In this issue... Features 4 Baking with Eugenia Cheng We get technical in our show-stopping interview
9
The magnetic pendulum James Christian and Holly Middleton-Spencer oscillate
22
An invitation to category theory Tai-Danae Bradley is a morphism
33
What’s the point of intersection? Elizabeth A Williams falls off a log
38
The ELHP Adam Atkinson gets artistic
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The Fields medals 2018 Albert Wood tells us who is outstanding in their field
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Topological tic-tac-toe Alex Bolton knots and crosses
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Too good to be Truchet Colin Beveridge tiles his bathroom
Somewhere over the critical line Tanmay Kulkarni tells a story
Regulars 3 Page 3 model 8 What’s hot and what’s not 15 I ♥ maths by David Richeson
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36 1
20 26 28
Dear Dirichlet
32 35 36 54 56
What type of number are you?
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Roots: Gerolamo Cardano by Emma Bell
61 69
Soap update
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Top ten: units
Puzzles Significant figures Gerda Grase tells us about Katherine Johnson Reviews Hilbert’s hotel: the board game Prize crossnumber How to make... a Möbius surprise
On the cover by Tom Rivlin
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chalkdust The team Rob Becke Chris Bishop Carmen Cabrera Arnau Hugo Castillo Sánchez Atheeta Ching Thùy Dương “TD” Ðặng Eleanor Doman Ed Goldsmith Sean Jamshidi Nikoleta Kalaydzhieva Antigoni Kleanthous Emily Maw Sam Porri Tom Rivlin Sally Said Mahew Scroggs Belgin Seymenoğlu Yiannis Simillides Adam Townsend
d chalkdustmagazine.com c contact@chalkdustmagazine.com a @chalkdustmag b chalkdustmag l chalkdustmag f chalkdustmag n @chalkdustmag@mathstodon.xyz e Chalkdust Magazine, Department of Mathematics, UCL, Gower Street, London WC1E 6BT, UK.
Happy Tuesday, and welcome to issue 08 of Chalkdust! This issue, we are excited to share with you some brand new features, including a mathematical fiction set in a divided society (pp 16–19), and a board game that will keep you entertained until the release of issue 09 and beyond (pp 36–37). We also have a wide-ranging interview with Eugenia Cheng (pp 4–7), as well as an introduction to her field of research (pp 22–25). Eugenia is a strong believer in the idea that mathematics is for everyone; a mantra that we also subscribe to at Chalkdust. As this issue is released, we will be in the middle of Black Mathematician Month—a celebration of the work and life of black mathematicians in recognition that maths, like other sciences, has a diversity problem. We will be following this with an outreach event in November, details of which can be found on our website, alongside more related articles. While you’re there, you will also find tons of other content, puzzles, and our brand new podcast. Finally, it would be remiss of us not to mention our summer holidays. While the rest of the UK baked in a month-long heatwave, a few Chalkdust members were lucky enough to aend the International Congress of Mathematics in Brazil, the largest meeting of research mathematicians in the world and the scene of 2018’s most dramatic awards ceremony moment. We hope that you enjoy this issue as much as we’ve enjoyed puing it together, and if it inspires you to write or do anything then we would love to hear from you! Our DMs are forever open. The Chalkdust team
Acknowledgements We would like to thank: our sponsors for allowing us to continue making Chalkdust; Helen Wilson, Luciano Rila, Helen Higgins, Robb McDonald and the rest of the staff at the UCL Department of Mathematics for their continued support; Giancarlo Grasso for help organising the BMM event; Alison Clarke for reviewing a book; Das Baskill for suggesting Möbius strip cuing for How to make; Dimitrios Voulgarelis for suggesting the ponytail model for the Page 3 model; Ellen Webborn for channelling Dirichlet for us; all this issue’s authors for sending us yet more excellent articles; and you for bothering to read the acknowledgements. 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 2018. 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
chalkdustmagazine.com
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Hugo Castillo Sánchez
T
HE ponytail is probably the world’s greatest hairstyle. But trying to work out what shape someone’s ponytail will be has puzzled scientists and artists since Leonardo da Vinci.
In 2012, mathematicians at Cambridge and Warwick used the ponytail shape equation to unravel some of the mysteries of the ponytail.
This equation can be used to find R, the radius of the ponytail, in terms of s, the arc length along the ponytail. The length at which gravity bends the hair is l, L is the length of the ponytail, P is the pressure due to the hairband, A is the bending modulus, and ρ is the hair’s density.
Deflection
The Rapunzel number, Ra, of a ponytail is the ratio L/l. This dimensionless number determines the effect of gravity on hair.
Ra < 1
Ra > 1 Ra
When Ra < 1, the hair doesn’t bend much, leading to a thin, straight ponytail. When Ra > 1, the hair bends strongly under gravity leading to a wide, bushy ponytail. So next time you want to look good at a party or a maths conference, simply calculate your Rapunzel number and pop on a hairband that exerts the correct pressure. 3
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In conversation with. . .
Eugenia Cheng COD Newsroom, CC BY 2.0.
Chris Bishop
W
meet Eugenia Cheng a couple of hours before she’s scheduled to give a talk at City University, where she’ll make another stop on her journey to “make abstract mathematics palatable” in the public consciousness. With over 10 million views on YouTube, three best-selling books in How to Bake Pi (2015), Beyond Infinity (2016) and The Art of Logic in an Illogical World (2018), and interviews ranging from the BBC to late night US television, it’s safe to say Cheng has made incredible progress on her mission.
Not ‘just’ a mathematician In talking to Cheng, you quickly realise that she is always trying new activities, pursuing further study and pushing herself to understand more of the world around her. She read voraciously growing up, but describes mathematics as “the only subject to stand up to [her] desire for rigour”. In aempting to satisfy her “curiosity for asking why things are true”, Cheng learned of category theory, the study of relationships between similar themes and concepts in different branches of mathematics. She describes this field as particularly abstract, but remarks that it could be viewed as a prerequisite to most undergraduate courses in the way it identifies links in areas of mathematical study. In her view, category theory does for mathematics what mathematics does for the world. This concept takes a lile bit of mental gymnastics, and Cheng takes it a step further in her current research in higher-dimensional category theory. She studies the relationships between the relationships themselves, adding “an extra layer of subtlety” to her understanding of mathematics. chalkdustmagazine.com
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chalkdust It’s clear that Cheng’s research has influenced the way she approaches her other passions, which include food, music, and teaching. An avid baker, Cheng’s first book, How to Bake Pi, begins each chapter with a recipe for the reader to try. This method mirrors the conversations that introduce chapters in Douglas Hofstadter’s seminal work Gödel, Escher, Bach: An Eugenia Cheng Eternal Golden Braid, which she credits as a “very influential Mille-feuille is a type of pastry made with hundreds of layers book” for her approach to writing about mathematics. Cheng believes that the general public is suffering from a severe case of “maths phobia”, and that introducing mathematics in recognisable, accessible ways is far more effective than teaching times tables and long division. On her appearance on The Late Show with Stephen Colbert in 2015, Cheng introduced the concepts of exponentials through making millefeuille live with the host. Children as young as seven have, through How to Bake Pi, gained an understanding of complicated abstract mathematics, and she recounts a story of a five-year-old calling out to her “I’m your biggest fan!” at one of her outreach talks. By giving younger and younger children an appreciation of what mathematics can do and how it is expressed in the world around us, Cheng believes that the fear of mathematics so many schoolchildren feel will erode and disappear over time. Cheng is the founder of the Liederstube, an environment for classical musicians to come together and enjoy performances in a relaxed seing, based in the Fine Arts building in Chicago. A talented pianist, Cheng performs alongside her busy schedule writing books and giving talks. When asked about whether performing in concert halls is more nerve-wracking than giving talks at the Royal Institution, Cheng doesn’t hesitate: “Compared with playing the piano, public speaking is easy! You can say whatever you want, and you don’t have to say particular things in a particular order.”
