In this issue... Features 4 Breakfast at Villani’s
We feel underdressed next to Cédric
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Cardioids in co ee cups Dominika Vasilkova loves covfefe
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Literary mathematics Roberto de la Cruz Moreno shows us a matrix in verse
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Geographic pro ling Michael Stevens and Sally Faulkner catch killers and malaria
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How to communicate science Marcus du Sautoy shares his blueprint for success
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Pretend numbers Wajid Mannan is outsmarted by his niece again
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Three- ngered maths Robert Low flips one upside down
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Maryam Mirzakhani’s maths Nikoleta Kalaydzhieva explains
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Pretty pictures in the complex plane Emily Clapham explains your 29 favourite fractals
Regulars 3 Page 3 model 8 What’s hot and what’s not 15 Which software are you? 26 Dear Dirichlet 36 Chalkdust comic
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by Tom Hockenhull
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Signi cant gures Patricia Rothman tell us about Sophie Bryant
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Reviews
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How to make... a hyperbolic surface
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On the cover by Peter Randall-Page
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Top ten: geometry instruments
Prize crossnumber Roots: Blaise Pascal by Emma Bell
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chalkdust The team Rob Becke Chris Bishop Hugo Castillo Sánchez Atheeta Ching Thuy Duong “TD” Dang Sean Jamshidi Nikoleta Kalaydzhieva Antigoni Kleanthous Rudolf Kohulák Rafael Prieto Curiel Tom Rivlin Mahew Scroggs Belgin Seymenoğlu Yiannis Simillides Adam Townsend Cartoonist Tom Hockenhull Cover adapted from Euclidean Egg III by Peter Randall-Page
d chalkdustmagazine.com c contact@chalkdustmagazine.com a @chalkdustmag b chalkdustmag f chalkdustmag e Chalkdust Magazine, Department of Mathematics, UCL, Gower Street, London WC1E 6BT, UK.
We oen think of mathematics as the process of solving equations and using (sometimes terrible) notation and symbols to obtain a result. But perhaps one of the most valuable experiences that being part of Chalkdust has given me is the awareness that mathematics is also about the people who do the maths. This issue was inspired by people who pushed back the frontiers of our mathematical knowledge. Inside, you will find our interview with Cédric Villani, tips and suggestions by Marcus du Sautoy to effectively communicate maths, and the life and legacy of Maryam Mirzakhani. Mirzakhani broke a glass ceiling by being the first woman and the first Iranian to be awarded the Fields medal, and continues to be a role model to future generations of mathematicians. Being part of Chalkdust has allowed me to meet and work with a fantastic team of volunteers who invest their time and patience to create a magazine, to paint a large map of Africa to be used in a science exhibition or to melt a fun substance to create (or not) hot ice. In order to keep things fresh and interesting and to ensure that the project survives, we need to pass the torch to the next generation. Thank you to those who saw this project grow, from the day I brought my copies of Laberintos e Infinitos from Mexico, to this magazine for the mathematically curious. From now onwards I will only be an avid reader of its articles and an enthusiast for its great contents. Rafael Prieto Curiel Editorial director
Acknowledgements We would like to thank our sponsors for allowing us to make Chalkdust happen. We would like to give particular thanks to Luciano Rila for his help with the cover, Michael Singer and Momchil Konstantinov for assisting with the Maryam Mirzakhani article, Michael Perry for his helpful feedback on the interview, and Alison Clarke for her help with the leers page. We would like to thank Robb McDonald and the rest of the staff at the UCL Department of Mathematics for their continued support. As always, we thank Sam Brown for his work on our LATEX templates. This issue would not exist without its excellent authors, who continue to send us top quality articles. Most of all, we would like to thank you, our readers, for your continued enthusiasm, and challenge you to write an article for issue 07… ISSN 2059-3805 (Print). ISSN 2059-3813 (Online). Published by Chalkdust Magazine, Dept of Mathematics, UCL, Gower Street, London WC1E 6BT, UK. © Copyright for articles is retained by the original authors, and all other content is copyright Chalkdust Magazine 2017. All rights reserved. If you wish to reproduce any content, please contact us at Chalkdust Magazine, Dept of Mathematics, UCL, Gower Street, London WC1E 6BT, UK or email contact@chalkdustmagazine.com
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Being part of a crowd is something that we all have to experience from time to time. Whether it’s in a busy shop or commuting to work, the feeling of being swept along by those around us is all too familiar. The ubiquity of the situation, and the huge amount of data available from CCTV footage, makes crowd dynamics a favourite subject for mathematical modelling. One popular method is known as the social force model, which applies Newton’s second law to each member of the crowd. Each individual accelerates to maintain their ‘desired velocity’, and this is balanced against forces from physical obstacles as well as the social force that maintains polite distance between people—a mathematical interpretation of personal space! Huge simulations of up to a million pedestrians have been run, which show the model’s remarkable powers. If groups of people want to travel in opposite directions along a bridge, for example, lanes of alternating direction naturally form to minimise “bumping”.
Dirk Helbing and Peter Molnar
Lanes naturally form when people walk in opposite directions
Dirk Helbing and Peter Molnar
When two crowds meet at a gap, the walking direction oscillates
Some of the results are more unexpected. For example, if people try and move too fast then it can actually slow them down via an increase in ‘friction’ that results from pushing. Further, it can be shown that two narrow doors are a more e ective way of leaving a room than one big door, so putting a bollard in the middle of an exit actually speeds people up! Still, not much solace when you’re stuck in a Christmas scramble at Woolworths... References Helbing D and Molnar P (1997). Self-organization phenomena in pedestrian crowds. In: Schweitzer F (ed.) From individual to collective dynamics, 569–577.
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In conversation with. . .
Cédric Villani Yiannis Simillides, Matthew Scroggs and TD Dang
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on a February morning, we’re standing outside one of the many trendy cafes in Fitzrovia. Down the street we spot a man striding our way, wearing a full suit, a hat, a giant spider brooch and hastily tying a cravat. It could only be superstar mathematician Cédric Villani.
Cédric is passing through London on his way back from the US, but this is no holiday. In his two days here, he is aending a scientific conference, giving a public lecture, and taking part in a political meeting. His packed schedule leaves the increasingly-busy Fields medallist just enough time to join us for breakfast.
Fields medal One aernoon in early 2010, Cédric was in his office at the Henri Poincaré Institute in Paris, geing ready to pose for publicity photographs. The photographer, from a popular science magazine, was seing up his tripod when the office phone rang. Cédric leant over and picked it up. It was Lázló Lovász, president of the International Mathematical Union. Fields medal ceremonies are held every four years, and six months before each ceremony, the winners are alerted by telephone about their success. During these six months, they are sworn to secrecy, but with the photographer in the room, Cédric suddenly realised that he might be in possession of the shortest-kept secret ever. By some miracle, the tripod had proved sufficiently chalkdustmagazine.com
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chalkdust interesting for the photographer, or perhaps he didn’t follow the English conversation, and the secret remained safe. Cédric had first realised that winning the Fields medal was If you try too hard to win a a possibility at some point in 2004, when he was 31. Fields Fields medal, you will fail. medals are only awarded to mathematicians under the age of 40, and until the phone call arrived, Cédric only placed his chances of winning at around 40%. “The prospect of winning the medal does put some pressure on you during your 30s. But everybody knows—it’s part of the common mythology—that if you try too hard to win it, you will fail.” In August 2010, Cédric was officially awarded the medal at the International Congress of Mathematicians in front of 4000 mathematicians and journalists. Finally, he was allowed to celebrate: he did so by taking a dozen colleagues to a fancy restaurant in Germany, thereby relieving himself of half the $15,000 prize money.
The Boltzmann equation and Landau damping While enjoying a hearty breakfast, Cédric explains his research to us. “In this room, we are surrounded by air. You can use the Navier–Stokes equations to describe this air. But at higher altitudes, where the atmosphere is more dilute, these equations do not work so well. Here, it is beer to use the Boltzmann equation.” The Boltzmann equation describes the statistical behaviour of a gas, and Cédric has worked on two areas related to this equation: the influence of grazing collisions, where two particles pass very close to each other; and on the increase in entropy as time passes.
The Boltzmann equation can be applied up where the air is clear less dense
Cédric’s other work, completed with Clément Mouhot, looked at the mechanics of plasmas: highenergy soups of electrons and positively-charged ions which are formed by superheating gases. Roughly speaking, if a plasma is exposed to a brief electric field, then the electric field will become very small as time goes by. This decay effect is called Landau damping. In the 1940s, Lev Landau proved that this damping occurs for a linearised approximation of a plasma. Cédric and Clément proved this result for the full non-linear system of equations. It was the work in these two areas that led to Cédric being awarded the Fields medal, although he has worked in other areas as well. Imagine you have a large pile of sand and a hole to fill (with the same volume as the sand). How should you go about moving the sand to fill the hole, while minimising the total work you have to do? This is an example of an optimal transport problem. He used the ideas of entropy from his study of the Boltzmann equation and applied them to this problem, and used this to establish a link between the non-Euclidean curvature of a manifold and properties of the entropy. This led to a “whole bunch of research related to non-Euclidean geometry”.
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Career choices If the young Cédric had had his way, his research life would be very different. “When I was a kid, I wanted to go into palaeontology. I recently had a great discussion with Jack Horner, the world’s most famous expert on the subject—‘Mr Dinosaur’, and it was like reconnecting with my youth.”
Academia is where my heart belongs.
So is he happy in mathematics academia? “Academia is where my heart belongs. I like industry, and I sit on the advisory boards of several companies, but I’m an academic guy. My research has not had an application so far that I am aware of. But, with applications, when they come it will be much later.” Traces of Cédric’s early passion can still be spoed though. He owns a cuddly toy dinosaur called Philibert, and leaves maths books open to keep him entertained. Years later, he found that Alan Turing, one of his greatest heroes, used to do the same with his teddy bear at university. In fact, Turing is the hero in his recently-penned graphic novel, Les Rêveurs Lunaires. Excited readers will be disappointed, however, as “even though England is everywhere in the book, English publishers have not yet been interested in making an translation.” This is a doubleshame, as you will remember from Chalkdust issue 04 that comic books about maths are ‘hot’.
Grumpy Gauss, oil on canvas, Christian Albrecht Jensen (1840)
He is, however, less sure whether he would like to travel back in time to work with Turing or other mathematicians. “People like Gauss— so fascinating, so superhuman. But he was known for being rather grumpy; maybe it would not be so pleasant! Then take Riemann—a genius! But a bit depressive; maybe he was not so fun to work with. I’m not sure if he would want to see me.”
A day in the life of a Fields medallist Life is rarely routine for Cédric. In a usual year he travels to 20–25 countries, and has roughly 30 different appointments each week. When he can, he enjoys a quiet family breakfast at home. The contents of this breakfast have not changed since he was a child, and include bread, jam and hot unpasteurised milk. For today, however, he makes do with a full English with scrambled eggs. Dairy products seem to feature heavily in Cédric’s day-today life. Impressively, he is able to visualise every shelf in his favourite cheese shop and name, in turn, every item on sale. This is very important to him, as otherwise he could return home from grocery shopping to find himself without one of his many favourite cheeses.
I never give fashion advice. I always tell peopleν “find your own way”, as I did find my own way.
He is in London to give a lecture to the public, something that he spends a large amount of time doing these days, “much more so than to mathematicians. But both are good: different feelings, chalkdustmagazine.com
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chalkdust different preparations.” Overall, since winning the Fields medal and gaining fame, Cédric claims that his time for research has been “divided by hundreds”. Indeed, the public lecture is not his only commitment in London. He is currently aending a meeting at the Royal Society about the numerical abilities of animals. This meeting included great revelations about the mathematical abilities of frogs—evidenced through their calls involving sounds of varying number and length—as well as fish, bees and chimpanzees. “One of the crazy things that emerged from this conference is that the tendency to put small numbers on the le and large numbers on the right is not merely a side effect of how we write numbers. You can also find this—in some sense—in newborn chicks and fish.” When in France, Cédric is recognised everywhere he goes, and is (still) posing for photographs. He is regularly featured on the covers of science magazines, and is oen confronted by giant billboards of his face. If you are planning on winning a Fields medal, do not panic: he assures us that you will quickly get used to this.
Politics When we meet Cédric, the French election is in full flow. As part of his stay in London, he is aending a meeting for the candidate he describes as the “young, centrist guy”. He is one of seven scientists on a board that provides scientific policy advice to the European Commission. However, he doesn’t recommend becoming too involved in politics, as he thinks there is no way to find time to pursue both a serious research career and a serious political career. “The current political climate is far from science in Cédric enjoying a popular maths magazine general. Science, as a field, is much more respected by society than politics. So there is reputation to be lost by going into politics. But the most popular politician in French history is Napoleon, and he was keen on mathematics, and a big protector of mathematicians and scientists. He was elected to the academy of sciences, aending when he could, and enjoyed discussions with many of the best mathematicians of his time. But he was always late…” Keen not to be late himself, Cédric finishes his eggs and heads off to his next commitment. It would seem, however, that Cédric does not always listen to his own advice: in June he became an elected member of the French parliament, as a member of the young, centrist guy’s party. Yiannis Simillides, Matthew Scroggs and TD Dang Yiannis, Mahew and TD are students at University College London. Yiannis has not seen the 1967 James Bond spoof Casino Royale, starring David Niven, Peter Sellers, Woody Allen, and Orson Welles as Le Chiffre. Mahew and TD have, but wish they hadn’t.
a @YSimillides a @mscroggs d mscroggs.co.uk a @televisionduck 7
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WHAT’S
& WHAT’S
HOT NOT HOT
Sudoku
Crossnumbers
Everywhere now, including the Times and pp 52–53 & 58.
Grade 9
NOT HOT
If you nd the new grades confusing, just remember the conservation of grade law: new GCSE + O-level = 10.
They’re all exactly the same. I say Su-no-ku.
Maths is a ckle world. Stay à la mode with our guide to the latest trends.
Agree? Disagree? a @chalkdustmag b chalkdustmag
HOT
Hacker camps
Grade A*
Who collects these any more? About as current as Pokémon Go.
NOT
A long weekend of talks, workshops, re pong and a cocktail making robot? Count me in!
HOT
String theory
A great eld that proves that physics is (occasionally) not terrible.
Law of Australia
HOT
The only law that applies in Australia.
Knot theory
What happens when string theory gets tangled.
HOT Rule 30
As seen at Cambridge North station.
