feel there are any that you click with? If so, why not try and introduce yourself to them? The more contact you have, the more likely it is that you connect with something that can really help. If you don't feel like you're getting anywhere, perhaps think up some new and exciting things to try. Ask your friends for their recommendations! The less you focus on this one aspect of the problem, and the more you focus on just enjoying your time together, the more rewarding you will find it. Failing all that, of course, you can always just copy last year's solutions.
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Dear Dirichlet, From the moment I met my husband, I kne w my soulmate search was over. We caught each other's eyes at a coffee shop eighteen months ago, got married six months later, and have barely been apart since. But last month I found out that my job is posting me to a cruise liner for three months, while he is still here in Britain. Na turally, I am worried that the distance and intermittent contact will tea r at the bonds between us. How do I know that our relationship effect s will be continuous at sea? DIRICHLET SAYS: Your relationship f(x) will be continuous at c if for every ε > 0, there exists δ > 0 such that |x − c| < δ =⇒ |f(x) − f(c)| < ε.
Dear Dirichlet, rs, and I can't imagine being I've been with my girlfriend for three yea two years before moving to with anyone else. We lived in France for h her in it. I've made the England, and every day is made better wit ally quite competent but am easy decision to propose to her. I'm norm having difficulty choosing a ring. Any advice? DIRICHLET SAYS: Have you considered the Gaussian integers? This ring is related to Pythagorean triples (romantic), and which numbers are the sums of squares. Furthermore, the only invertible integers are ±1 and ±i, symbolising the give and take that every relationship needs. Bonus exercise: Show that there are no elements in the Gaussian integers whose norm (defined as the square of the modulus) is of the form 4n+3. Hence show that if p=4n+3 is prime in Z, then p is also a Gaussian prime.
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Music Playlists, Facebook and Twier R P C, U C L
Have you ever browsed one of your friend’s playlists and discovered that you both share almost the same music? Maybe you have some additional songs and perhaps he has a band that you have never heard of before, but basically you have the same music. Is there a way to get a proper measure of how similar your music is? If you could compare your music with every person you know, is there a way of ranking playlists according to which is most similar to yours? It is possible that the playlist that is closest to yours contains songs that you have never heard before but that you will really enjoy. To define the problem a bit more formally, let A be the set of songs that you have, A = { a1 , a2 , . . . , an }, where ai is each individual song, which means that you have n songs; and let B be your friend’s playlist, with B = {b1 , b2 , . . . , bm }. There is no need for you and your friend to have the exact same number of songs, so n might be different to m. chalkdustmagazine.com
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chalkdust From those two sets, A and B , we want to define a measure that tells us how similar they are. If{we focus first}on the songs that you both have in common, we let C = A ∩ B = c1 , c2 , . . . , c p , which means that you share p songs (where p might be zero, in which case you have no music { in common) } and then the songs that you both have in total, D = A ∪ B = d1 , d2 , . . . , dq , where q is the total number of distinct songs that you have. Let’s assume that between both of your playlists you have at least one song, so q > 0. From this we define the similarity between your music preferences as Sim(A, B) =
|A ∩ B| |C| p = = , |A ∪ B| |D| q
where |A| denotes the number of elements in A, that is, the number of songs that you have. This measure of the similarity between your music has very interesting properties. First, as the number of songs you both share is obviously not negative, then the measure is also not negative; and also, the number of songs that you share has to be less or equal than the total number of songs that you both have, so the similarity is smaller than one. Wrien in a more mathematical way, we say that 0≤p≤q
⇒
0≤
p ≤1 q
⇒
Sim(A, B) ∈ [0, 1] .
A second property of this measure is that if you and your friend have exactly the same music, then C = D , which means that p = q, and so the similarity of your music is exactly one. In other words, Sim(A, A) = 1. A third property that we get from measuring in this way is that Sim(A, B) = Sim(B , A), in which case we say that the measure is commutative. A final property is the effect of acquiring a new song. If you add that new song to your collection but your friend does not, then the size of the intersection remains the same whilst the size of the union increases as a result of that brand new song and so the similarity would decrease. On the other hand, if you both buy the same song, then the size of the intersection p increases by one, and so does the union. For that reason, unless the similarity was already one (which means that your music was identical) then the measure increases thanks to that new song. 27
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chalkdust With this measure you could compute the similarity between your music and every person that you know and then rank them from the one with the most similar music to the one whose taste is most different.