Cheng modelling Chalkdust T-shirt
the
new
As the scientist in residence at the Art Institute of Chicago, Cheng teaches abstract mathematics to undergraduate art majors. She enjoys teaching mathematical ways of thinking to socially conscious students, giving them the ability to “frame social issues in a mathematical sense”. Perhaps these students represent the ultimate cases of maths phobia, but according to Cheng, there is a lot more in common between those that COD Newsroom, CC BY 2.0 study mathematics and the students she sees weekly. Through Cheng was a featured speaker her courses, Cheng is learning as much from her students as at Stem-con 2017 the other way around. She had not realised the applications of mathematical thinking as a “framework to agreeing on the world, something badly needed in today’s public discourse”. The parallels between Cheng’s passions and her research are immediately apparent. She finds the mathematical similarities between music, food, and teaching in the same way she identifies connections between areas of mathematics in category theory. 5
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Women in mathematics As a prominent woman in mathematics, especially in popular culture, the question of the gender gap inevitably came up. By Cheng’s own admission, she “was initially reluctant to address the issue” at the beginning of her career. A firm believer in the meritocracy of academia, she is certain she’s “able to achieve anything a man could in mathematics”. Cheng says that “when women in academia are young and not treated with much respect, they think it’s because they are young COD Newsroom, CC BY 2.0 and junior. But as they progress, they continue to noEugenia Cheng is a role model for many tice the lack of respect, making it clearly a feminist isyoung women interested in maths sue.” Cheng has also realised how important it is to act as a role model, and has embraced the challenge of becoming more visible to young women in mathematics. For Cheng, the most important message to communicate to these students is that “they are good enough”. Students that are struggling are “not finding it difficult because they don’t understand. They are seeking a deeper level of understanding—exactly the kind of person needed in higher level mathematics.” Cheng sees a marked difference in how the average female student approaches applying to PhD courses compared to their male counterparts. She admits that if she had not been offered the opportunity to study for her doctorate at her first choice (Cambridge), she would have given up pursuing a career in higher level mathematics, taking the rejection as a comment on her ability. She believes that in applying for mathematical postgraduate positions, women have to take the same persistent approach that men do, applying for any opportunity that allows them to follow their passions.
Most people think that selfconfidence is the most important part of being a mathematician… I believe that self-criticism is far more important.
Cheng also feels that too much self-confidence in one’s own abilities doesn’t make the best students. “Most people think that self-confidence is the most important part of being a mathematician, whereas I believe that selfcriticism is far more important. I’d much rather work with a student that underestimated their own abilities, than the other way around.”
In fact, Cheng believes there needs to be a reframing of the whole argument, proposing new words to replace masculine and feminine, as “we shouldn’t prescribe behaviours to genders”. For masculine, Cheng suggests ‘ingressive’. “Ingressive—it’s all about geing the right answer, being competitive, being first, exactly the way we teach mathematics at a young age.” Even the way we test is ingressive: “Exams are an ingressive thing too,” Cheng says. “You have to get as much done as you can, as right as you can, as quick as you can.” But research isn’t like that at all. “Research is congressive,” Cheng explains, using her replacement word for feminine. “You’re trying to discover deeper insight, you have to work collaboratively. It takes time and it takes patience.” And therein lies the problem, according to Cheng. “We’re selecting ingressively for a subject that should be very congressive in nature,” she concludes. “I suspect we’re losing a lot of talent this way.” chalkdustmagazine.com
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The art of logic Just before Cheng has to nip off for her talk, we move onto the subject of her new book, The Art of Logic in an Illogical World. “This really grew out of teaching art students because the ones I teach are so socially conscious, and want to change the world,” Cheng says. “It was a bit like when I used baking to perk up my mathematics undergraduates. If I talked about a social issue from a mathematical point of view, then they were all completely alert.” The book is a summary on the “insights mathematCheng’s new book tells us how mathematical thinking gives me on social and political issues.” ical thinking can be applied to questions In summary, the book is about “the nature of disabout social issues such as LGBT rights agreement.” We later aend one of Cheng’s talks at the Royal InstiMathematics is a way of betution, based on her book, and it’s fascinating to see how ing clear and unambiguous, mathematical thinking could be applied to questions of and we need that today. LGBT rights, racial privilege and political disagreement. “I always read the comments below news articles,” Cheng said, prompting a sympathetic laugh from the audience, “You have to! You have to know how people see the world so you know how to talk to them.” Cheng’s belief that mathematical principles allow us to cut through overly complicated debate is infectious and so clear, you wonder how anyone could possibly disagree. “Mathematics is a way of being clear and unambiguous, and we need that today,” Cheng concludes. Through her writing, talks and outreach work, there’s no debating the important work she’s doing for the subject, curing cases of maths phobia every day. If you’re curious as to just how category theory works, turn to pages 22–25. Chris Bishop Chris Bishop is a third year mathematics with economics student at UCL with a keen interest in applying mathematics in optimisation problems in real-world scenarios.
A triangle and four squares by Catriona Shearer
The area of the boom le square is 5. What is the area of the blue triangle?
5 You can find Catriona and loads more of her puzzles on Twier a @Cshearer41. 7
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WHAT’S
& WHAT’S
HOT NOT Placeholder text
HOT
Writing actual content
Lorem ipsum dolor sit amet, consectetur adipiscing elit...
Yawn.
Maths is a fickle world. Stay à la mode with our guide to the latest trends.
Agree? Disagree? a @chalkdustmag b chalkdustmag
NOT
HOT The Great British Bake-Off
The Big Internet Math-Off
Ultimate non-mathematical battle where audience members compete to not fall asleep.
Ultimate mathematical battle where passionate amateur mathematicians compete to be crowned the World’s Most Interesting Mathematician.
Winning a Fields medal
HOT
Rewarding years of hard work.
NOT
Stealing a Fields medal
Rewarded with years of prison.
NOT
Proof by contradiction
HOT
Suppose that proof by contradiction is not hot. Therefore 1 = 0. E
HOT
Maths podcasting A great way to enjoy maths on the go.
Proof by induction
It is obvious that Proof0 is boring. Assume that Proofn is boring. Then we can deduce that Proofn+1 is boring. By induction, this proof is boring.
NOT
Maths podracing
Now this is not hot.