Laws of mathematics
Great place to lose your phone while having a rubbish time.
NOT Game of life
As seen at no train stations.
NOT
Pictures: Background (illuminated dancefloor): Flickr user Seraphim Whipp, CC BY-SA 2.0; Cambridge North: Hugh Venables, CC BY-SA 2.0; Kangaroos: WikiMedia commons user Dellex, CC BY-SA 3.0; Sudoku: Héctor Rodríguez, CC BY-NC 2.0; Electromagntic Field: Sheila Thomson, CC BY 2.0.
“Very commendable” but not good enough.
More free fashion advice online at d chalkdustmagazine.com
NOT
chalkdustmagazine.com
KNOT
Music festivals
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Cardioids in coffee cups Flickr user fdecomite, CC BY 2.0
Dominika Vasilkova
P
the scene. It’s 1am and you’re up late working on some long-winded calculations. The room around you is dark, a desk lamp the only source of light. Your eyelids start to droop. But the work must get done! Time to fall back on the saviour of many a mathematician: coffee. But as you sit back down at your desk, you notice something weird. The light from your lamp is reflecting oddly from the edges of the cup, creating bright arcs—and it looks suspiciously like a cardioid curve! Time to investigate… Work forgoen, you pull out a clean sheet of paper and— well, dear reader, you may have been more sensible than me and just gone to bed at this point, or finished the work you were meant to be doing. For me, though… well, let’s just say that sleep would be impossible until this mystery was resolved. So. We have a cup. We have a light. We have an enigmatic looking curve. What’s going on? 9
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Let’s shed some light on this To keep things simple, we’re going to model the base of the cup as a perfectly reflecting two-dimensional circle, and limit the incoming light to the plane that the circle is in. We can also set the radius of the cup to be 1 without loss of generality. Since the cup and the lamp are reasonably far apart, a decent assumption to make is that the light coming in is collimated, which means that all the light beams are parallel to each other, as if from a point source at ‘infinity’. If we look at a single light ray coming in parallel to the x-axis, we know that the angle of incidence and The paths of two collimated light rays and the angle of reflection are equal, as measured from their different angles of reflection a line normal to the circle. If a second ray comes in parallel to the first, it will hit the surface at a different angle, so must reflect off at a different angle to the first, and so the reflected rays will no longer be parallel. Instead, they will overlap as shown above. A rule of ray tracing tells us that all light rays parallel to the axis will go through the focal point of a curved surface. This is the point where the two light rays cross in the diagram above. We can therefore expect that to be the brightest point of the paern that we see in the coffee, since it is hit by the most light. But what about the rest of the curve? If we draw in a few more light rays, as in the diagram to the le, we start to see areas where many different rays overlap and can build up a picture of the curve. Treating the incident light rays as a family of curves, the bright paern seen is their envelope. The envelope As we build up light rays, the shape of the is a curve that at every point is tangent to one of the envelope begins to emerge. The only differincoming rays, and so by moving along its length we ence between this diagram and the physical system is that here, the overlap makes a move between the different members of the family. dark envelope, whereas in the cup the overDue to the tangent property, it is also the boundary lap makes a bright envelope of the most dense area ‘swept out’ by the curves, so in many cases this corresponds to the curve you’d get by joining up all the points of intersection. If we can find the equation of this envelope, we can find out exactly what shape is being formed in the boom of the cup.
Calculus to the rescue! A hint about how to find the equation we need comes from the definition: we need to find a curve chalkdustmagazine.com
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chalkdust that is tangent to our family of curves at every point. So, perhaps unsurprisingly, a good way to do this is to use calculus. For a smooth family of curves, we first find a general equation for the curves by expressing them in terms of some parameter, say a. We can then find the equation of all their tangents by differentiating with respect to a. Since we need these two equations to match at every point along the envelope, to find the equation of the envelope we solve them simultaneously. For example, say we wanted to find the envelope of straight lines that enclose equal area between them and the axes—picture this as a ladder propped against a wall, but sliding down it. The equation of a straight line in terms of both axis intercepts has the form x y + = 1, a b where a and b are the intercept points. These are both parameters that describe the family of curves, so we use the fact that we want to keep the area A = ab/2 constant to eliminate b: x ay + = 1. (1) a 2A Differentiating this with respect to a gives y x = 0. (2) − 2+ a 2A So equations (1) and (2) are what we want to solve simultaneously. In this case, it’s possible to do this by eliminating a between the two equations, giving the equation of the envelope as √ A √ , xy = 2 or, if we limit ourselves to the first quadrant, A . 2 This is the equation of a hyperbola. An example case for A = 1 is shown to the right. In this case, we were able to eliminate the parameter from the equation to leave it only in terms of x and y, but as we will see later, it may be easier to leave envelopes in their parametric form. xy =
The curves and envelope formed for A = 1
But what about the co ee? Now we know how to find the equation(s) for an envelope, we can apply this method to our cup scenario. We know the equations of the lines coming in, since they’re all just straight lines parallel to the axis, but we need to find out the equation of the reflected light beams. It turns out that this can be done by taking advantage of the fact that the cup is circular and throwing some trigonometry at it. 11
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chalkdust Consider a light beam coming from the right and striking the cup at a point (x, y). The beam is at an angle θ to the normal of the surface, so using angle of incidence = angle of reflection, we know it will reflect at the same angle, as shown in the top right diagram. Using the fact that the red triangle is isosceles, the point (x, y) is therefore at an angle θ from the negative x-axis, so we can parametrise the point as (− cos θ, sin θ) as the radius of the cup is 1. The slope of the reflected ray is then − tan 2θ and the equation of the line, in terms of θ, is y − sin θ = − tan 2θ (x + cos θ),
We can use basic geometry to express our family of curves in terms of θ
or, aer some identity jiggery-pokery, x sin 2θ + y cos 2θ = − sin θ.
(3)
A plot of a few of these curves, with different values for θ, is shown in the middle right diagram and looks similar to what we see in the cup. That’s a good sign! Differentiating the above equation with respect to θ gives 2x cos 2θ − 2y sin 2θ = − cos θ. (4) Now, eliminating θ between these would be downright disgusting, so expressing (3) and (4) in matrix form and solving for x and y gives ( ) ) ( 1 cos 3θ − 3 cos θ x = . y 4 3 sin θ − sin 3θ
Plot of equation (3) for a few different values of θ. Note that these are just the reflected rays
Ploing this, it matches up very nicely on one side of the cup. But, unlike the real paern in the coffee cup, this one has an extra bit of curve! And this equation, alas, doesn’t describe the cardioid we’d hoped for—this is the equation of a nephroid. To add insult to injury, by the time I’d worked all this through, my coffee was ice cold. Yuck. In fact, the extra bit of curve appears due to all values of θ being allowed. As the sides of the cup will block about half the light, this imposes a restriction on θ, so we only chalkdustmagazine.com
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A plot of equation (3) with the calculated envelope drawn on
chalkdust see half the curve in our coffee. And a nephroid is, actually, correct—shapes like these that form when light reflects off a curved surface are called ‘caustics’, from the Greek word for ‘burnt’, as they can be used to focus sunlight to start fires. Both the nephroid and the cardioid belong to a larger family of curves called epicycloids, which are categorised according to the number of ‘cusps’ (sharp bits) that they have.
But I wanted a cardioid, dammit! If nephroids aren’t your thing, it’s possible to get a cardioid caustic in a cup if we change the setup slightly. Instead of having a point source at infinity, let’s put the point source on the rim of the cup and see what happens. The geometry of the incoming and outgoing rays is shown to the right. This gives the equation of the line as y (1 + cos 3θ ) + x sin 3θ = sin θ − sin 2θ, and differentiating, we get
−3y sin 3θ + 3x cos 3θ = cos θ − 2 cos 2θ.
The angular setup for a point on the rim. To keep the parametrisation the same as the last case, the angles of incidence and reflection have been defined differently
Solving these simultaneously is a tad more fiddly than before, but working through gives the envelope as ) ( ( ) 1 cos 2θ − 2 cos θ x . = y 3 2 sin θ − sin 2θ This is a cardioid. Yay!
Reflected light beams from a point source located where the rim touches the positive x-axis
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Additional complexities What we see in a cup is unlikely to be exactly one of the two previous cases. If the caustic is bright enough to be visible, the light source is probably not far enough away to be at ‘infinity’, and people don’t tend to go around puing point sources on the rims of their cups. If we have a light source that’s a finite distance from the cup edge, the incident rays will be at an angle somewhere between the nephroid and cardioid cases, so the curve seen is somewhere between the two. There is another large assumption we’ve made here that renders the situation somewhat unphysical. Our cup is a twodimensional circle! And although that makes the maths nicer, it’s not great for holding coffee. The physical principles are the same in 3D, with an extra angle to worry about, so what you actually see in a coffee cup is the intersection of the surface in the diagram on the right with the boom of your cup.
Rich Morris (singsurg.org), CC BY-NC-SA 4.0
The cusp catastrophe
This surface is called the ‘cusp catastrophe’, and can be found using catastrophe theory, which, among other things, looks at the behaviour of manifolds with singularities in them.
A change of focus Since we can focus light using almost any process that changes its direction, reflection caustics (or catacaustics) such as the ones we have been considering here are not the only type possible. A common refraction caustic is the rippling paern of light seen on the boom of bodies of water, and a rainbow is a caustic caused by a combination of reflection and refraction. More exotically, gravitational forces bend space-time and therefore the light travelling through it, which means that gravitational lensing can give rise to caustics of astronomical scale. The shape of the caustic gives key information about the astronomical object, and this method has been used to identify and analyse exoplanets around distant stars. Although these physical systems look completely different at first glance, they’re linked by a single phenomenon. The same flavour of physics that describes how the light in your morning cuppa behaves also describes the behaviour of light on ridiculously huge scales in the universe. And that’s prey cool, don’t you think? So, the next time you sit down to enjoy a hot beverage, take a moment to appreciate the awesome things happening, quite literally, right under our noses. Dominika Vasilkova Dominika is an Imperial physics student by day, a recreational mathematician by night, and a full-time dragonologist. She is oen found seing things on fire (by accident!) or scaring off other students with overexcited ramblings about Fourier analysis.
c dv814@ic.ac.uk a @dragon_dodo chalkdustmagazine.com
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Pictures CC BY-SA 4.0 R logo: Hadley Wickham & others at Rstudio. CC BY 3.0 SageMath logo: Sage team. CC BY-SA 2.5 Fortran card: Arnold Reinhold. Commodore: Bill Bertram. CC BY 2.0 Background image: Nicolas Raymond. GPL Python logo. Public domain Matlab logo. Mathematica logo. Alien. Yellow paper. Pen.
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PYTHON
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Literary mathematics
Roberto de la Cruz Moreno
I
Paris, November 1960, a group called Oulipo—an acronym of Ouvroir de liérature potentielle (workshop of potential literature)—was formed by the writer Raymond eneau and the mathematician François Le Lionnais. The group was motivated by experimental literature, and developed new writing methods through collaboration between mathematicians and artists. eneau defined potential literature as “the seeking of new structures and paerns which may be used by writers in any way they enjoy”. In this article, we see a few representative examples of Oulipian writing.
The poetry book that you will never finish reading The title of eneau’s 1961 book, Cent Mille Milliards de Poèmes (A Hundred Thousand Billion Poems), is not an exaggeration. This book, pictured right, contains exactly this many different poems, on just 10 pages. On each page there is a sonnet (a poem with 14 verses), but each verse is on a separate strip. Using these strips, the reader can form new sonnets by choosing each verse from the 10 options. The original sonnets have the same rhyming sounds, so each sonnet formed rhymes correctly. The number of possible sonnets is 1014 , or a hundred thousand billion. How much time would you need to read all the sonnets? I leave this as an exercise for the reader. Clément Gault, CC BY-NC-SA 2.0
(S+7) method The (S + 7) method was devised by the poet Jean Lescure and involves replacing every noun of a pre-existing text with the seventh noun following it in a dictionary. Of course, we could also try variations of this: taking other types of words (adjectives, verbs, etc.), taking the umpteenth word, or using your favourite maths magazine instead of a dictionary. Using a similar idea, eneau translated David Hilbert’s book Foundations of Geometry into Foundations of Literature by replacing the words ‘point’, ‘line’ and ‘plane’ by ‘word’, ‘phrase’ and ‘paragraph’ respectively. chalkdustmagazine.com
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Word matrices In the book Meccano ou l’Analyse Matricielle du Langage (Meccano or the Matrix Analysis of Language), eneau uses matrix multiplication to generate poems. He starts with a simple example to show how it works: cat ( ) the has the eaten = the · cat + has · eaten + the · mouse mouse And he continues constructing more elaborate creations, such as: end summit edge side 1 1 1 1 highway Annapurna ocean fencer ( ) rising standing bathing standing (On the) of the was the 1 of the black Tibetan mystical passionate sun Sherpa masseur bully 1 1 1 1 melancholy team Trinidad Marquise
The puzzle novel Lifeν a User’s Manual is the most famous novel of the writer Georges Perec. Perec defined his novel as a puzzle: “The whole book has been built as a house in which the rooms are joined to each other following the technique of the puzzle.”
The knight’s tour used in Lifeν a User’s Manual
The story takes place in an apartment building, divided into 100 rooms (10 floors of 10 rooms, so the building can be thought of as a 10 × 10 laice). Each chapter of the book takes place in one of these rooms and each room appears in just one chapter. The order of the rooms, however, is not random: it is a knight’s tour, mimicking a chess knight visiting each square exactly once. The author himself found this tour experimentally.
Roberto de la Cruz Moreno Roberto is a mathematician and a laboratory technician. He recently obtained his PhD in mathematical biology at the Centre de Recerca Matemàtica in Barcelona. He has loved maths and reading since he was a child.
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Geographic profiling: murder, maths, malaria and mammals © OpenStreetMap contributors. Cartography CC BY-SA 2.0
Michael Stevens & Sally Faulkner
I
you’re a police officer working on a huge case of serial crime. You’ve been handed the list of suspects, but to your horror 268,000 names are on it! You need to come up with a way of working through this list as efficiently as possible to catch your criminal. Along with the thousands of names, you’re also given a map with the locations of where bodies have been found (the map above). Given these two pieces of intel, how exactly would you prioritise your list of suspects? Have a go! Where exactly would you search for the criminal? We will reveal the answer at the end of article! Peter Sutcliffe, also known as the Yorkshire Ripper, was the name on a list of 268,000 suspects generated by this investigation in the late 1970s. But how were the team investigating these crimes meant to cope with such an overload of information? These are the fundamental problems that geographic profiling is trying to solve. How exactly does geographic profiling work? This article will introduce you to the fundamental ideas behind the subject. We will also look at the various applications, just like the Yorkshire Ripper case, along the way. These examples aren’t just in criminology though. The applications span ecology and epidemiology too!