S N The same idea on similarity could be applied to a social network like Facebook. In this case you may consider your set of contacts as the set to be compared, and the similarity is based on the amount of contacts that you have in common. Since Facebook provides the number of contacts that you and one of your friends have in common, then a formula that we can use is Sim(A, B) =
p , n+m−p
where n and m are, again, the number of contacts that each one of you have, and p is the number that you have in common. If you rank each one of your contacts based on your similarity, then the ones occupying the first places are likely to be your lifelong friends and/or family. The idea of similarity in a social network can be easily applied to create suggestions of other friends that you might share, places of interest or events. If we try to define similarity on a social network like Twier, then we have a difficult issue. On that social network you can have “followers” and people that you “follow” and the two are completely different. If we only focus on one of them, for example the followers that you have (which we can call the “Popularity” of a person) then you could have a high similarity with another account but never receive or read the same tweets. If we only focus on the accounts that you follow, then you could create an account and follow the same 164 Twier accounts that Katy Perry follows (who happens to be the most popular person in Twier) and chalkdustmagazine.com
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chalkdust conclude that your account is extremely similar to hers. However, it is very unlikely that you will end up with the 66.7 million followers that she has. For that reason we can define two different measures of similarity for this social network.
We can think of the first one as the “following similarity”, namely SimF (AF , BF ), which is based on the accounts that you and your friend decided to follow, expressed as AF and BF respectively, and computed the same way as we did with your contacts on Faceboook or your songs in your playlists. This function measures how similar your news feeds are, or, in other words, how likely it is that when you both sign in you will both receive the same tweets. On the other hand, we define the “popular similarity”, SimP (AP , BP ) which is based on the followers that you both have on Twier, with the higher the similarity, the more likely it is that the same people will read tweets from both of your accounts. Having two different measures of similarity is not very useful and we need a way to mix them. We have several options for combining these measurements, for example, a weighted combination of both, like Sim+ (A, B) = w SimP (AP , BP ) + (1 − w)SimF (AF , BF ), for a given parameter w ∈ [0, 1]. That number w can be interpreted as the relevance that the popularity has over the accounts that are being followed. If, for example, we take w = 1/2 then we are saying that they are both equally important, and if we take w = 0.9 then we place more importance on the popular similarity. Having to decide the value for w might be a bit arbitrary. A second option would be to define the similarity as √ Sim× (A, B) = [SimP (AP , BP )] [SimF (AF , BF )], 29
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Figure 1: Le: Sim+ (A, B), a weighted combination (with w = 0.7) of both of the similarities that assigns more weight to the following similarity than to the popular similarity. Right: Sim× (A, B), the product of both similarities.
which is simply the multiplication of both measurements. If you and your friend have a following similarity of 0.7 and a popular similarity also of 0.7, then the simple multiplication of the two numbers would be just under 0.5, so in order to avoid such small numbers, we incorporated the square root. It is easy to see that these new measurements that combine popular similarity and following similarity are also between 0 and 1. So you can now compare your similarity to one of your friends both on Facebook and Twier. Therefore, we have obtained a way to compare sets that might have different sizes, like your playlists, your Facebook friends or your popularity on Twier. This is very useful since obtaining a measure of your music similarity might help you find groups that you have never heard of before, and on social networks might result in you finding people that you had not been able to find before or posts and tweets that might interest you. Rafael Prieto Curiel is doing a PhD in Mathematics and Crime. You can follow him on Twier @rafaelprietoc or visit his blog rafaelprietoc.wordpress.com.