More free fashion advice online at d chalkdustmagazine.com
NOT chalkdustmagazine.com
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The magnetic pendulum A tabletop demonstration of chaos James Christian and Holly Middleton-Spencer
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have to go a long way to beat the magnetic pendulum for demonstrating the deep and profound nature of physics. One is pictured on the right. On the face of it, a swinging bob seems simple enough to understand. But as is so oen the case, simplicity is masking complexity and the motion possesses an almost magical quality. Here, we will take a glimpse into just how unpredictable the predictable can really be. Pull the pendulum back some distance in any direction, let it go, and prepare to be mesmerised by the way it darts back and forth in an erratic and seemingly random way. The bob is drawn simultaneously to all three base-plane magnets until, finally, it tends towards a precarious state of rest above one of them. Playing like this, it does not take long to become convinced of two immutable facts. Firstly, one can never know beforehand over which magnet the bob will stop. Secondly, and perhaps more subtly, the motion is not reproducible. No maer how hard one tries to replicate the initial displacement, the bob never follows the same winding path twice and the magnet that ultimately ‘wins’ seems to be governed by chance. How can we possibly find such randomness in a tabletop toy? 9
The magnetic pendulum: it comprises a bob with a small magnet suspended by a string above a base plane that contains similar magnets arranged with opposite poles to ensure araction autumn 2018
chalkdust Of course, the bob is not moving randomly at all and we are at best starting it off each time from only roughly the same initial conditions. Its motion is prescribed by purely deterministic equations, a combination of classical mechanics and electromagnetics, and at that level there is no randomness. Were we able to release the bob from exactly the same starting position each time, the subsequent motions would all be identical and they would always stop at the same place. No uncertainty. No unpredictability. No endless fascination! Since our first pull can never be repeated with infinite precision, what we are seeing is sensitive dependence on initial conditions or, more colloquially, the buerfly effect. Any change in input— any, no maer how imperceptibly small—can have a dramatic impact on output. That intriguing phenomenon turns out to be far more widespread and pervasive than one might first imagine. Moreover, it provides our working definition of chaos and crystallises what we mean by saying “a system is chaotic”. In this article, we will start to explore how the magnetic pendulum embodies chaos in the scientific (rather than the everyday) sense.
A phenomenological model It is not too difficult to devise a model that exhibits all the key qualitative features of the magnetic pendulum. Our approach is a phenomenological one, meaning that we are aiming to capture the essence of the motion using intuitive physical ideas rather than focusing on all the mathematical minutiae. A relatively easy way forward is to consider looking down on the pendulum from a plan view (see figure below). The bob’s trajectory in the three-dimensional space is projected downwards onto the horizontal base plane, and the origin of the (x, y) coordinates is fixed at the centre of an equilateral triangle. The magnets are subsequently located at vertices X 1 , X 2 , and X 3 , all of which lie along the circumference of a circle with a radius taken to be the unit length. The position vector of the bob may be represented by x (t) at time t. To account for gravity, it is sufficient for our purposes to consider a restoring force F grav ∝ −x whose influence, due to the minus sign, always acts to pull the bob towards the origin x = 0 (the constant of proportionality is set equal to 1, just to keep things simple). DissiLe: Schematic diagram of the magnetic pendulum pation might be introduced by way Right: Projecting the position of the bob (white square at of the standard velocity-dependent position x) onto the horizontal (x, y) plane. The base-plane damping force familiar from textmagnets (red squares 1, 2, and 3) are positioned at the vertices book physics. Here, we use F losses = of an equilateral triangle. −bu, where u ≡ dx /dt. For some constant b > 0, the effect of F losses is to drain kinetic energy from the motion through air resistance at low speeds. Finally, we must look to include the aractive forces due to the base-plane magnets. It is tempting to reach immediately for the inverse-square rule familiar from Coulomb’s chalkdustmagazine.com
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chalkdust law of electrostatics and Newton’s law of universal gravitation. We instead opt for a 1/distance4 rule as this form tends to describe the forces exchanged by magnetic dipoles. Since Newton’s second law of motion equates mass × acceleration to the combined forces of gravity, magnetism, and damping, we can write down a governing equation of the form 3
∑ d2 x dx Xn − x . +b +x = 2 2 2 5/2 dt dt n=1 (|X n − x | + h )
(1)
Pythagoras’s theorem has been deployed here, and so each contribution in the summation on the right-hand side of (1) corresponds to a ‘distance/distance5 = 1/distance4 ’ type of term. An additional parameter, h2 , has also appeared. Its role is to suppress unphysical (that is, infinite!) accelerations that would otherwise result whenever x approaches X n . We then interpret h as being related to the average height of the bob above the base plane. The equilibrium points are defined to be those positions x = x eq that are unchanging in time. Since the velocity and acceleration of the bob must be zero at those points (hence satisfying dx eq /dt = 0 and d2 x eq /dt 2 = 0, respectively), it follows from (1) that x eq =
3 ∑ n=1
(
X n − x eq
|X n − x eq |2 + h2
)5/2 .
(2)
Aer playing with the pendulum and noting the positions where the bob tends to stop, we might reasonably expect to find maybe three or at most four solutions (with x eq = 0 being the origin). It is worth mentioning that the nontrivial roots of (2) do not occur at x eq = X n , as one might initially suspect. Instead, they lie at the same angular positions as X n (as symmetry demands) but at a radial distance |x eq | that is slightly less than unit length. At these positions, the competing pulls from gravity and magnetism are perfectly balanced. Equation (1) does a surprisingly good job at mimicking the unpredictability so readily seen in experimental demonstrations; the le-hand side is just the damped harmonic oscillator problem from mechanics while the right-hand side sums over the pairwise magnetic-dipole interactions. On the one hand, any urges to seek analytical solutions should be kept in check. Even this strippeddown toy model confronts us with a formidable mathematical beast living in a four-dimensional realm whose axes are (x, y, ux , uy ). On the other hand, computing a numerical solution can be relatively quick and easy for given initial conditions, say x (0) = x 0 and u(0) = 0.
Basins of attraction Physically, we anticipate that the bob will almost inevitably come to rest at one of the non-trivial equilibrium points x eq [cf. (2)] as t → ∞. Those special points can be thought of as positions in the (x, y) plane that aract the trajectory, and accordingly they are oen referred to as fixed-point aractors. The idea now is to use a computer to carry out a systematic set of simulations, recording which magnet ‘wins’ (interpreted as the output) as we vary the starting point x 0 (taken to be the input). By associating the outcome of each computation with a colour (eg red for magnet 1, white 11
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chalkdust for magnet 2, and black for magnet 3), we can overlay the output on top of the input (x0 , y0 ) plane to produce a kind of abstract map. The set of all initial conditions lying in the red region is the basin of araction for magnet 1—that is, any x 0 lying on a red point will always end up at magnet 1 (and similar for the other colours and magnets), though the colour itself gives no information about the path taken by the bob to arrive at that point.