The rst model Geographic profiling uses the spatial relationship between crimes to try and find the most likely chalkdustmagazine.com
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chalkdust area in which a criminal is based; this can be a home, a work place or even a local pub. Collectively we refer to these as anchor points. The pioneer of the subject, Kim Rossmo, once a detective inspector but now director of geospatial intelligence/investigation at Texas State University, created the criminal geographic targeting model in his thesis in 1987. The criminal geographic targeting model aims to do exactly what we struggled with at the beginning of this article: prioritise a huge list of suspects. It starts by breaking up your map, populated with crime, into a grid, much like on the le. We assume that each crime that occurs, does so independently from every other. We then score each grid cell; the one with the highest score is likeliest to contain the criminal’s potential anchor point. How do we calculate this score? An important factor is the distance between crimes and anchor points. We choose to use the Manhaan metric as our measure of distance. In this metric, the distance between points a and b is the sum of the horizontal and vertical changes in distance. This is wrien as: Alistair Marshall, CC BY 2.0
d (a, b ) = |xa − xb | + |ya − yb |,
A gridded-up map
a = (xa , ya ),
b = (xb , yb ).
The Manhaan metric is so-called because it resembles the distance you have to travel to get between two points in a gridded city like Manhaan. This is the most suitable metric for our work, but it’s worth noting there are more that can be used (depending on the system you’re studying). Now we could just start searching at the spatial mean of our crimes and work radially outward from that point, however one rogue crime occurring far away from the rest could easily throw a spanner in the works. Instead we use something called a buffer/distance decay function. 5
score
4
k h, f (d ) = d kB g−h , (2B − d )g
3 2 1
anchor point
0 0
2
4
6
8
d >B d ⩽B
10
distance
A criminal isn’t likely to commit a crime close to an anchor point, out of fear of being recognised, so we place a buffer around it. In addition, to commit a crime far away from home is a lot of hassle, so the chance of a crime decays as we move away from the anchor point. This is why our buffer/decay function looks a bit like a cross-section of a volcano. The explicit function, f (d ), is wrien on the right, where k is constant, B is the buffer zone radius and g and h are values describing the criminal’s aributes, eg what mode of travel they use. With our distance metric, d, and buffer/decay function, f, we are now able to compute a score for each grid cell. 19
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chalkdust For n crimes, the score we give to cell p is S(p ) =
n ∑
f (d (p , c i )),
i=1
where c i is the location of crime i. So finally we have a score for each grid cell and we can prioritise our list! Ploing these scores on the z -axis produces a mountain range of values, like on the right. We can now prioritise by checking residencies at the peak of this mountain range and working our way down. Notice the collection of peaks around a particular area: this gives us an indication that perhaps the criminal uses more than one anchor point.
An example of the geographic profile created using the criminal geographic targeting model
An important question: how can we be sure this even works? Does it really identify anchor points efficiently? What do we even mean by ‘efficient’? This is answered with a quantity called the hit score. This is hit score =
number of grid cells searched before finding the criminal . total number of grid cells
So ironically, the lower our hit score, the beer our model performs. This is sensible, since we want to search as lile space as possible to catch our criminal.
The Gestapo case Oo and Elise Hampel distributed hundreds of anti-Nazi postcards during the second world war. The Gestapo’s intuition on where the Hampel duo might live was based on themes almost exactly the same as geographic profiling. Inspired by a classic German novel, Alone in Berlin, our group revisited the Gestapo investigation and published our findings in a journal that is so highly classified we are not able to read it. By analysing the drop-sites of the postcards and leers we were able to show that geographic profiling successfully prioritises the area where the Hampels lived in Berlin. Crucially, this study actually showed the importance of analysing minor terrorism related or subversive acts to identify terrorist bases before more serious crimes occur. chalkdustmagazine.com
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A statistical approach The criminal geographic targeting model is an incredibly useful tool and is used to this day by the CIA, the Metropolitan Police and even the Canadian Mounted Police. Mike O’Leary, professor at Towson University, Maryland asked why the criminal geographic targeting model only produces a score, when we require a probability. So he developed a way of using geographic profiling under the laws of Bayesian probability. O’Leary uses Bayes’ rule as seen on the right. How do we apply it to criminology? We want to know: what is the probability that an offender is based at an anchor point given the crimes they have commied? Using Bayes’ rule, instead we pretend we know where the anchor point is and ask; what is the probability of the crimes occurring given our anchor point? We use the formulation
Pr(c 1 , c 2 , c 3 , c 4 … | p ) =
n ∏ i=1
Wikimedia Commons user Mattbuck, CC BY-SA 3.0
Bayes’ rule is beer in neon
Pr(c i | p ),
where the equality derives from the assumption of independent crimes. Below, we can see a comparison between Rossmo’s criminal geographic targeting model and O’Leary’s simple Bayesian model. The problem with O’Leary’s model is he assumes that a criminal only has one anchor point. Unfortunately this is rarely the case. As we mentioned earlier, an anchor point could be a home, a workplace, a local pub or even all of the above. So we obtain a probability surface, but we only consider one anchor point. The criminal geographic targeting model entertains the idea that multiple anchor points exist, but doesn’t give us an explicit probability. What we really need is a way of combining both methods. Does such a method exist?
(a) The criminal geographic targeting model
(b) The simple Bayesian model
Examples of the geographic profiles created using the criminal geographic targeting and simple Bayesian models
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The elusive tarsiers South-east Asia, specifically Sulawesi, houses a huge number of endemic species. Oen habitat assessments of cryptic and elusive animals such as the tarsier (right) are overlooked, primarily due to the difficulties of locating them in challenging habitats. Traditional assessment techniques are oen limited by time constraints, costs and challenging logistics of certain habitats such as dense rainforest.
Callum Pearson
Using only the GPS location of tarsier vocalisations as input into the geographic profiling model we were able to identify the location of tarsier sleeping trees. The model found 10 of the 26 known sleeping sites by searching less than 5% of the total area (3.4 km2 ). In addition, the model located all but one of the sleeping sites by searching less than 15% of the area. The results strongly suggest that this technique can be successfully applied to locating nests, dens or roosts of elusive animals, and as such be further used within ecological research.
The best of both worlds The Dirichlet process mixture model is the best of both the criminal geographic targeting and the simple Bayesian models. So far we’ve only stated that we’re either working with one anchor point, or many. The beauty of the Dirichlet process mixture model is that we don’t need to specify the number of anchor points we are searching for. Instead, there is always some non-zero probability that each crime comes from a separate anchor point. So multiple anchor points can be identified while using a probabilistic framework. Introducing multiple anchor points is challenging since we need to know: 1. How are all the crimes clustered together? 2. In each cluster of crimes, where is the anchor point? Actually, what would be really useful is if we knew the answer to just one of these questions. If we knew how the crimes were clustered, finding the anchor points is easy (we use the simple Bayesian model to find the source in each cluster). But also, if we knew where the anchor points were, allocating crimes to clusters is easy (and of course we know where our criminal lives!). The solution to this problem is to use something called a Gibbs sampler. We use a Gibbs sampler in cases where we want to sample a set of events that are conditional on one another. In our case, anchor point locations depend on the clustering of crimes, but the clustering of crimes also depends on the anchor point locations. The steps the Gibbs sampler will take are: 1. Randomly assign each crime an anchor point (even though we don’t yet know where the anchor points are). 2. Find each anchor point by using the simple Bayesian model on each assignment. chalkdustmagazine.com
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chalkdust 3. Throw out the assignments of crimes to anchor points and now re-assign crimes but using the locations found in previous step. 4. Throw out the old anchor point locations and find new ones using this new assignment. 5. Repeat steps 3 and 4 many, many times. This produces a new profile like on the right below. We can now compare this to our other two models on the le. We can see the Dirichlet process mixture model displays fewer peaks than the criminal geographic targeting model, but that these peaks are tighter. This in turn will reduce the hit score of our search.
(a) The criminal geographic target- (b) The simple Bayesian model ing model
(c) The Dirichlet process mixture model
A comparison of the three main geographic profiling models
The malaria case Throughout history, infectious diseases have been a major cause of death, with three in particular (malaria, HIV and tuberculosis) accounting for 3.9 million deaths a year. Targeted interventions are crucial in the fight against infectious diseases as they are more efficient and, importantly, more cost effective. They are even more crucial when the transmission rate is strongly dependent on particular locations. For example, we were tasked with finding the source(s) of malaria outbreaks in Cairo by considering the breeding site locations of mosquitos. All accessible water bodies within the study area were recorded between April and September 2005, and 59 of these were harbouring at least one mosquito larva. Of these 59 sites, seven tested positive for An. sergentii, well-established as the most © OpenStreetMap contributors. Cartography CC BY-SA 2.0 dangerous malaria vector in Egypt. Using only the Water bodies with mosquito larvae spatial locations of 139 disease case locations as input into the model, we were able to rank six of these seven sites in the top 2% of the geoprofile.
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Applying the method We’ve done it! We now have a robust method for searching for our criminal. A list of 268,000 suspects is no longer so intimidating. Without this technique in 1975–1981, however, there was a lot more work for the team investigating the Yorkshire Ripper case. On top of a huge list of suspects, 27,000 houses were visited and 31,000 statements were taken during the investigation. If we apply our model to the crime sites we were given at the start of this article, we produce the contour map on the right. In this case the areas in white describe the © OpenStreetMap contributors. Cartography CC BY-SA 2.0 The geoprofile associated with the Yorkshire Ripper highest points on our probability surface, body dump sites (black dots). The anchor points of Pewhilst areas in red describe the lowest. In ter Sutcliffe are labelled as red squares addition to the contours, we also see two red squares right at the top of the map. These are the two homes Peter Sutcliffe resided at during the period of his crimes. The hit scores for his two residences are 24% and 5% respectively. So by searching only 24% of our total search area, we’ve managed to find both residences. This is far beer than a random search which would find them aer searching, on average, 50% of our area. Peter Sutcliffe’s homes are clearly marked on this map but we must remember an important point about geographic profiling: that it is not an ‘X marks the spot’ kind of model, but rather a method of prioritisation.
Investigating an old case We can’t talk about the Yorkshire Ripper without mentioning the notorious 1888 London serial killer, Jack the Ripper. The five locations around Whitechapel where bodies were dumped were studied using geographic profiling to try and gain a beer idea of where Jack the Ripper may have lived. The map overleaf shows us the associated geoprofile, with Jack’s suspected anchor point obtaining a hit score between 10–20%, much beer than a non-prioritised search! chalkdustmagazine.com
The Illustrated Police News
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Dramatic scenes covered the newspaper front pages, such as this from 1888
chalkdust This is just one example of many cases where we can utilise our new model to study cases from the past where such tools were not available.
© OpenStreetMap contributors. Cartography CC BY-SA 2.0
The geoprofile associated with the body dump sites (black dots) of Jack the Ripper. Jack’s anchor point (the red square) is suspected to be around Flower and Dean Street
Geographic profiling began in criminology, but now spans ecology (catching invasive species) and epidemiology (identifying sources of infectious disease) too. This means saving a hey chunk of time and money, as well as developing prevention strategies to minimise any negative impacts these problems may cause.
Michael Stevens & Sally Faulkner Michael and Sally are PhD students at een Mary University of London. Michael is geing under the bonnet of geographic profiling to improve the model’s performance. His interests are in mathematical modelling and data visualisation within environmental science. Sally is developing the geographic profiling model for use with biological data, addressing in particular conservation issues. Sally’s interests lie primarily in wildlife crime and conservation.
c m.stevens@qmul.ac.uk a @Mr_MCAS c s.c.faulkner@qmul.ac.uk a @tarsiussallius My least favourite notation Notation is great for saving time and effort when writing out maths. But bad notation can ruin even the greatest calculations. Throughout this issue, we share some of our least favourite pieces of notation. We’d really love to hear about yours! Send them to us at c contact@chalkdustmagazine.com, a @chalkdustmag or b chalkdustmag and you might just see them on our blog!
Einstein summation convention Sean Jamshidi
My least favourite notation is the Einstein summation convention. Normally, if you want to sum lots of similar terms you can use a handy symbol, Σ, which also tells you which terms you want to add together and how many of them there are. Using Einstein’s convention, however, you forget the Σ and leave your reader to guess these important details, or spell them out in words, both of which take longer than it would have done to write Σ in the first place! Not Einstein’s greatest contribution to maths. Maybe he wasn’t so smart aer all, says I (jk). 0ij /10
Largest triangle What is the largest area triangle that has one side of length 4 and one side of length 5? 25
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Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the prof’s help? Contact c deardirichlet@chalkdustmagazine.com
Dear Dirichlet, I’ve just started my PhD at a we ll-known university, and I’m tryi ng to make some friends. There are supposed to be 55 other students but nearly everyone in the PhD o ce refuses my o ers of tea, sits in silence, and will barely talk to me unless I whisper them some very speci c technical questions. I was hop ing there would be some people in the group wh o enjoy everyday things: biscui ts, beer, and just shooting the breeze. Is this rea lly what academia is like?
— Pearl among swine, Withheld
DIRICHLET SAYS: Most new employees feel like this initially, and
■
academia is no exception. You simply need to find a normal subgroup. Alas, it may not be very big: for a group of order 56, expect a normal subgroup of order 7 or order 8. It should be jolly good fun finding them, though: this subgroup is defined as being invariant under (ahem) conjugation. (Alternatively just follow them home from work as by definition, your colleagues who commute with each other - the centre - are a normal subgroup).
Dear Dirichlet,
It has processors a new kind of supercomputer! I’ve nally done it! I’ve invented nitely more do better than that. They’re in hey l—t alle par in rk wo t jus ’t which don s! I call it ultraparallel!! parallel than regular processor er Champ, London
— Comput
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DIRICHLET SAYS: i) This is not a question. I’m afraid, champ, that
your behaviour mimics an irritating crackpot at a plenary lecture who insists on making self-congratulatory statements after the esteemed professor at the front has spoken. ii) I’m not sure I believe these claims. I think you’re being a bit hyperbolic. chalkdustmagazine.com
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Dear Dirichlet,
I’m from continental Europe and I can’t make sense of all these roa d signs in America with their distances listed in miles! What’s going on? — Pedro, Cabal
lo Lake
■ DIRICHLET SAYS: The problem here, of course, is that distance is impossible to measure without a properly defined metric system, which America does not have. Maybe try asking someone from Imperial College?