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A Fields Medal at UCL: Klaus Roth J J, U C L
Be proud if you are studying Mathematics at UCL! Looking back, we have numerous famous alumni who later gained significant achievements in their field. One of them is Klaus Roth, who was once a research student at UCL, and later was a lecturer and professor at the university, during which time he won the Fields Medal. If you haven’t heard of the Fields Medal, it is seen as the equivalent of the ‘Nobel Prize’ in Mathematics (although unfortunately it has a much lower monetary reward) and is awarded every four years by the International Mathematical Union. The award is given to a maximum of four mathematicians each time, all of whom must be under the age of 40 and have made a great contribution to the development of Mathematics. Roth won the Medal in 1958, when he was 33 years old and still a lecturer at UCL (show more respect to your lecturers …you never know!), for having “solved in 1955 the famous Thue-Siegel problem concerning the approximation to algebraic numbers by rational numbers and proved in 1952 that 31
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chalkdust a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turán of 1935).” Born in Breslau, Prussia (now Wroclaw in Poland) in 1925, Roth moved to England at a young age. He aended St. Paul’s School in London, and then went to Cambridge for his BA degree. Aer graduating in 1945, he worked for a brief time at Gordonstoun School in Scotland, before coming back to London and starting his research in 1946 under the supervision of Theodor Estermann. Two years later he was awarded his master’s degree and two years aer that, in 1950, he achieved his PhD. He was working at the same time as an assistant lecturer and later he became a lecturer, a reader, and eventually a professor at UCL in 1961. In 1966, Roth was offered the post of chair of Pure Mathematics at Imperial College, which he accepted and retained until his retirement in 1988. He has won numerous awards on top of the Fields Medal, including the De Morgan Medal from the London Mathematical Society (LMS) in 1983 and the Sylvester Medal from the Royal Society in 1991. Roth is currently 89 years old, and residing in the north of Scotland.
R’ W Klaus Roth specialised in a branch of mathematics known as number theory, and most particularly in diophantine approximations, which look at approximating real numbers by rational numbers (those that can be expressed in the form p/q, where p and q are integers). The work for which he won the Fields Medal in 1958 dealt with approximating a particular class of real numbers known as algebraic numbers: those that are the solutions of √ finite polynomials with rational coefficients. For example, the irrational number 2 is an algebraic number as it is a solution to the equation x2 − 2 = 0; whilst such famous numbers as π and e are not the solution of any such polynomial and hence are called transcendental. When it came to approximating algebraic numbers, people knew that there were chalkdustmagazine.com
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chalkdust infinitely many rational numbers p/q such that
p
− a < 1 ,
q
q2
(1)
where a is an algebraic number. You can show this by writing a as a continued fraction, 1 a= . n1 + n + 1 1 2
n3 +...
However, the question was whether we could do beer: for a given algebraic number, could we find an exponent x greater than 2 such that there are infinitely many rational numbers (again of the form p/q) that approximate the algebraic number with an error less than q− x ? Or, expressing this more mathematically, for a given algebraic number a, let µ( a) be the upper bound of the exponents x such that there exist infinitely many rational numbers p/q satisfying
p
− a < 1 .
q
qx The question is then to find µ. Because of the result in (1), µ(r ) ≥ 2. In 1844, Liouville bounded µ from above, proving that if a was the solution to a polynomial of degree d (the highest power of the unknown in the equation is d), then µ( a) ≤ d. So there were certainly only finitely many rational numbers that approximated a with an error beer than q−d . A range of µ between 2 and d was, however, not good enough and various mathematicians narrowed it down before Roth’s decisive contribution. In 1908, Thue √ the upper limit down √showed that µ( a) ≤ d/2 + 1; in 1921, Siegel brought to 2 d; in 1947, Dyson improved it still further to around 2d. And then, in 1955, Roth showed that µ(r ) was actually equal to 2. In other words, if we want to improve the accuracy of our estimation to be beer than 1/q2 , then there are only finitely many rational numbers able to achieve this. In the words of Davenport, who presented Roth with the Fields medal, the theorem “seles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated.” Jessie Jing is a first-year undergraduate studying mathematics at UCL. With contributions from Pietro Servini.
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#1
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Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. One randomly selected correct answer will win £100. Three randomly selected runners up will win a Chalkdust pen. The prizes have been provided by G-Research, researchers of financial markets and investment ideas. Find out more at gresearch.com. To enter, email crossnumber@chalkdustmagazine.com with the sum of the across clues by 22nd July 2015. Only one entry per person will be accepted. Winners will be notified by email by 1st August 2015.
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D
A 1 D4 multiplied by D18. 5 A multiple of 101. 7 The difference between 10D and 11D. 9 A palindromic number containing at least one 0. 10 Subtract 24A multiplied by 24A backwards from 100000. 13 Subtract 8D from 35A then multiply by 17A. 15 Multiply this by 13D to get a perfect number. 16 The product of two primes. 17 A triangular number. 19 A factor of 6D. 20 30A more than the largest number which cannot be wrien as the sum of distinct fourth powers. 22 The sum of seven consecutive primes. 23 When wrien in Roman numerals, this number is an anagram of XILXX. 24 The largest prime factor of 733626510400. 25 A square number. 27 The product of all the digits of 7A. 28 A multiple of 107. 30 Unix time at 01:29:41 (am) on 2 January 1970. 32 When wrien in a base other than 10, this number is 5331005655. 35 The smallest number which is one more than triple its reverse. 36 All but one of the digits of this number are the same.