Figure 1. Basins of araction for the magnetic pendulum with b = 0.1 and h2 = 1/4. In the first pane, the grey squares denote the position of the base-plane magnets and the doed grey line is a circle with radius equal to the unit length. The second and third panes show successive magnifications.
Figure 2. An illustration of FSS in the magnetic pendulum. Two initial conditions that are arbitrarily close together can give rise to subsequent trajectories that will start to move away from one another aer a finite amount of time. The red path ends at magnet 1, while the black path ends at magnet 3.
Using this recipe, we discover a rather striking paern (see fig. 1). Regions around the origin appear relatively simple. There are large single-colour lobes which indicate that variations in x 0 tend to have lile impact on which magnet wins. Further out beyond the unit circle, there is much greater complexity and all three colours are intertwined in a beautifully complicated way. In those regions, the pendulum is extremely sensitive to changes in x 0 . Successive zooming-in suggests the intertwining survives down to smaller and smaller length-scales. That feature—proportional levels of paern detail persisting under arbitrary magnifications—is a defining characteristic of a fractal. chalkdustmagazine.com
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chalkdust A less obvious but equally fascinating property relates to the nature of the boundary separating two differently-coloured regions. We typically cannot cross from a region of red to an adjacent region of black without touching the white (similar is true for other permutations of colours). This situation is reminiscent of the delightfully strange Wada property from the field of topology.
Final state sensitivity The basins of araction have three-fold rotation symmetry about the origin, which is a consequence of the equilateral-triangle arrangement of magnets and the initial condition u(0) = 0. Their details also depend crucially on system parameters. One might consider what happens, for instance, when the level of damping is increased (see figure to the right) through b = 0.125 (first row), b = 0.150 (second row), b = 0.175 (third row), and b = 0.200 (boom row). The paern becomes less complex and, accordingly, the pendulum less sensitive to small fluctuations in x 0 . However, their key features remain intact: the persistence of selfsimilar structure (fractality) and complex boundaries that tend to involve all three colours. The Wada-type property is still present in the righthand column of the last two rows, but it is not obvious from these figures. Figure 3. Variation of the basins of arac-
A helpful way to quantify just how strongly the tion for the magnetic pendulum as damping long-term state of a system depends upon small is increased. The second and third columns fluctuations at its input is to estimate the fractal show successive magnifications. Other parameters and domains of the (x0 , y0 ) plane dimension 1 < D ⩽ 2 of the basin boundaries. are the same as those in fig. 1. One selects a set of NΓ points in a domain Γ of the (x0 , y0 ) plane and tests each of them in turn for the property of final-state sensitivity (FSS) by considering a triplet of initial conditions: say (x0 , y0 ), (x0 + ε, y0 ), and (x0 − ε, y0 ), where 0 < ε ≪ O(1) can be interpreted as an error or as a limit to our experimental precision. When the winning magnet is the same for all three trajectories, then the final state is independent of ε and (x0 , y0 ) is, accordingly, free from FSS. Alternatively, think of ε as the radius of a small disc centred on (x0 , y0 ), somewhere within which the ‘true’ initial condition lies. FSS, as demonstrated in fig. 2, appears whenever that disc impinges on a basin boundary and thus overlaps more than one colour. If the total number of points possessing FSS for a given ε is denoted by Nε , one finds that that Nε /NΓ ∼ ε α . The parameter α ≡ 2 − D is known as the uncertainty exponent and it satisfies the 13
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d log10 (Nε /NΓ ) d(log10 ε)
(3)
and a larger D is indicative of increased susceptibility to initial fluctuations. The paerns shown in the middle panes of figs 1 and 3 turn out to have dimensions in the range D ≈ 1.32 (for b = 0.1) to D ≈ 1.16 (for b = 0.2). It follows that the basin boundaries for lightly-damped pendula tend to be associated with higher values of D. More generally, we now see a connection between the dimension of an abstract fractal paern (which, crucially, need not be an integer such as 1 or 2) and the physical property of FSS.
Concluding remarks In this article, we have started to unpick some of the intriguing behaviour exhibited by what is, in reality, a simple toy—one that never fails to capture the imagination of university students in lectures and Ucas applicants (and their parents alike!) at open days. The apparently erratic swinging and unknowable terminus of a bob are not quite so ‘random’ as one might first suppose from a few naïve observations. The essential ingredient giving rise to all this rich and diverse behaviour is the interplay between the three constituent feedback loops (here, due to gravity, dissipation, and magnetism). Although we have considered the barest of bare-bones models (from pretending gravity provides a restoring force proportional to −x, to suppressing the fully-vectorial character of magnetic interactions), the beautiful complexity of nature survives and we simply cannot get rid of it. That, it seems to us, is a mind-blowing conclusion! James Christian and Holly Middleton-Spencer James is a lecturer in physics at the University of Salford with research interests covering various theoretical aspects of electromagnetics and fluids. When not writing papers or teaching, he spends most of his time playing with fractals and wishing he were a mathematician instead. Holly is James’s long-suffering graduate student and has just completed an MSc on waves scaering from fractal screens. She will shortly be starting a PhD in applied maths at the University of Newcastle, studying vortices in superfluids. In the end, she’ll be a real mathematician.
Did you know... …that if you walk randomly on a 2D laice, you will eventually end up back where you started, but if you do the same on a 3D laice you might not. Or, as Shizuo Kakutani puts it: “A drunk man will find his way home, but a drunk bird may get lost forever.”
Did you know... …that primes are so scary that there is a prime named aer one of the seven princes of hell: Belphegor’s prime (1000000000000066600000000000001). chalkdustmagazine.com
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I
MATHS David Richeson
Tired of hearing people ask which area of maths is your favourite? Get one of these tattooed on your forehead and you’ll never be asked again...
I I
TOPOLOGY SET THEORY COMPLEX ANALYSIS -1 I FRACTALS I r ≤ 1 – sin(θ) POLAR COORDINATES I RECREATIONAL MATHS I STATISTICS I (l° v)(e) FUNCTIONS AYE! VOTING THEORY I MATHEMATICAL BIOLOGY y 1 VECTORS x 1 I GRAPH THEORY INFINITE SERIES I KNOT THEORY A A A A
A A A A
I
n =1 2
n
David Richeson David is a professor of mathematics at Dickinson College and is editor of Math Horizons.
a @divbyzero d divisbyzero.com 15
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Somewhere over the critical line Tanmay Kulkarni
M
Polignac was a number. He was prime, and proud of it. One sleepy Sunday morning, with a cup of tea in hand, he opened the newspaper. The major headline shouted, : ! Primes had been subject to prejudice for so long even though primes had founded the society. The strong dislike stemmed from primes being factors of composites. Maxamillion sighed with despair, but resigned, he continued reading.
With this idea on his mind, he went to his friend, another prime named Bernhard Oblong. Maxamillion said “I think we should try and find the Critical Line—we both get something: you’re a factorial prime, hence with the Formula, you could find out what your n is; and I could find my twin prime!” “Are you sure the Formula exists?” Bernhard asked. “We have nothing to lose and everything to gain! Just like Pascal’s Wager.” Maxamillion reasoned. “I hope you are right because I really want to know what n is!” Bernhard agreed enthusiastically. They embarked on their journey, oblivious to the dangers ahead. Starting in their city, the Number Line, they eventually reached a large building, the RSA bank in the outskirts of town. They noticed a sign with big black leers: .