Dear Dirichlet,
I’m on sabbatical in Egypt and just doing some light archaeolo gy. Currently I’m trying to enter a sealed tomb hid den deep within the Valley of the Kings, but I can’t seem to enter without working out the meaning of an ancient code written on the door. Any advice?
— H. Jones Jr, Chicago DIRICHLET SAYS: Ah, Egypt.
One of the many countries where one cannot find badgers. I once spent an afternoon in de-Nile, but ended up in de-Mediterranean Sea. Anyway, to your question... Do you not have the key? Is there not a back door? Sounds like you need to brush up on your knowledge of crypt-ography. ■
Dear Dirichlet,
ential future jobs have to come up with some pot we and ool sch at ek we eer car It’s ms a bit of a uning of going into teaching but it see for ourselves. I’ve been thinkin involves travel. should choose something that spired choice. My friends say I g me to become ior ecclesiastical leaders electin What are the odds of all of the sen — Sophie Willock, Bedford College the Pope?
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DIRICHLET SAYS: Well, given that the number of cardinals is infinite, I’d say
quite unlikely! ... But I don’t want to pontificate. More Dear Dirichlet, including two seasonal specials, online at d chalkdustmagazine.com 27
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Pretty pictures in the complex plane Wolfgang Beyer, CC BY-SA 3.0
Emily Clapham
S
of the greatest works of art in history have been produced by mathematicians. One fascinating source of mathematical artwork is fractals: infinitely complex shapes, with similar paerns at different scales. Fractal geometry has dramatically altered how we see the world. Technology has many uses for fractals, one of which is the production of beautiful computer graphics. These prey pictures are used to present a large amount of information about a function in a clear and comprehensible manner, and the simplicity of the maths involved in producing these pictures is fascinating. Modern art studies have oen been dismissive of the power of beauty in mathematics, with the idea that “beauty is not in itself sufficient to create a work of art”. Mathematics produces rigid and inflexible answers, whereas art is free-moving and open to interpretation. However, it is undeniable that these prey pictures demonstrate true beauty, not only in the images but also in the mathematics behind them.
Prey pictures in the zplane are widely used as computer graphics, book covers and even sold as works of art.
The mathematics behind the pretty pictures Extremely simple functions can be used to produce these pictures. For example, let’s consider the quadratic function f (x ) = x 2 + c, for some constant c. 29
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chalkdust An iterative method is applied to the function. First, a seed (let’s call it x0 ) is selected to be the initial value for iteration. The solution of the function is then subsequently recycled as the new input value, x. In this way: x 1 = x 20 + c, x 2 = x 21 + c = (x 20 + c)2 + c, x 3 = x 22 + c = · · ·
and in general, x n = x 2n−1 + c.
We continue until the iteration either converges to a fixed point or cycle, or diverges to infinity. The orbit is the sequence of numbers generated during the process of iteration: x0 , x1 , x2 , x3 , . . . , xn . If we only apply real numbers to the quadratic function we limit the graphical representation of the iterations to a line. To produce pictures in the plane, we use complex numbers instead. Through the process of iteration, each seed will either converge or diverge, and so for a given function we can divide the plane into an escaping set Ec = {z0 : |zn | → ∞ as n → ∞} (that is, all the seeds that end up at infinity) and prisoner set, where the iteration tends to a point or becomes periodic.
The abundant beauty in the plots is somehow increased when the simplicity of the mathematics is understood.
The Julia set of a function To go from the iterative procedure described above to the vivid images to the right, we need to introduce the idea of the Julia set of a function, named aer the French mathematician Gaston Julia. Julia was an extraordinary man, who tragically lost his nose while fighting in the first world war. Despite the substantial injury, he made immense progress in the field of complex iteration and published the book Mémoire sur l’itération des fonctions rationnelles in 1918, which began the study of what we now call a Julia set.
Connected and unconnected Julia sets of the quadratic function for different values of c
The filled-in Julia set is the collection of points in the complex plane that form the prisoner set of a function, while the Julia set itself is the boundary of this region. The points within the filled-in Julia set remain bounded under the iteration since their orbits converge to an aracting point or cycle. Conventionally, when pictures of the Julia set are shown, the filled-in Julia set is shaded black and varying colours are used to show the rate at which the escaping set diverges to infinity. The Julia set is therefore the edge of the black region. Maps 1–7 above show the Julia sets of the quadratic function for different values of c, with the escaping set colour-coded as follows: red areas represent chalkdustmagazine.com
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chalkdust points that slowly escape from the set, while blue areas signify points that quickly escape to infinity. The value of the complex constant c influences the shape of the Julia set. Maps 1, 4 and 5 all have black centres, which indicate that the Julia set is connected, while maps 3, 6 and 7 demonstrate unconnected sets. For these images, the Julia sets have no black regions and instead the pictures are just flurries of colour. It is not always easy to spot whether a Julia set is connected, however. In map 2, there is no obvious black region, but neither are there colourful individual flurries and instead we see a spiky line. In fact the set is connected, it is just that the filled in Julia set is so slender that the black line points are not visible in the image. During the initial study of these sets, a fascinating criterion for connectivity was discovered concerning the critical point, z0 = 0. If the critical point is used as the seed, we produce the critical orbit, which is bounded if and only if the Julia set is connected. Fractal paerns appear in all plots, apart from when c = 0 or −2. The picture below displays examples of magnified sections of the fractals, for c = −0.7 (maps 9–12), c = −0.12 + 0.75i (maps 13–16), c = 0.1 + 0.7i (maps 17–20) and c = −0.1 + 1i (maps 21–24). Each enhancement of a section produces what appears to be copies of the whole section, not just in overall shape but also with smaller embellishments on every ‘limb’. For connected plots, these fractals appear as loopy ovals and circles or thin, almost stick-like, sections. For disconnected plots, however, the fractals are grouped together in intricate floral paerns, revealing the same shape and paern with each level of magnification. Prior to computer technology, Julia had to rely on his imagination and manually carry out the iterations by hand. Fiy years later, another mathematician applied modern computing power to plot these prey pictures, finally showing the sets in all their beauty…
Magnified sections of fractals for different values of c
The Mandelbrot set The Mandelbrot set is named aer the Polish mathematician Benoit B Mandelbrot, known for being the founder of fractal geometry. The word fractal is derived from the Latin fractus, which means broken, and describes the shape of a stone aer it has been smashed. Mandelbrot discovered that fractals appear not only in mathematics but also in nature, through crystal formation, the growth of plants and landscapes, as well as in the structure of the human 31
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chalkdust body. In 1945, Mandelbrot read Julia’s 1918 book. He was fascinated and, with the aid of computer graphics, was able to show that Julia’s work contained some of the most beautiful fractals known today. To create the Mandelbrot set, each complex value of c is used as the constant term in the quadratic function f (z ) = z 2 + c and iterated with the critical point z0 = 0 as the seed. If the orbit escapes to infinity, the number of iterations taken for the modulus of the function to exceed a specified value is used to decide on the colour of the map at that point, c. Otherwise, when the orbit converges, the point is coloured black. The Mandelbrot set is the set of black points. For example, if we let c = −0.15 + 0.3i then we have the complex quadratic function f (z ) = z 2 − 0.15 + 0.3i. We start with z0 = 0 as the seed and the sequence of iteration (to 5 significant figures) is as follows: z1 = 02 − 0.15 + 0.3i
⇒
2
z2 = (−0.15 + 0.3i) − 0.15 + 0.3i
⇒
z1 = −0.15 + 0.3i,
z2 = 0.2175 + 0.21i,
z3 = −0.14679 + 0.20865i,
z4 = −0.17199 + 0.23874i, z5 = −0.17742 + 0.21788i.
Continuing to 30 iterations, the orbit has not escaped to infinity and instead converges to the point z = −0.17082 + 0.22361i (again to 5 significant figures). Therefore, c = −0.15 + 0.3i is within the Mandelbrot set and is coloured black. On the other hand, if we take c = −1.85 + 1.2i, and hence the complex quadratic function f (z ) = z 2 − 1.85 + 1.2i, then the sequence of iterations (to 5 sf) is as follows: z1 = 02 − 1.85 + 1.2i
⇒
2
z2 = (−1.85 + 1.2i) − 1.85 + 1.2i
⇒
z1 = −1.85 + 1.2i,
z2 = 0.1325 − 3.24i,
z3 = −12.33 + 0.3414i, z4 = 150.06 − 7.2189i, z5 = 22465 − 2165.4i.
If the modulus of z exceeds 100, then it has been proven that the orbit escapes to infinity. This occurs on the fourth iteration, so the colour chosen to represent the value of 4 would be ploed at the point (−1.85, 1.2) in the complex plane. The resulting image is shown in map 8 on page 30, and also in the picture to the le.
The characteristic segments of the Mandelbrot set chalkdustmagazine.com
The largest segment of the set is called the cardioid due to its heart-like shape. Aached to this are adornments called bulbs, upon closer inspection of which it is possible to see many smaller, somewhat similar, embellishments. 32
chalkdust The bulbs are not completely identical, although most exhibit a similar shape, and the main differences can be seen in their filaments. The filaments are the thin strings of bounded points that sprout like sticks from the tops of the bulbs. These sticks are extremely narrow and they appear to be coloured red, which would indicate they are not part of the set. However, if we were to zoom in closer on these regions, we would actually see black lines!
The self-similarity of the Mandelbrot set
The Mandelbrot set is self-similar, consisting of miniature Mandelbrot sets within the boundary of the largest set. By enhancing the filaments, smaller copies of the overall set appear in ‘Russiandoll’ like fashion, as seen in maps 26–30 above. Closer inspection of map 27 shows many more selfsimilar sets within the filaments around the perimeter of the Mandelbrot set. Magnifying the small copies of these Mandelbrot sets would yield infinite layers of self-similar sets. Other fascinating and intricate shapes occur, for example the ‘seahorse valley’ that is visible in maps 31–34 above. By enhancing the plot within this region we see two rows of seahorse shaped embellishments, each with ‘eyes’ and ‘tails’. Further magnification of the ‘eyes’ reveals spiral constellations of more ‘seahorses’.
Connection between Julia sets and the Mandelbrot set The orbit of the critical point z0 = 0 can be used to test the connectivity of the Julia set, and the Mandelbrot set shows the boundedness of these critical orbits. Hence, the Mandelbrot set itself indicates the connectivity of the Julia sets of all the different complex quadratics. The Mandelbrot set can be described as M = {c ∈ C | Jc is connected}, where Jc is the Julia set of the function z 2 + c. The Julia set is a connected structure if c is within the Mandelbrot set, and will be broken into an infinite number of pieces if c lies outside the Mandelbrot set. The cardioid-shaped main body contains all values of c for which the Julia set is roughly a deformed circle (page 34; maps 35, 37, 38 and 40). The values of c which lie in a bulb of the Mandelbrot set produce a Julia set consisting of multiple deformed circles surrounding the points of a periodic 33
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chalkdust aractor. The number of subsections sprouting from a point on the Julia set is equal to the period of the bulb in the Mandelbrot set (below; maps 36, 39, 41–44). The nature of the convergence of points within the Mandelbrot set depends on the segment in which the point resides. Seeds within the cardioid converge to an aractive point, whereas orbits starting in the bulb lead to an aracting cycle. Three particularly interesting cases of Julia sets are shown below. The first is when c = 0, where the filled-in Julia set comprises of all the values within the unit circle (circle of radius 1, centred on the origin) and each of these points converges to 0 when iterated. The Julia set is the boundary of the circle, the points of which, when iterated, remain on the boundary. The second interesting case is when c = i. Here, the Julia set is a dendrite, meaning there are no interior points. Instead, the set is just a branch of points. For this complex constant the dendrite is a single line in an almost lightning-bolt shape. The final case is c = −2, where the Julia set is a dendrite that lies directly on the horizontal axis between −2 < x < 2.
Explore the sets yourself
A specific Julia set can be defined by a point in the Mandelbrot set
Three remarkable examples of the Julia set with c = 0, c = i and c = −2
I hope to have displayed the beauty behind these pictures by emphasising the extraordinary quantity of information contained in such a simple procedure, as well as through highlighting the complexity of each image, in the variety of fractals and colours visible, which further enhances the beauty. If this article has sparked an interest in fractals, then why not try exploring these sets for yourself? You could do this by magnifying different sections of the Mandelbrot set to explore the countless shapes and paerns that exist within. You could also go deeper into exploring individual orbits. All of these pictures are generated using simple quadratic formula. However, the Julia and Mandelbrot sets can be produced for a wide variety of functions in a similar manner to obtain countless prey pictures. chalkdustmagazine.com
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chalkdust These images are already becoming dated, having been taken for granted for so many years since they were first produced on the big bulky computers of the 1980s. The Julia set of the quadratic function, and the corresponding Mandelbrot set, could be inspiration for prey pictures which are yet to be fully explored, or even discovered. Largely, the discoveries discussed here have been recorded in recent years. Furthermore, there could still be vast amounts of information within these sets that are yet to be discovered. Could you be the one to make a discovery? Emily Clapham Emily is a mathematics graduate from Sheffield Hallam University. She is taking a year out to write GCSE study revision guides, before pursuing a career in teaching. Emily is hoping to interest more children into taking STEM subjects.
c emclapham@hotmail.co.uk
Powers by Daniel Griller c
Some numbers can be wrien in the form ab , where a, b and c are all whole numbers greater 2 than one. For example, 256 can be wrien as 23 . In how many different ways can each of the following numbers be wrien in this form? (i) 16
(ii) 89
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(iii) 23
You can find more puzzles by Daniel in his (highly recommended) book Elastic Numbers and on his blog d puzzlecritic.wordpress.com
My least favourite notation
The d’Alembert operator, □ Tom Rivlin
In physics, the d’Alembert operator is the special relativistic, 4-vector equivalent of the square of the del/nabla/grad operator, ∇: ∂2 ∂2 ∂2 1 ∂2 □=∂ ∂ =g ∂ ∂ = 2 2 − 2 − 2 − 2 c ∂t ∂x ∂y ∂z 2 1 ∂2 1 ∂ = 2 2 − ∇2 = 2 2 − Δ . c ∂t c ∂t …Except it’s not, is it? It’s a square. Do you know what a square means? It means ‘character not found’. When I see a d’Alembertian, I don’t know whether to sum over spacetime or check I have my Unicode reader configured correctly. □/10 … Oops, sorry, let me format that correctly: 0/10
Write for Chalkdust!