1 700 less than 3D. 2 The sum of this number’s digits is equal to 16. 3 A Fibonacci number. 4 This is the same as another number in the crossnumber. 5 A square number containing every digit from 0 to 9 exactly once. 6 This number’s first digit tells you how many 0s are in this number, the second digit how many 1s, the third digit how many 2s, and so on. 8 The lowest prime larger than 25A. 10 The largest prime number with 10 digits. 11 A multiple of 396533. 12 If you write a 1 at the end of this number then it is three times larger than if you write a 1 at the start. 13 Multiply this by 15A to get a perfect number. 14 The factorial of 17D divided by the factorial of 16A. 17 The answer to the ultimate question of life, the universe and everything. 18 A multiple of 5. 21 The number of the D clue which has the answer 91199. 26 The total number of vertices in all the Platonic Solids (in 3D). 27 Two more than 29D. 29 The first and last digits of this number are equal. 31 A multiple of 24A. 33 Each digit of this number is a different non-zero square number. 34 A square number.
(10) (3) (10) (5) (5) (5) (2) (2) (2) (3) (7) (3) (2) (2) (2) (5) (5) (5) (10) (3) (10)
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(3) (5) (3) (5) (10) (10)
(2) (10) (10) (5) (2) (7) (2) (5) (2) (2) (5) (5) (2) (3) (3)
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Re-inventing 2D: What a Difference a Matrix Makes L R N K, U C L
Manhaan is a Cartesian plane brought to life. The original design plan for the streets of Manhaan, known as the Commissioners’ Plan of 1811, put in place the grid plan that defines Manhaan to this day. Using rectangular grids in urban planning is common practice but Manhaan went as far as naming its streets and avenues with numbers: 1st street, 42nd street, 5th avenue and so on. The idea of a system of coordinates was first published in 1637 by René Descartes (hence Cartesian) and revolutionised mathematics by providing a link between geometry and algebra. According to legend, school boy René was lying in bed, sick, when he noticed a fly on the ceiling. He realised that he could describe the position of the fly using two numbers, each measuring the perpendicular distance of the fly to the walls of the room. Voilà! The Cartesian plane was born. It is only natural then that we interpret the Cartesian plane in terms of spatial coordinates but what happens if we take the same old Cartesian plane and reinvent the 2D space as a space of functions?
A C S F Consider the following family of functions: y = ae x sin x + be x cos x where a and b are real. chalkdustmagazine.com
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chalkdust We call it a family of functions because different values of a and b will result in different functions. For example, for a = 2 and b = −1, we have y = 2e x sin x − e x cos x . Now let’s make a rule that whenever we write down a member of this family of functions we will ALWAYS write the term in e x sin x first and then the term in e x cos x . If that’s the case then we only need a pair of numbers in order to define a function belonging to this family since we have agreed that the first number refers to the coefficient of e x sin x and the second number to the coefficient of e x cos x. Do you see what I’m geing at? Once that is established, y = 2e x sin x − e x cos x can ). ( be ) represented by (2, ( −1) a 2 . So we have that We might even choose to write a and b as the vector b −1 (√ ) √ 2 is in fact y = 2e x sin x − e x cos x and is y = 2e x sin x + πe x cos x and π so on. If we(represent y = e x sin x as y = e x sin x + 0e x cos x, its vector form is then sim) ( ) 1 0 ply and similarly y = e x cos x is , exactly like i and j in vector notation. 0 1 x x We can think of e sin x and e cos x as the building blocks of our functions or, to use mathematical terminology, the components of our functions. Our Cartesian plane is now used to describe a function space rather than spatial coordinates.