He aimlessly scanned the newspaper until something caught his eye: : ? The article speculated that the Critical Line was a fabled golden brick road that could lead to the Formula connecting all primes. It was supposedly hidden in the zeta landscape: an untouched land that defined the real and imaginary axes. The zeta landscape was complex and hard to navigate, and hence, the perfect place to hide the Formula. With the Formula, all the secrets underlying primes and how to find them would be revealed. Being a prime, Maxamillion did not have siblings, and living in a conflicted society, he felt alone, despite his many friends. But the Formula would give him a chance to find his twin prime who would make him feel complete. chalkdustmagazine.com
“Let’s withdraw some money for the journey to the zeta landscape,” suggested Bernhard. “Sure. But we beer be careful,” replied Maxamillion. A man with a baton stopped them. “Halt!” he said. “Can’t you read? ! We don’t have primes coming this side of town.” Maxamillion and Bernhard had no choice but to leave and try the next village. Another man, wearing a smug smile, saw the commotion through the glass doors of the 16
chalkdust As they were walking, Friedrich explained to them: “The UPS stands for United Prime Service. We are dedicated to protecting the rights of primes against the relentless prejudice of the composites. Join us. We are with the primes, we will continue to be so until the end.”
bank. “Hmm… primes. Interesting! Primes usually avoid banking with us because we use them to encrypt our credit cards. I know an opportunity when I see one. Let me see if I can capture them,” he thought to himself. The slippery man phoned the most notorious prime-hunter in the zeta landscape—the Mersennary.
“Consider us members,” Maxamillion said. “Even though you are not as sturdy as us, we could use your help. Find the Formula and put an end to this!”
“Hello?” Mersennary crackled. “I want you to capture two primes headed your way. Bring them back dead or alive. $1,000,000 in credit cards,” the man ordered.
Before she sent them on their quest, the leader gave them a fascinating relic: ζ(s). “This is the zeta function,” the leader explained. “Use this once you reach the zeta landscape: it will help you navigate your way along the Critical Line.”
“It’s a deal!” Mersennary replied. Their pockets empty, Maxamillion and Bernhard soon reached a small village. A sign hung over the entrance: . “Wow. Those are some extreme views!” exclaimed Bernhard. Not soon aer, they found themselves surrounded by hundreds of primes who Maxamillion recognised immediately: Sophie Germain primes. They resisted any composite prejudice. They were well-built primes (2p + 1) rebelling against the system and hoping to teach composites that primes are the building blocks of society, deserving equality. “Who are you? Why are you here? Are you really primes or are you composite sympathisers?” The leader pelted questions faster than the duo could process.
Progressively, the scenery morphed into a barren land: the zeta landscape. The two dimensions defined the real and imaginary axes. Using the relic, they navigated across the imaginary hills and the complex terrain. Even with hypothetical fog layering the land, they could clearly make out the glowing pathway. There it was—the Critical Line! “Whoa. It really is real. Really real.” Bernhard gasped. However, there was a dilemma ahead, for the Critical Line split into three paths.
“We are Maxamillion and Bernhard. We are trying to find the Critical Line. We are primes and certainly not composite sympathisers.” Maxamillion responded swily. “Then you must be admied into the UPS at once,” the leader proclaimed. “Follow Friedrich into the tent.” The two friends did as they were told. 17
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chalkdust Suddenly, a prime emerged from the fog. His sunken eyes added to the barren landscape. He whipped out his weapon, the square function. With it, the prime could square Maxamillion and Bernhard and turn them into composites. He advanced towards them, armed and dangerous. The duo trembled as beads of icy sweat trickled down their backs. “I am the Mersennary. I am paid to hunt down primes like you,” he rasped. Maxamillion tried to plead with him: “Why are you trying to break something that can’t be broken? We are all primes here. We have a rich history. Primes have been the dominant species in the whole of maths for hundreds of years. We ruled because we could not be broken down into other numbers. When we multiplied ourselves together, we created composites. Even though the composites have oppressed us, we remain strong and resistant. Primes will never be split. You are one of us, so are you a traitor?”
“Let’s go x/ln(x)!” Mersennary said. “No! Let’s go Li(x)!” Bernhard replied. Maxamillion urged: “Stop arguing! How about we compromise? Let us go explore the stairway π(x). Maybe the Formula is hidden in the middle to stop people who aim too high or too low!” They began climbing the never-ending stairs. They were about to give up hope when they saw the Formula. It was π(x). When Maxamillion touched it, it surrounded him with a blue light. Full of excitement, he asked the Formula to find his twin prime. But his enthusiasm didn’t last long as the formula would not give an answer. He sighed in desperation, but then he had an idea. He asked it the value of Bernhard. It answered 26951. Then he asked it the value of himself. It answered 26953.
“Sorry. It is nothing personal, just business.” Mersennary responded coldly and inched closer. In a desperate aempt, Maxamillion tried again: “Wait! You are in it for the money, right? War is not a steady business, and I am sure you would earn more at a new job. We want to get the Formula, which could help you learn more about yourself and other primes! You could use that to your financial advantage, eh?” Mersennary pondered and realised he had the wrong end of the number line. He decided to join them in the search.
“What⁈ We were twin primes all along⁈” Maxamillion shouted. “Wow!” Bernhard exclaimed. He then proclaimed: “With this, we can end prejudice! We could change the composites’ opinion of us by explaining all the secrets behind the primes and how intricate and beautiful we are!”
Together they looked at the three new paths: Li(x), π(x), and x/ln(x). Li(x) looked the most promising, because it went the highest, and it looked daunting enough to hide the Formula. x/ln(x) was very low, and it seemed like a place to start. chalkdustmagazine.com
“We could also start a bank that serves all number-kind! Then I would have a steady 18
chalkdust crashed to the ground with a satisfying BANG!
source of income!” declared Mersennary excitedly.
The Formula and the relic were placed in Museum Polytechnique. The conflict between the two sets was finally resolved as the composites realised that the Formula revealed the complexity behind the primes. They realised that primes are so complex that they deserve to be treated beer. Thus, the numerical landscape was changed forever!
As soon as they got back to Number Line, the trio started a bank: the Riemann bank. Soon it was booming and bought over the RSA bank. The first thing Maxamillion did as CEO was to demolish the sign saying, ‘ ’. The law that barred the primes went down with the sign and they both
Glossary Composites Logarithmic integral, Li(x) Mersenne primes Factorial prime Natural logarithm Pascal’s wager Primes Riemann hypothesis Riemann zeta function Sophie Germain primes Twin primes
Numbers that can be wrien as the product of 2 or more primes. An approximation of the number of primes until a certain given number, formulated by Gauss. Primes of the form 2n − 1. Primes of the form n! − 1. A logarithm with base e, not base 10. A wager that states that if you believe in God and God does not exist, you have nothing to lose. If God does exist, you have everything to gain. Numbers that do not have any factors beside 1 and themselves. Riemann’s conjecture deals with the locations of the solution to Riemann zeta function. It is the holy grail of mathematics. An infinite series used to investigate properties of prime numbers. Primes in the form 2p + 1 where p is a prime. n and (n + 2) are primes.