We’re always on the lookout for new and exciting ideas, and publish new articles online every week. If you have something to share, then get in touch with us at c contact@chalkdustmagazine.com 35
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Signi cant gures
Sophie Bryant North London Collegiate School
Patricia Rothman
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1884, Sophie Bryant’s paper, On the ideal geometrical form of natural cell structure, was published by the London Mathematical Society (LMS). It was ambitious, logical and descriptive: it looked at the phenomenon of the honeycomb.
Her insight was that the complex and beautiful honeycomb shape was a product of the natural activity of bees. All that was needed was for each bee to excavate its own cell at approximately the same rate as the others, and to use the excavated material to build up the walls of its cell. Bryant’s conclusion, that elongated rhombic semi-dodecahedra are the natural form of honeycomb cells, had been observed by Kepler. In the eighteenth century, it was believed that the honeycomb was the most efficient cell shape possible, but this is now known not to be the case. In 1964, the Hungarian mathematician Fejes Tóth observed in his paper, What the bees know and what they do not know, that there are in fact more efficient cell shapes which have yet to be determined. Kepler conjectured in 1611 that no packing of balls of the same radius in three dimensions chalkdustmagazine.com
A rhombic semi-dodecahedron can be made by puing square-based pyramids on the faces of a cube
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chalkdust has density greater than the face-centred cubic packing—the cannonball packing—with a density of about 74%. Bryant’s paper assumed this conjecture to be true, as it had appeared obvious for centuries and many had aempted proofs. The conjecture was eventually proved by Hales et al in 1998. Their computer-assisted proof was so huge that it took 12 referees to check it. Aer five years, the referees said that they were 99% sure that the proof was correct. Unusually, Annals of Mathematics published the paper in 2005 without complete certification from the referees. It was finally accepted as proven in 2014, and then only with the aid of massive amounts of computer time. Bryant’s approach to the subject was not unusual at that time. Abstract proofs, so essential to us now, were not as common as Wikimedia Commons user Greg L, CC BY-SA 3.0 general discussion of mathematical phenomena. She wrote “The Cannonball packing form of a natural structure is a logical result of its mode of genesis, and that form is ideal of which the mode of genesis is perfectly regular”. She states that there are only three possible arrangements without explaining why these are the only ones. Bryant’s paper is notable since it is the first published paper wrien by a woman member of the LMS. However, she was not the first woman to be elected to membership, being preceded by two remarkable women, Charloe Angas Sco and Christine Ladd Franklin.
Being rst Being the ‘first woman’ was not unusual for Bryant. She was the first woman to receive a DSc degree in England, studying what was then mental and moral philosophy, but today would be referred to as psychology and ethics. She was also one of the first three women to be appointed to a Royal Commission—the Bryce Commission on Secondary Education in 1894– 95—and she was one of the first three women to be appointed to the senate of London University.
Charlotte Angas Scott Though Bryant was the first woman member of the LMS to publish a paper, she was not the first woman member. That honour goes to Charloe Angas Sco (1858–1931), an algebraic geometer, who became a member in 1881. Sco had been aided in her mathematical education by an enlightened father. This resulted in her obtaining a scholarship to Girton College, Cambridge. However, women in Cambridge were not granted degrees until 1948 and she had to be content with the accolades of her peers. She was appointed a lecturer at Girton and received an external BSc degree from London University, and later a doctorate. Sco moved to the newly opened Bryn Mawr College for women in the USA where she was appointed head of mathematics, and remained for forty years.
While on the senate she advocated seing up a day training college for teachers, which eventually became the Institute of Education. Later in 1904, when Trinity College, Dublin opened its degrees to women, Bryant was one of the first to be awarded an honorary doctorate. In Cambridge, she 39
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chalkdust was also instrumental in seing up the Cambridge Training College for Women which eventually became Hughes Hall, the first postgraduate college for women in Cambridge. She was also, it seems, one of the first women to own a bicycle.
Beginnings and early widowhood Bryant was born in Ireland, and was fortunate to learn mathematics as well as other academic subjects with her five siblings in a very natural way from their father, the Rev WA Willock DD. A keen educationalist, he had been a fellow and tutor at Trinity College, Dublin and had gained high honours in mathematics and mental sciences.
Christine Ladd Franklin The second woman member of the LMS, who also joined in 1881, was Christine Ladd Franklin (1847–1930), an American mathematical logician. Though Johns Hopkins University was not open to women, UCL’s JJ Sylvester, then professor of mathematics, urged not only that she be admitted, but arranged for her to do graduate work under his supervision and to be granted a fellowship. As Johns Hopkins did not award degrees to women, she le without a PhD for her dissertation on symbolic logic.
When Bryant was about thirteen, her family moved to England and her family education continued until she aended Bedford College, where she was awarded the Arno scholarship for science in 1866. She sat the Cambridge local examination for girls in 1867 and was the only one to be placed in the first class of the senior division. In 1869, Bryant married the surgeon Dr William Hicks Bryant, only to be widowed the following year when he died of cirrhosis at the early age of 30.
Schoolteacher and doctorate
Aer a short interval, Sophie Bryant returned to her studies. While she had been siing her examinations, she was introduced to Frances Buss, the headmistress and founder of North London Collegiate School (NLCS), an excellent school then and still highly regarded today. It had been founded in 1850, the year of Bryant’s birth. She was finally awarded a PhD by Johns Hopkins forty-four years aer she submied her dissertation, when she was seventy-eight years old.
Bryant arranged to meet Buss who, in 1875, invited her to teach mathematics at NLCS and encouraged her to take a training course as well. Three years later, London University opened its degrees to women. As Bryant had not had a conventional education, she had to learn Latin and biology to matriculate before she could sit for her degree. In 1881, she earned a BSc degree, gaining a first class in mental and moral science and second in mathematics. In 1884, she received a science doctorate. The NLCS, where she had continued to teach, presented her with scarlet doctoral robes. Bryant was influential in improving the education system and introduced a scheme of enlightened and serious study. In 1885, Buss died and Bryant became the headmistress of NLCS until her retirement. chalkdustmagazine.com
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Psychology Meanwhile, she continued to publish ambitious papers. In her 1884 paper in Mind, The double effect of mental stimuli; a contrast of types, Bryant aempted to analyse the difference between reflex actions, which are performed without conscious thought, and consciously controlled actions. She was grappling with a contemporary problem: the understanding of consciousness. Unfortunately, her arguments are too diffuse to shed much light on the problem. In 1885, she published a paper in the Journal of the Anthropological Institute, Experiments in testing the characters of school children. This North London Collegiate School Bryant receiving her study, undertaken at the suggestion of Francis Galton, produced an doctorate early account of the use of open-ended psychometric tests to deduce personality types. Bryant claimed that her results agreed with the observations of teachers familiar with the children but did not provide any supporting evidence. Despite incomplete analysis, this was a pioneering study.
Later life Bryant was interested in Irish politics, and wrote books on Irish history and ancient Irish law. She was an ardent Protestant Irish nationalist and was active in the Home Rule movement, which pressed for Irish self-government within the United Kingdom. She wrote on women’s suffrage in 1879 but later advocated postponement until women were beer educated in politics. She enjoyed mountain climbing and she summited the Maerhorn twice. Her death in 1922 was both tragic and unexpected. Only four years aer retirement, she was on a mountain hike near Chamonix, in France, when she went missing. Her body was found thirteen days later with several injuries.
Epilogue Although Bryant’s direct contribution to mathematical scholarship was not substantial, her influence as a teacher and educationalist was immense. The rising number of women mathematicians today is a lasting tribute to her work. Patricia Rothman Tricia is an honorary research fellow in the Department of Mathematics at UCL.
c p.rothman@ucl.ac.uk d iris.ucl.ac.uk/iris/browse/profile?upi=PEROT30 Did you know... …the French word for pie chart is camembert? 41
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On this page, you can find out what we think of recent books, films, games, and anything else vaguely mathematical. Full reviews of many of the items featured here can be found at d chalkdustmagazine.com
Elastic Numbers
Mathematical socks
Daniel Griller “These puzzles will make you immediately reach for the nearest pen and paper and get solving.”
“Choice buffer between foot and shoe.”
ggggi
ggggi
∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
3Blue1Brown (YouTube)
Visualizing Mathematics with 3D Printing
“Very very good. Gorgeous animation.”
Henry Segerman
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Mathematical T-shirt
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“Expect ridicule for its poor mathematical content.”
Power-up
Matthew Lane “Highly enjoyable.”
ghiii
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Hidden Figures
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“A highly enjoyable and feelgood movie.”
Life is strange
“An excellent game all about chaos theory.”
ggggh
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The Winton Gallery
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at the Science Museum “Some really good stuff. And some dice.”
Bletchley Park
“A very good day out.”
ggggi
Ungry Young Man, CC-BY 2.0
∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
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National Museum of Computing
Chalkdust issue 06
“Flawless. Not even a single tpyo.”
“Surprisingly interesting.”
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ggggg chalkdustmagazine.com
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Pretend numbers Andy Maguire, CC BY 2.0
Wajid Mannan
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years ago when one of my nieces was quite small, she loved to play a guessing game with the family. She would think of a number (for us that means a whole positive number) and then we we would ask her questions with yes/no answers, trying to guess what her number was. For example, we could ask ‘is it even?’ or ‘is it bigger than 100?’, etc. Now this game was going on for some time and I was starting to wonder if she actually had a number or was she just pretending (this was by far my cheekiest niece). Pretending to have a number is not too hard when the questions are coming at you one at a time. You just need to make sure you don’t box yourself in. For example, if one of us asked ‘is it bigger than 100?’, then she would have to reply ‘yes’. Otherwise we could just go through all the numbers from 1 to 100 asking ‘is it 1?’, ‘is it 2?’ etc, and when she answered ‘no’ to all of them she would have been caught out. My niece could definitely pull it off.
Pretend numbers A much deeper question is whether she could have decided in advance the answer to every possible yes/no question, without actually having a number. In order for her to do that her answers would have to be consistent. That is, for any possible finite sequence of questions that we might ask, her answers could not contradict themselves. We will call such a consistent set of answers a pretend number (the technical term is a non-principal ultrafilter). 45
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chalkdust So do pretend numbers even exist? The answer turns out to be more philosophical than mathematical. A pretend number requires an infinite set of answers to every possible yes/no question. Thus their existence is tied up with the axiom of choice. The axiom of choice is traditionally explained in terms of pairs of identical socks. If I had 3 such pairs and I wanted to pick one sock from each pair, then I would have 2 × 2 × 2 = 8 ways of doing so. If I had 5 pairs of socks then I would have 32 ways of picking one from each pair. The more pairs of socks I have, the more ways of choosing one from each pair. So it seems reasonable that if I had infinitely many pairs there would certainly be at least one way of choosing one sock from each pair. However, this would require infinitely many choices to be If I add 100 to your number made and it is not clear that this is possible. Note that is the leading digit 2? Yes! the socks are identical, so I cannot for example just pick What if I add 1000? the larger one from each pair and explain all my choices Yes! in one sentence. The axiom of choice says that infinitely What if I add 8,971,231? many choices can be made. It cannot be proven or disYes! proven mathematically, so we are free to believe it or not. Whilst the sock example may make it sound reasonable, it has some prey unintuitive consequences if you believe it. For example, the freedom to make infinitely many choices would allow you to cut a solid object into such fuzzy and strange pieces, that you could put them back together and have twice the volume that you started with!
y
Moreover, if you believe that it is possible to make infinitely many choices then pretend numbers do exist. In fact, for any consistent set of answers to some yes/no questions there is a pretend number with those properties. So there is a pretend number U with leading digit 2 such that for any number n, if I ask, as shown above, ‘Does n + U have leading digit 2?’, the answer is ‘yes’. She must be making it up! What if she’s thinking of 20,000,000?
y
Note that no actual number has this property! If you had selected an actual number, say 2341, then it would fail as 1000 + 2341 does not have leading digit 2. However these answers are consistent, as for any finite set of values of n that I could ask about, there is always an actual number m that U could be. That is, n + m all have leading digit 2. For example, if I say ‘yes, 1000 + U has leading digit 2, and so does 123456 + U and 7101202 + U’, then I haven’t contradicted myself as U could be 20,000,000.
On the other hand, if we do not believe in the axiom of choice, then there need be no pretend numbers at all. The surprising and somewhat controversial thing is that both answers are mathematically correct. As my undergraduate lecturer put it, if you believe in a mind that can make infinitely many choices then you are more likely to believe in the axiom of choice, whereas if you only believe in the concrete and tangible, you may be more inclined not to. My niece never told us what her number was. Did she have a number? Was she tricking us? Had she found a way around the philosophical ambiguity of the axiom of choice and come up with a chalkdustmagazine.com
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chalkdust pretend number? We will never know.
Properties of pretend numbers Given the philosophical and even religious ambiguity that surrounds their existence, one can perform surprisingly concrete operations on pretend numbers, such as arithmetic. If I want to add an actual number n to a pretend number U then I get a pretend number n + U. But what is n + U? Recall that a pretend number is just a collection of consistent answers to every yes/no question about it, so to say what n + U is I just need to be able to answer yes/no questions about it. But a question about n + U is really just a question about U, and we know all the answers to yes/no questions about U, since U is a pretend number. For example, if you ask ‘Is 3 +U a square number?’, then the answer is just the same as the answer to the question ‘Is U three less than a square?’. Now for the mind-bending part. How do I add two pretend numbers? What is U + V? Again I just need to be able to answer yes/no questions about U + V. Consider a yes/no question: ‘Does the number have property A?’ (eg A could be the property of being prime or being a square). For any actual number u, we have just seen how to answer for u + V. So we can ask ‘Is U one of the actual numbers u, such that u + V has property A?’ We know the answer, as by definition we know the answer to all yes/no questions about U.
Mark Morgan, CC BY 2.0
Some examples of numbers presented in a very colourful way
So we say that U + V has property A precisely when U has the property of actual numbers u that u + V has property A.