M D M It’s all very well but so what? One very interesting feature of this family of functions becomes apparent when we differentiate y = ae x sin x + be x cos x. Using the product rule, we have y′ = ae x sin x + ae x cos x + be x cos x − be x sin x and, collecting like terms, we obtain y′ = ( a − b)e x sin x + ( a + b)e x cos x. 37
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chalkdust It turns out that y′ is also composed of one component in e x sin x and one in e x cos x, and therefore belongs to our family of functions. In this case we say that the function space is closed under differentiation. ( ) a Given a general function y = in the family, its derivative is then given by b ( ) a−b y′ = . a+b This is a surprising result and one that doesn’t apply to all families of functions. If you consider y = axe x + b cos x, you can check that its derivative does not belong to the same family. But in our family derivative of ( ) it so happens that the ( ) 1 0 both components, i.e. y = e x sin x = and y = e x cos x = , belong to 0 1 the family as well. That means that we can build a matrix that will differentiate any function in the family, just like a transformation matrix. We wish to find a 2 × 2 matrix Mdiff such that ( ) ( ) a a−b Mdiff = b a+b ( It’s not hard to see that Mdiff =
1 −1 1 1
) will do the trick.
For y(= 2e)x sin x − e x cos x, we have y′ = 3e x sin x + e x cos x. In vector notation, 2 y= and its derivative is given by −1 ( )( ) ( ) 1 −1 2 3 ′ y = = 1 1 −1 1 which is equivalent to y′ = 3e x sin x + e x cos x. Of course this should always work since we have used the general form y = ae x sin x + be x cos x of the function to come up with the differentiating matrix. In our function space, differentiation is but a linear transformation of a function vector. chalkdustmagazine.com
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I M C Our next question is: would that work for integration as well? One might expect that it would, since integration is the inverse operation of differentiation, i.e. if you integrate the derivative of a function, you get back to the original function (disregarding the elusive +C). Probably, then, integration would generate functions within the family. We can always check. Let’s use integration by parts on each of the components and then combine them to integrate y = ae x sin x + be x cos x. Starting with ∫
e x sin xdx = e x sin x −
∫
e x cos xdx
and using integration by parts again, we have ∫
e sin xdx = e sin x − (e cos x + x
x
x
∫
e x sin xdx ).
This leads to ∫
2
e x sin xdx = e x sin x − e x cos x
and therefore ∫
e x sin xdx =
1 x 1 e sin x − e x cos x. 2 2
(1)
Similarly we can show that 39
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chalkdust ∫
e x cos xdx =
1 x 1 e sin x + e x cos x. 2 2
(2)
Using (1) and (2), ∫
(
( ae x sin x + be x cos x )dx = a
) 1 x 1 e sin x − e x cos x 2 2 ( ) 1 x 1 x +b e sin x + e cos x 2 2
and, re-arranging into components, we reach ∫
1 1 ( ae x sin x + be x cos x )dx = ( a + b)e x sin x + (− a + b)e x cos x. 2 2
As expected, our function space is also closed under integration. Similarly to differentiation we can therefore build an integrating matrix as follows: Mint =
1 2
1 2
− 12
1 2
Here comes the punchline. And this really blows me away so I hope you will feel the same. If integration is the inverse operation of differentiation, does it follow that the inverse of the differentiating matrix is the integrating matrix for our family of functions and vice-versa? Of course you know what the answer is: a resounding yes! ( ) 1 −1 We found that the differentiating matrix is Mdiff = . Therefore its 1 1 ) ( 1 1 1 1 inverse is M− diff = 2 −1 1 and so we have a precise equation chalkdustmagazine.com
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Mint =
1 M− diff
1 = 2
(
1 1 −1 1
)
between the integrating matrix and the inverse of the differentiating matrix. Even though we created a system whereby functions are represented as vectors, and differentiation and integration become linear transformations in the plane, the inherent invertibility of differentiation and integration is reflected in the invertibility of their corresponding matrices. Another fascinating result arises when We ( ) we consider higher order ( derivatives. ) ( ) a a a − b ′ found that the derivative of y = is given by y = Mdiff = b b a+b ( ) 1 −1 with Mdiff = . If we now want to find the second derivative of y, which 1 1 in turn is the first derivative of y′ , we only need to use Mdiff to differentiate y′ as follows: y′′ = Mdiff y′ = Mdiff Mdiff y which leads to y′′ = M2diff y. Following a similar inductive argument, we can conclude that the nth derivative of y will be given by y(n) = Mndiff y. So if we raise Mdiff to the power of n we construct a matrix that calculates the nth derivative of any member of our family of functions!