Tanmay Kulkarni Tanmay is 10 years old and lives in Washington state. He loves exploring all sorts of maths and the trails along the Cascade Range. He dabbles in blues music on the acoustic guitar.
Astrid and Borg by Daniel Griller i) Astrid thinks of a positive integer n. She then tells Borg the sum of the two largest factors of n (excluding n itself). Is it always possible for Borg to deduce the value of n? ii) Borg thinks of a positive integer m. He then tells Astrid the product of the two largest factors of m (excluding m itself). Is it always possible for Astrid to deduce the value of m? You can find more puzzles by Daniel in his (highly recommended) book Elastic Numbers, on his blog d puzzlecritic.wordpress.com, and on Twier a @puzzlecritic. 19
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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, The annual village fete is fast approaching, and every year I embarrass myself at ‘guess the number of sweets in the jar’. My exasperated wife end s up telling me to just say a number, and I always panic. Last year my guess was i − π. Maybe I was just hungry.
— Hungry hungry hippo, Gospel Oa
k
■
DIRICHLET SAYS: It’s important not to get stressed in these situa-
tions. Sure, you have countless choices, but you’ve just got to think rationally, stay positive: don’t make it overly complex. The sweets in the jar stand is an integral part of any fete, and who wouldn’t want some tasty gummy d’Alemberts, strawberry Lovelaces or shilbert lemons? If in doubt, round up, and you can’t go too far wrong.
Dear Dirichlet,
g for hospital staff. ss selling standard issue clothin I recently started my own busine from cusehow get countless complaints som I nth mo ry eve but ers ord I get lots of g. What can I do? tomers who never receive anythin bing up, Warwick
— Scrub
■ DIRICHLET SAYS:
I’m afraid the all or nothing delivery is an inherent problem for a uniform distribution business. The normal concerns businesses have are that all customers receive something but hardly anyone receives everything they need. To increase your probability of success, simply create more stock and distribute it more widely. Else send partially complete orders to more people, but that could lead to more complaints. At least you’re doing much better than my friend Dirac. His company, Delta, always produced the full quota for orders, but it would get sent out in one go to a single address. His distribution just didn’t function. chalkdustmagazine.com
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Dear Dirichlet,
I run a chain of stores that imp orts a lot from Europe. I’m sure that Brexit will be a disaster for us and I’ve been figh ting loudly to try and stop it. Ho wever, it appears that my campaigning has led to my stores in Basildon, Rochester and the Kran National Park in Togo suddenly not making any profit. Any idea wh y?
— Not another one, Snettisham
■
DIRICHLET SAYS: After marking your store locations on the antique
globe I keep in my study, I’m afraid I have bad news. Your stores with zeros as their profits all lie on the line half-a-degree east. It does appear that your valiant efforts have condemned your business to suffer in this way: a direct corollary of the Remain hypothesis.
Dear Dirichlet,
al maths section but I I am not sure if you have a flor took in my garden towould like to enter this photo I garden? day. What do you grow in your
— Green fingers, Outdoors
■ DIRICHLET SAYS: I always enjoy receiving pictures of flowers. At my country house, I grow hya-sinh and cro-cosh-es. In the city, I just keep herbs: some sparsley, teragon, and a small pot of Sage (because it’s cheaper than Mathematica). In fact, I recently uploaded some photos of them to the ar-Chive. I did try growing integeraniums once but they were eaten by the badgers.
Dear Dirichlet,
visiting them twin daughters. I’m excited to be to h birt en giv ly ent rec has er My sist worried I will strugg them for the first time but I’m at home next week and meetin ell tips? — Aunt you happy to see me, Sandw gle to tell the twins apart. Any
■ DIRICHLET SAYS:
A twin f:IR–>IR is differentiable at home, ΩϵIR, if the derivative f(Ω+h)-f(Ω) exists. f’(Ω) = lim h–>0 h More Dear Dirichlet, including two seasonal specials, online at d chalkdustmagazine.com 21
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An invitation to category theory Martin Damboldt
Tai-Danae Bradley
E
in our mathematical education, we learn about a strong interplay between algebra and geometry—algebraic equations give rise to graphs and geometric figures, and geometric features can be encoded in algebraic expressions. It’s almost as if there’s a portal or bridge connecting these two realms in the grand landscape of mathematics: whatever occurs on one side of the bridge is mirrored on the other. So although algebra and geometry are very different areas of mathematics, this connection suggests that they are intrinsically related. Incidentally, the ‘bridge’ that spans them is a but a dim foreshadow of much deeper connections that exist between other branches of mathematics that also may, a priori, seem unrelated: set theory, group theory, linear algebra, topology, graph theory, differential geometry, and more. And what’s amazing is that these relationships—these bridges—are more than just a neat observation. They are mathematics, and that mathematics has a name: category theory.
What is a category? Martin Kuppe once created a wonderful map of the mathematical landscape (see facing page) in which category theory hovers high above the ground, providing a sweeping vista of the terrain. It enables us to see relationships between various fields that are otherwise imperceptible at ground level, aesting that seemingly unrelated areas of mathematics aren’t so different aer all. This becomes extraordinarily useful when you want to solve a problem in one realm (topology, say) but don’t have the right tools at your disposal. By ferrying the problem to a different realm (such as group theory), you’re able to see the problem in a different light and perhaps discover new tools that may help the solution become much easier. In fact, this is precisely how category theory came to be. It was birthed in the 1940s in an aempt to answer a difficult topological question by recasting it in an easier, algebraic light. Thinking back to the mathematical landscape, you’ll notice that each realm consists of some objects (set theory has sets, group theory has groups, topology has topological spaces…) that relate to each other (sets relate via functions, groups relate via homomorphisms, topological spaces relate via continuous functions…) in sensible ways (such as composition and associativity). This common thread weaves throughout the landscape and unites the various fields. The mathematics of category theory formalises this unification. More concretely, a category is a collection of objects with relationships between them (called morphisms) that behave nicely in terms of composichalkdustmagazine.com
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Martin Kuppe, by permission
Martin Kuppe’s map of the mathematical landscape
tion and associativity. This provides a template for mathematics, and depending on what you feed into that template, you’ll recover one of the mathematical realms: the category of sets consists of sets and relationships (ie functions) between them; the category of groups consists of groups and relationships (ie group homomorphisms) between them; the category of topological spaces consists of topological spaces and relationships (ie continuous functions) between them; and so on.