This takes a bit of geing your head around, but is not as bad as it seems. Let us look at an example. Suppose Umar is playing the guessing game and is thinking of the number 2, and Vicki is playing the game and thinking of the number 3. Then if we add their numbers via the process just described we would hope to get 5 (so our notion of addition agrees with addition of actual numbers). Let’s see if that is indeed the case: The sum has a property A precisely when Umar’s number has the property of numbers u that u + V has property A (where V denotes Vicki’s number). As Umar is thinking of the number 2, he would answer yes precisely if 2 + V has property A. This is the case precisely when V has the property of actual numbers v that 2 + v has property A. As Vicki is thinking of 3, she would answer yes precisely if 2 + 3 has property A. In summary, the sum of Umar and Vicki’s numbers has a property precisely when 2 + 3 = 5 has that property. In particular, if you ask if the sum is 5, then the answer is yes. Thus the sum of Umar and Vicki’s numbers is indeed 5 as we had hoped. 47
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chalkdust Addition of pretend numbers behaves quite sensibly in many ways. For pretend numbers U, V, W we have (U + V ) + W = U + (V + W ) as you would expect. Also for an actual number n we have n + U = U + n. The big shock is that there exist pretend numbers U and V with U + V ̸= V + U. For example, we have already seen that we can have a pretend number U with leading digit 2, such that for any actual number n the leading digit of n + U is still 2. Similarly we can have a pretend number V with leading digit 5, such that for any actual number n the leading digit of n + V is still 5. So what is the leading digit of U + V ? For any actual number u the leading digit of u + V is 5, so if you ask if U has this property, I would have to say ‘yes’, or you would have caught me out as not having an actual number. So the leading digit of U + V is 5. But by the same logic the leading digit of V + U is 2, so they cannot be the same!
Once we accept the axiom of choice we are able to assume the existence of pretend numbers…
Once we accept the axiom of choice we are able to assume the existence of pretend numbers with desired properties and do calculations with them as above. However we should bear in mind that we can never actually specify a pretend number. We cannot specify consistent answers to all the infinitely many yes/no questions one might ask about it. To actually produce a pretend number would contradict the fact that it is not mathematically wrong to say that there are no pretend numbers.
Applications Given this intangibleness, it is natural to assume that pretend numbers are an intellectual curiosity, of no relevance to anything else. In fact nothing could be further from the truth. They play a prominent role in many fields such as mathematical logic, non-standard analysis, point set topology, topological dynamics, geometric group theory, combinatorics and field extensions and products of algebras. One especially amazing application is in the field of Ramsey theory. Ramsey theory considers the painting of structures (such as the connections in a network or whole numbers) with different colours. In particular, it determines which structures are guaranteed to be found in a part painted a single colour. The usual idea is that if so lile of the structure is painted one colour so that what you are looking for cannot be found in that colour, then enough must be painted in the remaining colours that what you are looking for can be found in one of them. A simple example is that if you paint 11 balls each either red or green, then 6 of them must be the same colour. You do not know if there are 6 green balls or 6 red balls, but if you don’t have one then you must have the other. In order to appreciate the application of pretend numbers to Ramsey theory, I invite you to solve the following puzzle before reading on: chalkdustmagazine.com
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chalkdust In the box below the numbers from 1 to 30 have been coloured with different colours. Can you find three distinct numbers that are all the same colour and all the same colour as any sum of some of them?
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Find distinct numbers a, b, and c so that a, b, c, a + b, a + c, b + c and a + b + c are all at most 30 and all the same colour.
Hopefully that wasn’t too hard. Now suppose that the numbers from 1 to 1,000,000 are coloured red, green, blue and yellow. Could you guarantee that no maer how they are coloured, you can still find three distinct numbers, all the same colour and whose sums are all the same colour? It is much harder when you do not know how the numbers are coloured, as you must account for every possible colouring. Now suppose that all the numbers, 1, 2, 3, . . . , are coloured with a finite number of colours. Instead of finding just three numbers, could you guarantee that there is an infinite set of numbers, so that if you add any finite combination of them together, the result is always the same colour? In 1970 this was an open problem that combinatorialists were seeking a solution to. Around this time a young Charlie Nguyen, CC BY 2.0 mathematician, Steven Glazer of Berkeley, thought of a University of California, Berkeley short and elegant proof that however all the numbers are coloured, such an infinite set can always be found. However his proof would only work if there were a pretend number U with U + U = U. Not knowing if a pretend number with the required property existed, Steven Glazer did not publish his work. In 1974 Neil Hindman finally proved that such an infinite set can always be found, via a much more complicated proof. The result now bears his name: Hindman’s Theorem. A year later Steven Glazer mentioned his proof to another mathematician, Fred Galvin. Now, Fred Galvin knew about compact semigroups. These structures arose in an area of mathematics known as dynamical systems, which grew out of the study of how physical systems evolve over time—far away from the world of colouring in whole numbers, one might think. Don’t worry if you don’t know what a non-empty compact semicontinuous semigroup is. What is important is that the pretend numbers are one (assuming the axiom of choice) and Fred knew it. Further he knew that in any such semigroup, one can solve U + U = U. 49
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chalkdust Note no actual number has this property, as when you add an actual number to itself it gets bigger. However, using mathematics from this very different field, one can deduce that there is a pretend number with this property. Thus Steven Glazer’s proof was valid.
Glazer’s proof Suppose all the positive whole numbers have been coloured with some finite set of colours. Let U be a pretend number with the property that U + U = U. We can ask if U is red. If not we can ask if it is green. Proceeding through all the colours, the answer must at some point be yes (or the answers would not be consistent and the pretend number U would be revealed as fake). Suppose then that U is blue. As U + U = U, it is not saying anything extra to add that U + U is blue. Thus U satisfies the property of actual numbers u that ‘u is blue and u + U is blue’. Thus there must actually be a number x1 with this property (or U is revealed as fake). Now we know the following are blue: U, U + U, x1 , x1 + U, x1 + U + U. Rearranging we have the following all blue: U, U + U, x1 , U + x1 , U + x1 + U. Let FS of a sequence be the set of sums of distinct terms. So we may make the above statement more succinctly by saying that the FS(x1 , U, U ) are all blue. So U has the property of actual numbers u that the following are all blue: u, u + U, x1 , u + x1 , u + x1 + U. Again there must actually be such a number x2 , or U would be revealed as fake. Thus FS(x1 , x2 , U ) are blue. We proceed in this way by induction. Suppose we have established that FS(x1 , . . . , xn , U ) are all blue. As U = U + U, we know that FS(x1 , . . . , xn , U, U ) are all blue. Thus there must actually be a number xn+1 so that FS(x1 , . . . , xn , xn+1 )
and
FS(x1 , . . . , xn , xn+1 + U )
are all blue. We already knew that FS(x1 , . . . , xn , U ) are all blue, so combining all three sets we get that FS(x1 , . . . , xn , xn+1 , U ) are all blue. Thus we obtain an infinite sequence x1 , x2 , x3 , . . ., whose sums of distinct terms are all blue. Either we have a subsequence of infinitely many distinct xi or some value x gets repeated infinitely oen. In the laer case all finite sums of distinct terms of the sequence x, 2x, 3x, 4x, . . . are blue. Either way we have infinitely many distinct numbers, with any sum of distinct numbers being blue and the proof is complete. Steven Glazer found out that his solution works, one year aer the problem had been solved by someone else. Nonetheless his proof is much celebrated by mathematicians for its shortness, its elegance and how it demonstrates the importance of mathematicians sharing ideas from different chalkdustmagazine.com
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chalkdust fields. In the words of George Bernard Shaw: “If you have an apple and I have an apple and we exchange apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas.” Wajid Mannan Wajid is a lecturer at een Mary University of London. He works in low dimensional topology, focusing on the question of when it is possible to completely flaen solid objects.
d maths.qmul.ac.uk/people/wmannan My least favourite notation
sin−1 x
Matthew Scroggs
sin2 x means (sin x)2 , so surely sin−1 x means (sin x)−1 or doesn’t, that would be too logical.
1 sin x ,
doesn’t it? No, of course it –1/10
Two circles by Daniel Griller A
In the diagram, the red circle passes through the centre of the blue circle and blue circle passes through the centre of the red circle.
C
C is a point on the circumference of the blue circle and inside the red circle. ACD is a straight line. D B
Prove that the lines BD and CD are the same length.
You can find more puzzles by Daniel in his (highly recommended) book Elastic Numbers and on his blog d puzzlecritic.wordpress.com
My least favourite notation
÷
Chris Bishop
I struggle to think of a more redundant sign in arithmetic. More oen than not, we’d rather multiply by the reciprocal, or draw a horizontal line between what we’re dividing and what we’re dividing by, allowing for easier cancellation. Division is important—don’t get me wrong—I just think we can do beer than this double doed abomination. 0 ÷ 10 51
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Rules Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc is advised for some of the clues. To enter, send us the sum of the across clues via the form on our website (d chalkdustmagazine.com) by 8 January 2018. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 22 January 2018. One randomly-selected correct answer will win a ÂŁ100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Nerd DVD, a utilities puzzle mug and much, much more. Three randomly-selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping. Find out more at d mathsgear.co.uk chalkdustmagazine.com
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Across
1 This number is divisible by both the sum and the product of its digits. 3 All the digits of this number are different. 5 The product of 1A and 5D. 6 The difference between 22D and 23D. 11 A prime number. 12 A square number. 13 The maximum number of pieces a cube can be cut into with 36 planar cuts. 15 A factor of 7D. 17 When wrien in hexadecimal, this number spells a common English word. 18 This number and 19D are a pair of triangle numbers whose sum and difference are also triangle numbers. 20 A palindrome. 23 A multiple of 7 whose digits add to 7. 24 A factor of 33D. 25 The largest number whose square has 12 digits. 29 A factor of 12 more than 22D. 30 A multiple of 26D. 31 The sum of the digits of 29A. 32 The product of 36D and 12A. 33 A Fibonacci number. 37 A palindrome that is a multiple of 1111. 41 An odd number. 42 Each digit of this number is one more than the previous digit. 43 This number and 29D are a pair of triangle numbers whose sum and difference are also triangle numbers. 44 The mean of 45D and 41A. 46 This number and 47A are a pair of triangle numbers whose sum and difference are also triangle numbers. 47 see 46A.
(2) (4) (3) (5) (2) (2) (4)
(4) (4)
(2)
(5) (4) (2) (6) (2) (3) (2) (3) (2) (7) (2) (4) (3)
(2) (7)
(7)
Down
1 Take this number’s digits as the first two numbers in a sequence. Form a Fibonaccilike sequence. This number is in this sequence. 2 This number is divisible by both the sum and the product of its digits. 4 The square root of 3A. 5 The product of the first two digits of 35D. 7 Not a prime number. 8 Ten times the sum of the across clues in this crossnumber. 9 A factor of 45D. 10 The product of 11A and 12A. 11 Three more than a multiple of four. 14 A multiple of 45D. 16 The product of 11A and 36D. 19 see 1λA. 21 A factor of 20A. 22 A multiple of 36D. 23 A multiple of 44A and 28D. 24 An integer x that satisfies x 2 + 1 = 2y 4 , for some integer y. 26 Not equal to 44A. 27 The price of a £100 coat aer discounting 10% then adding 10% tax. 28 The price of a £100 coat aer adding 10% tax then discounting 10%. 29 see 43A. 33 A multiple of 17A. 34 A multiple of one more than 15A. 35 An even number. 36 The mean of 11A, 12A and 36D. 37 The sum of the digits of this number’s cube, sixth power and seventh power are all equal. 38 In a base other than 10, this number can be wrien as 110001. 39 The product of the digits of 34D. 40 This number is equal to the product of its digits plus the sum of its digits. 45 A multiple of 5. 53
(2)
(2) (2) (2) (5) (8) (2) (3) (3) (7) (3) (2) (2) (6) (6) (3) (2) (2) (2) (3) (5) (5) (3) (2) (2)
(4) (2) (2) (3)
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chalkdust
Have you been taking your weekly dose of chalkdust?
Can maths help win auctions?
How can you stay dry in the rain?
Why do croissants taste so good?
Can you reach absolute zero?
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Auction: Cedim news, CC BY 2.0. Rain: nur_h, CC BY NC-SA 2.0. Croissant: Anthony GergeďŹ&#x20AC;, CC BY 2.0. Roundabout: Holger Ellgaard, CC BY-SA 2.0. Free kick: Akshay Davis, CC BY 2.0.
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Mathematics for the three-fingered mathematician Robert J Low
W
’ all familiar with using a couple of different bases to represent integers. Base ten for almost all purposes when we do our own calculations, and base two, or binary, for geing computers to do them for us. But there’s nothing special about ten and two. We could equally well use any integer, b, greater than two, so that the string of digits dn dn−1 dn−1 . . . d0 , where each di is positive and less than b, represents the integer n ∑
di bi .
i=0
Some bases are slightly more convenient than others for doing arithmetic. Bases eight and sixteen are both used in various computer applications, and there is an active society, the dozenal society, devoted to using and promoting the arithmetical advantages of base twelve. Much less common, but far more interesting, is base three. With base three, the digits are all 0, 1 or 2. But I want to look at a variation on this. Instead of using 1 and 2, I’ll use 1 and −1; but it’s not convenient to have minus signs in the middle of our numbers, so because of this and for reasons of symmetry I’ll represent them with 1 (for 1) and (for −1). Base three is ternary, and this variation of it is called balanced ternary. 1
1 001
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chalkdust Then the first few integers are 1,
1 ,
10,
1
0,
11.
Challenge 1
Convince yourself that any integer has a unique representation in this notation. But why would anybody use this notation? The reasons are simplicity and symmetry, which combine to make the system really quite beautiful. The first thing we can notice is that the negative of any number is obtained by simply swapping the 1 and symbols. There’s no actual need for a minus sign: if the leading digit is 1, the number is positive; if it is , the number is negative. A nice consequence of this is that we don’t need to have a subtraction algorithm: we can just change the sign and add. 1
1
Next, there are fewer basic rules of arithmetic to learn: the addition and multiplication tables of single digits are very simple, and have some very prey symmetry to them.
×
1
0 1
0 1 1 0 0 0 0 0 1
1
1 0 0 1 1 1
1
1
1 1 1
0
0
1
1 0 1
1
1
+
Challenge 2 Convince yourself that these tables are correct. With these rules in place, the standard procedures for addition and multiplication work just as before, but now there is no real difficulty in dealing with negative numbers.