B P These ideas can be extended to higher dimensions as long as we define a family of functions that is closed under differentiation and integration. For example, y = a sin x + b cos x + ce x is composed of three components and forms a function space which is closed under differentiation and integration. As this is a 3-dimensional space, its differentiating and integrating matrices are 3 × 3. And we don’t have to stop here. Think of y = a sin x + b cos x + ce x + de x sin x + f e x cos x — note that I used f instead of e to avoid confusion with e x . 41
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The connection between inverse operations and inverse matrices suggests that, given a function space closed under differentiation, the differentiating matrix will unfailingly render an inverse matrix that represents integration. However, this is not always the case. Consider the family of functions defined by y = a + bx. If we ( re-write y as y = ax0 + bx1 , we can use vector notation to represent y as ) a y = . This function space is closed under differentiation since y′ = b = b ( ) b 0 1 bx + 0x = and hence y′ belongs to the function space. The differentiating 0 ( ) 0 1 matrix is therefore given by Mdiff = . Since det Mdiff = 0, Mdiff has 0 0 no inverse. Does that mean that it is not possible to integrate ∫ y? Of course not; we know how to integrate this function and the integral is ( a + bx )dx = ax + 1 2 2 bx (+C ). So why does M diff have no inverse? Integration adds a quadratic term which is not a component of our family of functions. Therefore our function space is not closed under integration, i.e. integration generates a function that is outside our family of functions. As a result, Mdiff is non-invertible. If Mdiff had an inverse then that would be the integrating matrix and hence the function space chalkdustmagazine.com
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chalkdust would be closed under integration, which is not the case. We have re-invented the 2D space as a function space. In our new construct, vectors represent functions and we can find matrix transformations that map functions to their derivatives and vice versa. Inherent properties of invertibility are preserved and we can even extend all these concepts to higher dimensions. Function space appears in various areas of mathematics from set theory to topology, but this particular use of matrices is not that well known. This seems the right place to profess our debt to Prof F E A Johnson who teaches this topic at UCL. I wonder if this more abstract approach to the Cartesian plane would be of any use in urban planning. Perhaps coordinate pairs could represent subjective qualities such as excitement and joy instead of addresses. Just imagine what Manhaî&#x20AC;źan would look like. Dr. Luciano Rila works in the Department of Mathematics at UCL and is also an Area Coordinator for the Further Maths Support Programme. He is keen to promote the beauty of mathematics and he also teaches static trapeze. You can follow him on Twiî&#x20AC;źer @DrTrapezio. Nikoleta Kalaydzhieva is an undergraduate studying maths at UCL.
# 2 -- Circles Pictured are: a quarter circle with radius 2r, and two semi circles with radius r. Prove that the pink area is equal to the green area.
Source: MathsJam Answers on www.chalkdustmagazine.com
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Chalkdust can only thrive with your thoughts, ideas and mathematical curiosities. In return, itâ&#x20AC;&#x2122;s designed to be a place where you can publish articles away from the constraints of mainstream journals. If you find something interesting, we believe that it should be shared. Articles should be sent to contact@chalkdustmagazine.com either as a Word document or LaTeX file. facebook.com/chalkdustmag @chalkdustmag
# 104 -- Two Triangles In the diagram, the three sides of a triangle have been split into thirds, then the lines have been added. What is the area of the smaller shaded triangle as a fraction of the area of the large triangle? Source: MathsJam Answers on www.chalkdustmagazine.com
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The Symposium of the Muses M M, UCL
This issue’s cover picture is a creation of Anthony Lee, a young British artist, who has always been fascinated by exploring the possibilities of creating images through light. In Anthony’s eyes, this experimental process is the result of “the idea of an ephemeral substance or state, the idea that the captured moment was never intended to last or be repeated. In my light images neither the light nor the shape can last and yet they stay captured in the image I present.” It is interesting to notice where both the artistic and scientific processes intersect and interact with each other — and where they do not. The artist, Anthony, is looking for a way to use scientific knowledge to express his personal emotions and inner thrills; and the resultant art is the outcome and purpose that elevates and distinguishes the science. And yet Anthony is bending and filling reality with his own meanings — his “ephemeral” ideas of light and shape — that are changeable and unique to him. Contrast this with the aims of scientists, who look for permanent truths that affect every observer, irrespective of their uniqueness in this space-time continuum. 45
spring 2015