The analogy between a category and a template is due to Barry Mazur from his wonderfully written, non-technical introduction to category theory, When is one thing equal to some other thing? In it he writes, “This concept of category is an omni-purpose affair…There is hardly any species of mathematical object that doesn’t fit into this convenient, and oen enlightening, template.” Indeed, as category theorist Eugenia Cheng so aptly put it in her treatise Higher-dimensional category theory, “category theory is the mathematics of mathematics”. 23
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It’s all about relationships One of the main features of category theory is that it strips away a lot of detail: it’s not really concerned with the individual elements in your set, or whether or not your group is solvable, or if your topological space has a countable basis. So you might be thinking, “Eh, category theory sounds so abstract. Can any good come from this?” Happily, the answer is yes! An advantage of ignoring details is that our aention is diverted away from the individual objects and turned towards the relationships—the morphisms—that exist between them. And as any category theorist will tell you, relationships are everything. Indeed, one of the main maxims of category theory is that a mathematical object is completely determined by its relationships to all other objects. To put it another way, two objects are essentially indistinguishable if and only if they relate to every object in the category in the same way. This theme (which is a consequence of a famous result called the Yoneda lemma) isn’t too different from what we observe in life. You can learn a lot about people by looking at their relationships— their Facebook friends, who they follow on Twier, who they hang out with on Friday nights, for instance. And if you ever meet two people who have the exact same set of friends, and whose interactions on social media are exactly the same, and who hang out with the exact same people on Friday nights, then you might jokingly say, “You can’t even tell them apart”. Category theory informs us that, all jokes aside, this is actually true mathematically! So you might wonder “Hmm, if mathematical relationships are that important, then what about relationships between categories? Do they exist?” Great question. The answer is: absolutely! In fact, these particular relationships have a name—they are called functors. But why stop there? What about the relationships between those relationships? They, too, have a name: natural transformations. In fact, we can keep on going: “What about the relationships between the relationships between the relationships between the…?” Doing so will land us in higher-dimensional category theory, which is where much of Cheng’s research lies. As abstract as this may sound, these constructions—categories, functors, and natural transformations— chalkdustmagazine.com
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chalkdust comprise a treasure trove of theory that shows up almost everywhere, in mathematics and in other disciplines! Since its inception, category theory has found natural applications in computer science, quantum physics, systems biology, chemistry, dynamical systems, and natural language processing, just to name a few. (The website d appliedcategorytheory.org/workshops contains a list of applications.) So even though category theory might sound a lile abstract, it is highly applicable. And that’s no surprise. Category theory is all about relationships, and so is the world around us!
Conclusion Categories are a lile bit like anchovies: some folks love ’em, while for other folks they’re an acquired taste. So yes, it’s true that category theory may not help you find a delta for your epsilon, or determine if your group of order 520 is simple, or construct a solution to your PDE. For those endeavours, we do have to put our feet back on the ground. But thinking categorically can help serve as a beacon—it can strengthen your intuition and sharpen your insight—as you trek through the nooks and crannies of your favourite mathematical realms. And these days it’s especially hard to escape the pervasiveness of category theory throughout modern mathematics. So whatever your mathematical goals may be, learning a bit about categories will be well worth your time! This article has been adapted from “What is Category Theory, Anyway?” first published on 17 January 2017 at d math3ma.com Tai-Danae Bradley Tai-Danae Bradley is a PhD candidate in mathematics at the CUNY Graduate Center, where she spends much of her time thinking about category theory, topology, machine learning, and quantum physics. When not thinking on these things, she’s probably writing (and doodling) about them on her blog, Math3ma.
a @math3ma d math3ma.com My favourite function Maths is full of functions. We’ve spread some of our favourite, and not-so-favourite, throughout this issue. We’d really love to hear yours! Send them to us at c contact@chalkdustmagazine.com, a @chalkdustmag or b chalkdustmag and you might just see them on our blog!
The example function Tom Rivlin
My favourite function is what I like to call the example function. I write it as eg(x). You know this function. It’s the one your lecturer squiggles on the chalkboard when they’re explaining how calculus works or something. When someone is talking about the properties of functions in general, and they An instance of eg(x) I drew to need some random example, a few simple squiggles will do. explain… I dunno… integrals Unlike other functions, it can take various forms (any form, in or something? fact). It always looks different, but it always gets the job done. 25
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Puzzles
Looking for a fun puzzle but not got time to tackle the crossnumber? You’re on the right page.
Sudoprime by Peter Chamberlain
7
In this crossnumber grid: 1. Each of the eleven answers is a prime. 2. No number begins with 0. 3. No digit appears more than once in any row, column or either of the two long diagonals. 4. No even digit appears more than once anywhere in the grid.
3
Arrange the digits Put the numbers 1 to 9 (using each number exactly once) in the boxes so that the sums are correct. The sums should be read le to right and top to boom ignoring the usual order of operations. For example, 4 + 3 × 2 is 14, not 10.
× ×
+
−
=2 × = 20
×
+ = 4
× ÷
+ −
=2
−
= 39
= 48
Mini crossnumber by Humbug As usual, no numbers begin with 0. Whenever a clue asks for a multiple of another clue, the two clues are not equal: eg 2D is not equal to 1D.
3
1
2
4
5
7
chalkdustmagazine.com
Across
2 The last three digits of 3A. (3) 3 A multiple of 2A whose first three digits are a (6) palindrome. 7 An integer. (2)
6
Down 1 2 3 4 5 6
A prime number. A multiple of 1D. A multiple of 2D. A multiple of 3D. A multiple of 4D. A multiple of 5D. 26
(2) (2) (2) (2) (3) (3)
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Codeword In the crossword grid below, each leer of the alphabet has been replaced by an integer. Once you have decoded the crossword, the leers in yellow can be arranged to make a word related to three other words appearing in the grid. 8
14
13
5
17 6
13
10
22
18
8
10
5
9 13
24
5
13
3 4
22
14
12
12
19
23
14
24
4
17
5
18
6
19
7
20
8
21
9
22
10
23
20
11
24
19
12
19
12
25
17
7
13
26
17
21
G
10
8
26
10
8
19 2
5
14
24
O 5
24
17
22
13
22 17
14
14 8
22
20
19 14
15 16
14
12
2
O
3
17
22
14
13
12
17
1
3
10
14
10 15
17
21
8
25
11
1
12
22
17
16
5
14
7
7
22
20
7
10
17
22 5
5
8
20
21 17
10
13
3
17 25
25
Making elevens Put the numbers 1 to 9 (using each number exactly once) in the boxes so that the sums are correct. The sums should be read le to right and top to boom ignoring the usual order of operations. For example, 4 + 3 × 2 is 14, not 10.
14
+
−
= 11
+ ×
= 11
−
+ − + = 11
G
−
+
A N B O C P D Q E R F S G T H U I V J W K X L Y M Z
− = 11
+ = –11
= 11
More puzzles including Christmas conundrums and 3D sudoku online at d chalkdustmagazine.com 27
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Significant figures
Katherine Johnson NASA Langley, CC BY-NC-SA 2.0
Gerda Grase
T
year, on 26 August, one of the most memorable and well-known mathematicians, Katherine Coleman Goble Johnson, celebrated her 100th birthday. This is a tribute in honour of her life so far.