Challenge 3 Write out the numbers from zero to twenty in this notation to see the paerns, and do some addition and multiplication to see it all working.
Non-integers and more symmetry There’s no problem with dealing with non-integers. Just as with binary and decimal, those with a terminating expansion are those which are an integer divided by the base—in this case three—and all others do not terminate. Also, just as with binary and ternary, there is a unique representation for integers, but not for all numbers. In the case of base ten, we know that any decimal expansion ending in an infinite string of nines represents a number with a terminating decimal expansion. This is encapsulated in the fact that 0.999 . . . = 1. 1 01
1
1
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Challenge 4 Convince yourself that 111
0.111 . . . = 1.
...
where the two expressions both represent the number one half. So, we don’t get a unique representation in this notation any more than in decimal; but the nonuniqueness also acquires a pleasant symmetry. We could do something like this with base ten, of course, but we have a decision to make: should we use digits from negative four to five, or from negative five to four? This is an unfortunate consequence of the fact that ten is an even number. With an odd base, we don’t have this problem, and get all the symmetry of balanced ternary. So balanced ternary has very slightly more basic operations than binary, but this is compensated for by the extra symmetry to do with negative numbers.
In real life? I admit, I have not converted to using balanced ternary for my everyday arithmetic. But it is great fun to play with. It also has not been influential in the history of computing. The only cases I know about are a wooden mechanical calculator from the 19th century built by Thomas Fowler (and so preceding the binary electrical and electronic computers of about a century later), and the Setun computers, the first developed in 1958 at Moscow University (which showed advantages of efficiency over competing binary computers), followed up by a newer model in 1970. I have one last confession/challenge. You have probably noticed that I have mentioned addition and multiplication, but not division. That’s partly because the long division algorithm is just a bit more fiddly than the usual one. So I’ll close with:
Challenge 5 Work out a long division algorithm.
Challenge 6 Decide whether you do actually find balanced ternary more pleasing than binary, decimal, or any of the other competing number systems.
Robert J Low Robert teaches maths at Coventry University.
c mtx014@coventry.ac.uk a @RobJLow d robjlow.blogspot.co.uk 1 010
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Two of each by Oyler In this crossnumber, each of the ten digits 0 to 9 (inclusive) appears twice. No entry starts with zero and all are distinct. 1
2
6 9
10
3
4
7
8
11
12
13
14
5
Down Across 2 multiple of 13A 1 prime 4 prime 3 square 5 square 6 it has 8 factors 6 factor of 5D 8 triangular 7 palindrome 9 square 10 prime 11 triangular 12 prime 13 factor of 2D 14 6A − cube
You can find more crossnumbers set by Oyler in the book Challenging Crossnumber Puzzles and the journal Crossnumbers arterly. Find out more at d crossnumbersquarterly.com
My least favourite notation
Interval notation Atheeta Ching
In a couple of papers, I have seen ]a, b[ being used to denote the open interval (a, b), and even [a, b[ to denote [a, b) which looks a bit awkward. At first I thought it was due to the papers being old, but apparently this is still commonly used in some parts of the world and was introduced by a group of French mathematicians who worked under the pseudonym of Nicolas Bourbaki. max(]-1,0])/10
Send us your articles! Got an exciting idea? Want to share it with us? We publish new articles every week online, as well as every six months in our printed magazine. So send us your articles! All submissions are welcome, and if you’re new to writing, we’re happy to work with you to hone it into something special. Who will your article be rubbing shoulders with in the next issue of Chalkdust? Get in touch with us at
c contact@chalkdustmagazine.com chalkdustmagazine.com
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Blaise Pascal Emma Bell
T
influence of Blaise Pascal is most keenly felt in his work on probability and the binomial theorem, illustrated by the famous Pascal’s triangle. It cannot be denied that Pascal’s triangle is a thing of mathematical beauty. However, this array of numbers was not discovered by its namesake, rather its applications and importance were highlighted by Pascal in his work, akin to Pythagoras’ theorem, which was certainly not invented by Pythagoras himself. However, Pascal’s legacy to mathematics goes further than the instantly recognisable triangle… Blaise Pascal grew up in 17th century France—a time much romanticised by Alexandre Dumas and his musketeers. The story of the Pascal family reads just like a chivalric novel, with famous names from mathematical and social history peppered throughout. Blaise showed a great aptitude for the sciences despite his father, Étienne, believing that the boy should not be taught formal maths until aer the age of 15. Legend has it that Blaise formulated some of Euclid’s proofs by drawing in the dust on a floor. As a result, his father relented, and Blaise began his mathematical studies. Étienne Pascal was a tax collector and lawyer, himself interested in maths and science. Indeed, the ‘Limaçon of Pascal’ curve is named for Étienne, not Blaise, and Étienne served the French government, Wikimedia Commons user Janmad, CC BY 3.0 siing on commiees to examine scientific propositions. He corBlaise Pascal responded with Mersenne and Roberval, debating the studies of Descartes and Fermat. Étienne was the lone parent for Blaise and his sisters aer his wife Antionee died in 1626. He moved the family to Paris in 1631, selling his interest in a vast area of land in the Auvergne region for a considerable amount of money. Étienne invested this money in government bonds, certain 59
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chalkdust that his fortune would be safe. It was not. The thirty years’ war was raging across the continent, and Cardinal Richelieu’s economic policies meant that the government could not honour the bonds. Étienne found himself at odds with a government he had trusted and helped, and made his anger known. Richelieu threatened Étienne with imprisonment, and so he fled Paris to escape the Bastille. His exile was relatively short lived thanks to Jacqueline Pascal, Blaise’s artistic sister. Aer performing a play for Richelieu, she convinced the cardinal to forgive Étienne. She was so convincing that he also gave Étienne a promotion, installing him as king’s commissioner for taxes in Rouen in 1639. The promotion was tough. Thanks to the wars and civil uprisings, Rouen’s finances were in chaos, and Étienne worked through the night, every night, to aempt to bring order to them, writing in 1643:
I have never been in a tenth part the perplexity that I am in at present. Blaise had accompanied his father to Rouen. He saw how hard Étienne was working, and set his mind towards making the job of organising the city’s accounts less labour-intensive. Blaise invented a calculating machine. The Pascaline. Previous aempts at effort-saving devices did not suit Blaise’s plans. He needed a machine that would add up large sums of numbers automatically, or repeatedly subtract. Napier’s Bones, created by John Napier (1550–1617) were efficient for multiplying and dividing numbers. Based on the multiplication table, they drew upon a laice form of multiplication, and by aligning the bones, or rods, in a certain way, addition could be done instead of multiplication, and subtraction in place of division.
Bernd Gross, CC BY-SA 3.0
A set of Napier’s bones, on display in Stugart, and the laice method used for multiplication
Edmund Gunter (1581–1626) invented the ‘Gunter’s Scale’, a ruler-like device which allowed navigators to quickly move between different forms of measurement. His contemporary William Oughtred (1574–1660), developed this further, bringing together the Gunter Scale and Napier’s other invention, the logarithm, to make the slide rule. A calculating clock had been invented by Wilhelm Schickard in the 1620s, based on Napier’s bones. It used pinwheels to ‘carry’, and was designed to be able to add, subtract, multiply and divide. It was described in leers from Schickard to Johannes Kepler, but was never seen working. Later chalkdustmagazine.com
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chalkdust examination of the descriptions in the correspondence suggest that the mechanism would jam, especially if several carrying actions happened simultaneously. Pascal’s Pascaline was special. All previous devices either needed the operator to ‘do the maths’, rather than having the calculation happen automatically, or plainly did not work.
Rama, CC BY-SA 3.0
Six-digit Pascaline from 1652
Blaise opted for a simple design based on addition. The machine would be set to zero, and then the operator would use a stylus to enter an initial number by using the digits around the circumference each dial as reference points. The stylus would reach a stop bar, as the dial was turned, just like on a rotary telephone dial.
Each separate dial represented a digit using place value. The operator could then add on another number in the same way, and keep repeating this until they had added all the required numbers. The ‘accumulator’ at the top of the machine displayed the final total. The machine automatically carried when each position reached nine, using a mechanism that Blaise designed himself. Subtraction was carried out on the apparatus using ‘complements to 9’. The bar at the top of the machine where the accumulator is situated hides a second set of numbers. If 82953 were displayed on the accumulator, the bar would be hiding the digits 17046 (8 + 1 = 9, 2 + 7 = 9, 9 + 0 = 9, 5 + 4 = 9 and 3 + 6 = 9). The bar could slide up and down, revealing the 9 complement of the total. By using this 9’s complement, subtraction could be carried out via the same operating technique as addition. The calculation of 40732 – 6549 is demonstrated below. Étienne made use of his son’s invention to aid his work. Blaise, however, was never completely satisfied with the Pascaline. He spent a great deal of time refining it, making machines with more wheels, decimal machines, machines for currencies and distances which needed different bases on each wheel. He wanted to sell the calculators for widespread use, but they proved to be very expensive due to the intricate parts. Instead, the Pascaline was seen more as a status symbol for European nobility. Even though fewer than 20 were sold, the Pascaline was the first working automatic calculator, inspiring later inventors and mathematicians. Mechanical calculators were used around the world well into the 1970s, when electronic versions started to take over. Out of frustration at seeing his father labour over calculations, Blaise Pascal laid the foundations for a device which people today still count on. Emma Bell Emma is the maths enhancement manager at Grimsby Institute of Further and Higher Education.
a @El_Timbre c belle@grimsby.ac.uk 61
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a hyperbolic plane You will need Triangle paper, scissors, sticky tape
Instructions
1
Cut out a hexagon and a triangle from the triangle paper.
Cut along one of the lines from a corner of the hexagon to the centre.
3
Tape the triangle between the two edges of the cut you just made. There is now more than 360° around the point, so the surface will not be at.
Continue to tape more triangles to the surface, making sure there are always seven triangles at each point.
5
2 4
Congratulations! You have made a hyperbolic surface.
Tube map platonic solids, FrĂśbel stars and slide rules: more How to make at d chalkdustmagazine.com chalkdustmagazine.com
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chalkdust
On the cover
Euclidean Egg III Peter Randall-Page
Peter Randall-Page
T
my life I have made an informal study of natural phenomena, through drawing or just looking, in a spirit of curiosity. This long but unsystematic practice has given me an impression of the world around us as a dynamic and fertile system, driven by a ubiquitous tendency for spontaneous paern formation (best understood in terms of the laws of physics) mitigated by an equally strong tendency for seemingly random variation. It could be argued that the evolutionary process itself is driven by this tension between paern and randomness, structure and chaos, order and disorder, theme and variation; without random mutation there would be stasis. In nature, we oen see this ordering principle manifest itself as various kinds of symmetry or repetition. Most animate creatures exhibit external bilateral symmetry; insects, crustaceans, fish, birds and animals including ourselves all tend to be bilaterally symmetric.
Rosa Pineda, CC BY-SA 3.0
A bilaterally symmetric scorpion
In common with other sentient creatures, we humans navigate and comprehend the world both spatially and temporally through paern recognition, and being highly social creatures we are particularly auned to reading expression and meaning in faces and bodies. It is therefore no surprise that bilaterally symmetric shapes seem to have a unique sense of potential meaning and 63
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chalkdust emotional impact for us. Whilst mirror image symmetry gives structure, the actual paern being reflected is oen far more chaotic. Like a kaleidoscope, the coloured shards are arranged at random; order is created by repetition of these random arrangements. Think of the paerns on moths, buerflies, shield bugs, ladybirds and beetles, there is oen very lile order in the arrangement of marks on one half, the exquisitely satisfying order of the whole is created by reflection. In the ‘Euclidean Egg’ series of drawings, as with much of my other work, I have chosen to use a working process which has an inherent element of chance and randomness. There are two ordering principles in these drawings: one is bilateral symmetry, the other is Euclidean geometry. I constructed a series of geometric egg shapes in such a way as to create a seamless curve where two arcs meet. The result is a faint line drawing of an egg shape together with the construction lines needed in order to create such a taut and smooth curve. These geometric eggs by their very nature have mirror image symmetry around a vertical axis. Folding the paper along this vertical axis and using paint introduces an element of chance. Using Peter Randall-Page a pipee dropper, I spread ochre paint onto one of Euclidean Egg III, our featured cover art this the areas between the construction lines on one half issue of the drawing. Folding the paper in half along the axis of symmetry creates two identical blobs of paint which, whilst roughly contained within the construction lines, inevitably have a somewhat random outline, reminiscent of the inlets and peninsulas of a Scandinavian island. I then add another blob of paint and continue the process, gradually building the drawing; blot by blot, fold by fold. This process is akin to the psychoanalytic evaluation technique developed by the Swiss psychoanalyst Hermann Rorschach in 1921. Rorschach’s theory was predicted on our psychological sensitivity to bilateral symmetric shapes. He developed a series of 10 mirror image ink blots which are shown to the subject, who is then asked to say what they see in them. Their observations are then used as a way of analysing the subjects subjective response to what are effectively totally random, but highly symmetric, shapes. Rorschach’s ink blot test has gone in and out of favour as a psychoanalytic tool during the last century but for me, our reaction to his ambiguous symmetric forms reveals something about the way in which our perception of the world is driven by subjective projection of feeling as well as objective analysis and observation. We read meaning into the world as well as taking meaning from what we perceive. chalkdustmagazine.com
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chalkdust My fundamental concern in making art is an exploration of what makes us tick, the emotional subtext to our everyday experience. The world enters our consciousness as emotion and expression as well as information and knowledge. We respond to shapes and colours, forms and spaces, poetry and music in ways which can be difficult to analyse or quantify. Whilst we have so many ways of communicating with one another (not least language itself), the medium of visual art is uniquely capable of exploring these oen intangible emotional responses.