chalkdust estions about what light is and how it plays a vital role in the existence of life have driven the development of scientific knowledge from its earliest days. In particular, in the last two centuries a deep understanding of light led James Clerk Maxwell to formulate the Theory of Electromagnetism — one of the four fundamental interactions in nature — and, through his equations, to completely unify the electrical, magnetic and optical phenomena. Thanks to his obsession with light, Albert Einstein overcame the inconsistency between Newtonian Mechanics and Maxwell’s Equations to create a new paern where space and time became completely inseparable: the space-time of Special Relativity. His further considerations about the photoelectric effect, combined with Planck’s work on black bodies, finally opened to us the revolution of antum Mechanics. Since light has played this central role in modern physics and mathematics, we have decided to make use of Anthony’s image to represent a new magazine that ‘comes to light’. The bizarre yet intriguing form that appears on the front cover has been created by using a single strip of Light Emitting Diodes (LEDs), set to a continuous loop of varying colour, and then capturing these sequences with a camera on a slow shuer speed while moving one’s body. As he describes in his fine art project, Anthony feels that the inclusion of his body in the creation of these images makes them not only individual and unique but also representative of him as a person. chalkdustmagazine.com
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chalkdust Returning to the science, the creation of the blue LED by Professors Isamu Akasaki, Hiroshi Amano and Shuji Nakamura in the early 1990s led the Nobel commiee that awarded the trio the Nobel Prize for Physics in 2014 to declare that “the 21st Century will be lit by LED lamps” — more energy efficient and environmentally friendly than the existing incandescent light bulbs. Whilst red and green LEDs had been around for years, the absence of blue meant that it was not possible to combine the three colours to produce white light. Our cover is art’s small homage to this scientific breakthrough: red, green and blue come together to give birth to a dazzling white. The shape from which the white springs forth resembles a shell, an object also present in an iconic painting of the Italian Renaissance: Boicelli’s The Birth of Venus, painted between 1482 and 1485. Here, it is a nude Venus who emerges from the shell, floating on waves that have been stirred by the breath of the god Zephyrus, who is being clasped by a female figure — possibly the nymph Chloris — in a representation of procreation. Art scholars argue that the artist was influenced by a Platonic Academy present in Florence at the time, which pushed for a revival of Plato’s philosophy, and that the nude Venus is an embodiment of Plato’s ideas of divine love as a vivifying energy and driving force of nature. In this view, one can use Venus as a representation of the creative drive that unites both artists and mathematicians. In art, it inspired Boicelli to produce one of the greatest aesthetic creations of the Renaissance period; whilst in mathematics, it motivates us to explore more deeply the beauties of the subject, from which we obtain either a pure aesthetic pleasure or a thrill at having explained the mysteries of nature and social phenomena. Anthony Lee is a student of Sound Art & Design at the University of the Arts, London. For more of his art, go to anthonyleefineart.wordpress.com Maia Miglioranza is a PhD student at UCL, interested in Riemannian Geometry, Geometric Measure Theory, Art and Philosophy.
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spring 2015
chalkdust
This issue, Top Ten features the best haircuts held by mathematicians. To vote on the Top Ten symbols for use in algebra go to chalkdustmagazine.com.
Rumour has it that judges’ wigs were inspired by Leonhard Euler’s hair.
Peter Dirichlet loves answering your leers and caring for his hair.
Hypatia proves that ancient Greeks also had great hair.
If that beard was on top of his head, Johannes Kepler would be number one.
It can’t be simply a coincidence that It could be argued that Albert Einstein ”acre reindeer stash” is an anagram of is not a mathematician. It could not be ”Rene Descartes’ hair”. argued that his hair is not amazing.
Carl Gauss. Because sideburns are cool.
In close second, it’s Ada Lovelace.
Isaac Newton grew this to act as a helmet against falling apples.
Our runaway winner this issue is Gofreid Leibniz.
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UCL DEPARTMENT OF MATHEMATICS
Mathematical Modelling MSc
The Trefoil Knot and its Tantrix Curve © Mitchell Berger, 2006
The MSc programme includes modules in: • Advanced Modelling Mathematical Techniques • Computational and Simulation Methods • Frontiers in Mathematical Modelling and its Applications • Nonlinear Systems • Operational Research. Some of the current optional modules are: Asymptotic Methods and Boundary Layers, Biomathematics, Cosmology, Evolutionary Games and Population Genetics, Financial Mathematics, Geophysical Fluid Dynamics, Hyperbolic PDEs with Applications, Mathematical Ecology, Mathematics for General Relativity, Quantitative and Computational Finance and Theory of Trafic Flow. Find out more: Mathematics Department: http://www.ucl.ac.uk/Mathematics/ Apply now: http://www.ucl.ac.uk/prospective-students/