A construction of the simplest Euclidean egg
In this particular drawing I am aempting to reconcile order and randomness, Euclid and Rorschach. My aention is concentrated on making a satisfactory balance between the ‘theory’ of pure abstract geometry with the ‘practice’ of what happens in the real world (in this case, the viscosity of the paint as well as the texture and absorbency of the paper are all determining factors). Being preoccupied with my aempt to reconcile these polarities is strangely liberating. The task involves innumerable decisions and appraisals which is conducive to a spontaneous and playful approach. In fact, play is an important concept for me. Play can be unselfconscious and create fresh associations and ideas. In order to play well, however, one needs a playground. Football without rules and a finite pitch would neither be fun to play nor interesting to watch. Although rooted in a study of natural phenomena, my work is less concerned with reproducing existing forms A construction of the cover image than with trying to grasp the underlying dynamics which determine the shapes and forms we see around us and to use these dynamic processes to create new objects which are both novel and familiar. In the words of the philosopher and art historian Ananda K Coomaraswamy in his 1956 essay The Transformation of Nature in Art, “art is ideal in the mathematical sense like nature, not in appearance but in operation.” Peter Randall-Page Peter is a British artist and Royal Academician. During the past 25 years he has gained an international reputation through his sculpture drawings and prints. He has undertaken numerous large-scale commissions and exhibited widely.
a @PRPsculpture c contact@peterrandall-page.com d peterrandall-page.com Did you know... …that all prime numbers (apart from 2 and 3) end in 1 or 5 when wrien in base 6? 65
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We love it when our readers write to us. Here are some of the best emails, tweets and letters we’ve been sent. Send your comments by email to c contact@chalkdustmagazine.com, on Twitter a @chalkdustmag, or by post to e Chalkdust Magazine, Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
Dear Chalkdust, I’ve attached LATEX and pdf of a short article on balanced ternary (topic of my short talk at the MathsJam meeting last year). Hope it’s of use to you. Robert Low, Coventry a @RobJLow
Dear Chalkdust, I’m a math [sic] major in the United States and have been following your magazine for a while now, and I have to congratulate you once again on an amazing publication! It’s rare to nd a magazine that perfectly balances rigorous mathematics with easy to follow explanations and wonderfully motivating questions. Je rey Ayers, USA
It is with great excitement that I discovered your magazine this morning! Well done indeed ;)
I’m eating my slice of birthday cake while reading today’s birthdaythemed Chalkdust blog! Smoky, London a @SmokyFurby
Thomas Faulkenberry, Texas
Look what just arrived on Jeju. World domination will ensue. Thank you TD. And I very much like the Top Tens! Shambolic Librarian,
Golden ratio as the least favourite: what a sacrilege! It is the most irrational number and that fact alone makes it special!
a @JejuLibrarian1
Peter Karpov,
a @inversed_ru
Crossnumber completed. Always a good mental workout, and always teaches me something new! ...even if it’s only about the setter’s levels of maturity. Colin Beveridge,
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a @icecolbeveridge
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The mathematics of
Maryam Mirzakhani Assad Binakhahi
Nikoleta Kalaydzhieva
M
Mirzakhani, the first woman Fields medallist and an explorer of abstract surfaces, le us in the prime of her life. Rightly, the world press mourned her passing, but what I hope to do here is to write about the beautiful and difficult mathematics she loved working on. As a pure mathematician, she was usually driven by in-depth understanding of the different complex structures on abstract surfaces, rather than the search for application. Nevertheless, her work has been used in solving real life problems. But what I personally find fascinating about her is the courage and creativity she had in aempting and solving long standing problems and the variety of areas within mathematics she worked on; from complex geometry and topology to dynamical systems. Here is a more intuitive exposition of some of her achievements.
Surfaces I am sure that if I asked you to give me an example of a surface you would be able to do so straight away. You might say the surface of the Earth is obviously one, and you would be right. However, defining what we mean by a surface mathematically is a lile bit trickier. Let’s give it a go. A geometrical object is called a surface if, when we zoom in very closely at the points on the shape, we can see overlapping patches of the plane. If we were to use mathematical language we would say that a surface 67
Descubra Sorocaba, CC BY 2.0
A tasty surface autumn 2017
chalkdust is locally homeomorphic to the plane. You might not find this definition particularly helpful so let us consider a few more examples. Oranges, tomatoes, apples and, for more delicious alternatives, cakes, cupcakes, ring doughnuts and pretzels are all surfaces. Well, almost! In order for them to be surfaces we need to picture them hollow (or like a balloon), rather than solid. If we consider those objects geometrically, meaning that we differentiate between different angles and size lengths, we notice that there are infinitely A genus 1 surface many of them. This is why we consider them topologically. Using continuous deformations we can turn almost all of our examples into a sphere, except the ones that have holes in them. They are considered to be in a class of their own. Thus we can classify the surfaces up to deformations (topological equivalence) by the number of holes, which we call the genus. We can see that the sphere has genus 0, the torus (ring doughnut) genus 1 and the 3-fold torus (pretzel) genus 3, thus these are all inequivalent surfaces. Mirzakhani’s work was on Riemann surfaces. To turn a surface into a Riemann surface we need to give it additional geometric structure. For example, we can give the surface geometric structure that allows us to measure angles, lengths and area. An example of such geometry is hyperbolic geometry. It is the first example of non-Euclidean geometry; the only way it differentiates from Euclidean geometry is that given a line ℓ and a point P that is not on the line, we can draw at least 2 distinct lines through P that are parallel to ℓ . One peculiar consequence of this new axiom is that rectangles do not exist in hyperbolic geometry. Moreover, the angle sum for a triangle is always less than 180°. Mirzakhani’s early work was on hyperbolic surfaces, which are Riemann surfaces with hyperbolic structure. The problem with hyperbolic surfaces is that we cannot really visualise them, because the hyperbolic structure on the Riemann surface can’t be embedded in R3 . However, we can try and describe roughly how you put the structure on the surface. Imagine our Parallel lines and a triangle on a hypersurface is made out of rubber and we can bend it and bolic surface fold it in all dimensions. Now we add the hyperbolic structure, but for that we need, according to John Nash, 17 dimensions. If we next dip it in cement it becomes solid, and we can no longer stuff it into 3 dimensions, hence we can no longer visualise it completely. On these surfaces, Mirzakhani studied special objects called closed geodesics. Roughly speaking, a geodesic is a generalisation of the notion of a straight line that we have on the Euclidean plane. We can define a geodesic more rigorously as a path between two points on the surface, whose length cannot be shortened by deforming it. For example, on the sphere, the geodesics are called great circles. These are simply the intersections of a plane going through the origin and the sphere itself. chalkdustmagazine.com
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chalkdust A closed geodesic is a geodesic that starts and ends at the same point. The simplest example of a closed geodesic is a circle. We also allow intersections, that is, geodesics that look, for example, like a figure-of-eight and are much more complicated. Using the hyperbolic structure on the Riemann surface, we can compute the lengths of these closed geodesics. A natural question we can ask is how many such closed geodesics are there on any hyperbolic surface of length ⩽ L? The answer was established in the 1940s by Delsarte, Huber and Selberg and it was named the prime number theorem for hyperbolic surfaces, because of the resemblance to the prime number theorem. That is, the number of closed geodesics, denoted by π, satisfies
A sphere, sliced in half along a great circle
π (X, L) ∼ eL /L,
as L → ∞. Roughly speaking, the number of closed geodesics on a hyperbolic surface X of length ⩽ L gets closer to eL /L as L becomes very big. We can see that their number grows exponentially, meaning very quickly, but more importantly we also see that the formula does not depend on the surface we are on. The next question to consider is what would happen if we no longer allow our geodesics to intersect themselves? Would our formula change much? Will the growth rate be significantly different? That is, we wish to compute the number of simple closed geodesics (simple meaning no intersections are allowed) on a hyperbolic surface X of length ⩽ L, denoted σ(X, L). In 2004, Mirzakhani proved, as part of her PhD thesis, that σ(X, L) ∼ CX L6g−6 , as L → ∞,
where g denotes the genus of the surface X, and CX is some constant dependent on the geometry (hyperbolic structure) of the surface. It is important to make clear that surfaces of a given genus can be given many different hyperbolic structures. As a consequence the number grows much slower (polynomially) but it also depends on the surface we are on. The surface may be the same, but the different structure implies that we would have different geodesics and their lengths would also be different. Whilst she was computing σ(X, L), she discovered formulae for the frequencies of different topological types of simple closed curves on X. The formulae are a bit too complicated to explain here, but let us consider an example: suppose X is a surface of genus 2; there is a probability of 1/7 that a random simple closed geodesic will cut the surface into two genus 1 pieces. How cool is that⁈ Even though these results are for a given hyperbolic structure, Mirzakhani proved it by considering all structures at the same time. We know that we can continuously deform surfaces of the same genus g and they will be topologically the same, however geometrically they may be different. These deformations depend on 6g − 6 parameters, which was known to Riemann. We call these parameters moduli and we can consider their space, the so-called moduli space of all hyperbolic structures on a given topological surface. By definition, a moduli space is a space of solutions of geometric classificaFlight paths follow geodesics on the Earth’s surface tion problems up to deformations. This is a bit abstract, so let 69
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chalkdust us illustrate it with a simple example. Suppose our geometric classification problem is to classify the circles in Euclidean space up to equivalence. We would say that two circles are equivalent if they have the same radius, no maer where their centre lies. That is, our modulus (parameter) is the radius r of the circle, and we know that r ∈ R+ . Hence the moduli space will be the positive real numbers. So what can we do with these new spaces? Greg McShane observed that you can add a new structure; a so-called symplectic structure which, roughly speaking, allows us to measure volumes on moduli spaces. Mirzakhani found a connection between volumes on moduli spaces and the number of simple closed geodesics on one surface. She computed some specific volumes on moduli spaces and her celebrated result followed.
Dynamical systems In recent years, Mirzakhani focused her aention on dynamical systems on moduli spaces. A dynamical system is simply a system that evolves with time. Originally, dynamical systems arose in physics by looking at the movements of particles in a chamber or planets in the solar system. It was observed that these large systems are similar to smaller ones, and by studying toy models we might shed some light on the actual physical dynamical systems. One such toy model is the dynamical system of bil- Curtis Perry, CC BY-NC-SA 2.0 “Pot as many balls as you can” liard balls on a polygonal table (not necessarily rectangular). Bear in mind that in this version of billiards we only use one ball and it can travel forever on a path as long as it doesn’t reach a corner. The billiard balls will take the shortest paths, thus they travel via geodesics, and this is where Mirzakhani’s research come into play. As we know by now, she studied surfaces rather than polygons, but if you orient the edges of the table in pairs and glue them together then you can turn it into a surface. Even though billiard dynamics might seem simple, there are difficult problems that are still unsolved. One might ask if there are any periodic billiard paths, and if so, would the answer change if we change the shape of the table T ? This problem has been solved: it is known that there is always at least one periodic billiard path for a rational polygonal table (by a rational polygon, we mean a polygon whose angles are rational multiples of π). But what if we now ask what is the number of such periodic billiard paths of length ⩽ L on a table T, denoted by N (T, L)? It is conjectured that the following asymptotic formula holds. N (T, L) ∼
CT L2 , π Area(T )
where CT is some constant depending on the table. Alongside Alex Eskin and Amir Mohammadi, Mirzakhani made some progress towards this result. They showed that limn→∞ N(T, L)/L2 exists and is non-zero. Moreover, she showed that for the asymptotic formula to even exist in the form above, there exist only countably many numbers CT . Recently, Mirzakhani and Eskin’s work on billiard paths was applied to the sight lines of security guards in complexes of mirrored rooms. chalkdustmagazine.com
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chalkdust Another example of her impact on dynamical systems is her work on Thurston’s earthquake flow. Suppose we have a Riemann surface X of genus g, a simple closed geodesic γ on X and a real number t. Then we obtain a new Riemann surface Xt = twtγ (X ) by cuing X along γ, twisting it to the right by t and re-gluing. Then we can define the flow at time t to be
Geodesic laminations on hyperbolic surfaces
twt ( , X ) = ( , twt (X )) where is geodesic lamination. The definition of geodesic lamination is quite technical, so in this artiAaron Fenyes cle we can simply think of it as a disjoint collection of simple geodesics on X. Intuitively, we have a dynamical system like the movement of planets in time t, but in our case the objects that move with time are moduli spaces. We get some sort of periodicity, because if γ has length L, then XL+t = Xt . Mirzakhani showed something truly remarkable: The earthquake flow is ergodic. This means that if we follow the laminations along we would be very close to any point on the surface with probability 1. This came as a surprise, because until then there was not a single known example showing that the earthquake paths are dense. She might not have always wanted to be a mathematician, aspiring to be a novelist when she was younger, but she le a big mark on mathematics. As the first Iranian and first woman to win the Fields medal, I believe she has been an inspiration to many young girls and women, including me, to go into research and be optimistic when solving problems because the “beauty of mathematics only shows itself to the more patient followers”. Nikoleta Kalaydzhieva Niki is a PhD student, studying number theory at University College London. When she’s not playing with numbers, Niki can be found cuing paper, giving talks about Möbius strips, and Chalkdusting.
c zcahndk@ucl.ac.uk My least favourite notation
Bold notation for tensors Hugo Castillo Sánchez
There are many notations used to represent vectors and matrices, formally known as tensors. In physics, vectors are oen represented in bold, v, and matrices are wrien a lile bit bolder, A. I dislike this notation as it is hard to differentiate whether we are using a vector or a matrix. It’s so confusing! And how do we represent a higher order tensor? Make the symbol extra mega bold? Although this notation might be useful for simple equations, it becomes a nightmare when it is used in large systems of equations with lots of variables. I suggest moving to the underline notation v, where the number of times a vector is underlined is equal to the order of the tensor. 0/10 71
autumn 2017
This issue features the top ten geometry instruments. To vote on the top ten mathematical celebration days go to d chalkdustmagazine.com
At 10 this week, no-one’s favourite member of the Oxford geometry set: the 30° set square.
At 9, and mightier than the swordcil: the pencil.
At 8, and vastly superior to the circular protractor, because no-one ever needs to measure re ex angles: the semicircular protractor.
At 7, it’s the perfect tool for those who are overcon dent (or trying to annoy their maths teacher): the pen.
Bursting back into the charts at 5, due to its newly-found fame in The Emoji Movie: the 45° set square.
Down two places to 6, it’s a surveyors favourite* tool: a theolodite.
At 3, something you can use to make this white space ll up: your imagination.
As crucial to a mathematician as a book of PlayStation cheats to a 90s child: a copy of Euclid’s Elements is at 4.
If you don’t know what GeoGebra is, then stop reading Chalkdust now, go download it and play. Luckily this is the last sentence in Chalkdust, so you were nished anyway...
At 2, perfect for drawing a circle or locating north twice: a pair of compasses.
* source: a focus group of surveyors. Pictures: Set squares: Wikimedia Commons user Dnu72, CC BY-SA 4.0; Euclid’s Elements: Folger Shakespeare Library Digital Image Collection, CC BY-SA 4.0
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