Family life and first steps towards mathematics Katherine was born on 26 August 1918 in White Sulphur Springs, a small town in West Virginia. She was the youngest of four children and was always the smart kid—she finished high school at the age of 14 and earned her Bachelor of Science in mathematics and French from the West Virginia State University at the age of 18. This, in part, was thanks to her father, who moved the family closer to a school to help his children get a beer education. She still remembers her family dearly. Especially fondly, Katherine talks about her father’s stepmother, known as granny. They would visit her house to eat some of her delicious pancakes—just as anyone would with their grandmothers! The four children loved their parents very much: they thought of their mother as the preiest lady in the world and their dad as the most handsome man. Katherine says she was daddy’s girl, but she always remembers her father telling her, “you are as good as anybody in this town, but you’re no beer”. chalkdustmagazine.com
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chalkdust Katherine was always good at mathematics. She has memories from childhood very clearly linking her to the subject: “I counted everything. I counted the steps to the road, the steps up to the church, the number of dishes and silverware I washed… anything that could be counted, I did”. But mathematics wasn’t the only subject for her. Sure, she loved it, but she was just as good in English because it also felt logical to her.
A postcard of Katherine’s hometown
When asked why mathematics became the subject she was most fond of, she says it was because it was the subject you had to work hard for. It was the one subject with a right and a wrong, and once you got it right, it was right—unlike history! The next push towards maths came at university. In eighth grade she had a maths teacher who happened to teach at the university Katherine went to years later. One day, she happened to meet the teacher again, who told her “if you aren’t in my math class this semester, I’m coming aer you!” So Katherine had no choice but to go to maths class and her career in mathematics had begun. Later at university, her maths interests were taken care of by Mr Claytor. He added courses to the university almost exclusively for Katherine because he could see her potential. It was he who steered her towards research mathematics and eventually NASA.
Female mathematician at NASA
f
Katherine started her career in a rather unusual manner—she became a computer. Back then, these were the people who did the calculations for NASA’s predecessor NACA (the National Advisory Commiee for Aeronautics) before the space race began. Katherine was sent to the flight research division.
She counts her character as one of the key things that contributed towards her career as a NASA mathematician. She remembers how her siblings and parents always used to try to shush her because she was always so assertive and stubborn. Aer she was invited to join the men doing the mathematics behind the calculations she would perform, she started fighting for her own place within the team. She demanded to see all the data, and asked to join the confidential meetings NACA held, slowly gaining respect in the midst of the all-male team. Especially noted in the recent film Hidden Figures are the times when Johnson overcame and baled racism, as the only female and the only African American in the department. Katherine herself always says, though, it was never anything special: she just did her job and was appreciated for that, not her sex or skin colour. On 5 May 1961, Freedom 7 was to be launched. Modern technological computers were already running the numbers and everything was being prepared for the mission that would make Alan Shepard the first American in space. The astronaut, however, was feeling uneasy. He made the call to NASA to speak to Katherine Johnson. Would she redo the calculations?, he asked. Because if 29
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chalkdust she got it right, he knew it was right—he would feel safe to go on the mission. Katherine approved the computer’s calculations. She admits the NASA team was much more worried about Shepard never making it back to Earth. If he missed the trajectory by a few degrees or tried entering at a different velocity, he would never return home. The mission was successful and Katherine later also checked the trajectory for the Apollo 11 and Apollo 13 missions, further proving her incredible mathematical skills. As a token from NASA, Katherine received an American flag that flew to the moon. At this point you may wonder—wait, you’re telling me about all these amazing things she has done, but you aren’t sharing any of the maths. Unfortunately, many of her articles aren’t Aerial view of Apollo 11 available to the public. However, the three that are, are described below. I must warn you—all contain sophisticated mathematics and will most certainly take quite a while to wrap your head around. However, I can give you some insight into what the three papers contain. The first paper (Skopinski & Johnson, 1960) focuses on calculating the azimuth angle when placing a satellite over a predetermined position to ensure safe landing. The second paper (Westrick & Johnson, 1962) is an analysis of the data from the Echo 1 satellite. It contains a lot of very nice graphs—it’s worth having a look just for the curves!
Skopinski & Johnson, 1960
The third paper (White & Johnson, 1964) probably has the toughest maths, but similar to the first one, it focuses on finding solutions of some variables for the landing of a satellite. Arm yourself with some patience—the papers are worth your time even if they might seem a bit daunting at first!
After NASA She retired from NASA in 1986, but she still has her hands full. For 50 years she enjoyed singing in a church choir. She loves playing bridge and other mathematical games, and she plays the piano and enjoys spending time with her six grandchildren and 11 great-grandchildren. She has authored or co-authored 26 research papers, and she has worked on the space shule and Project Apollo’s lunar lander. For her achievements and lifelong work, she received the Presidential Medal of Freedom in 2015. In 2016, the BBC named Katherine in their 100 Women 2016 as one of the most inspiring women alive. And then there’s the aforementioned film Hidden Figures, which revealed the story behind the three brilliant mathematicians Dorothy Vaughan, Mary Jackson, and of course Katherine, which has since conquered my and many other mathematicians’ hearts. chalkdustmagazine.com
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chalkdust Katherine Johnson has been an inspiration for mathematicians all around the world, showing how one person can change so much. From gaining respect in one of the most prestigious research facilities in the world in times of unimaginable discrimination, to creating mathematics which helped many astronauts find their way back home; from simply being a wonderful person to being an incredibly talented mathematician—here’s to Katherine Johnson on her 100th birthday!
Katherine Johnson in 2008
Gerda Grase Gerda is a Latvian third year undergraduate mathematics student at the University of Edinburgh. Her favourite Platonic solids are currently the icosahedron, closely followed by the tetrahedron. In her free time, Gerda enjoys dancing in the university’s salsa society and looking aer her ever-growing collection of plants.
c gerdagrase@gmail.com The desert problem by Alaric Stephen and Alex Mayall 3. 5 l
3l
2l
2l 1l
Out in the desert, there are 4 containers of water containing a total of 10 litres of water doed around a circular hiking trail. The locations and sizes of the containers and the amounts of water needed for each segment of the trail are shown to the right. Is there a point on the trail where you can start the hike and make it the whole way round without running out of water?
3l
2l
3.5l If there were n containers spread around the trail containing a total of 10 litres between them (albeit not necessarily in an equal distribution or with equal spacing between them), will there be a point on the trail where you can start the hike and make it the whole way round without running out of water?
You can hear Alaric and Alex discuss puzzles like this on their superb podcast Odds and Evenings. Find out more at d oddsandevenings.com. 31
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Which number are you? START Do you like numbers?
NO
YES
NO YOU ARE
Are you an engineer?
Are you a mathematician?
22
NO YES
YES
YES NO YES
YES YOU ARE
Are they real?
YOU ARE
imaginary
irrational PAST IT
Are you in your prime?
NO YES
YES
! T? HA W
YOU ARE A
YOU ARE
Sophie Germain prime
perfect
IT’S POINTLESS
54
MATHS
YOU ARE
IT’S A GAME
How you doin’?
What’s the meaning of life, the universe, and everything?
42
Were you born in 1776?
NO
^^
WHAT’S MARMITE?
Do you like Marmite?
NO
Do you have any friends?
YOU ARE
square
YOU ARE YOU ARE A
p
YOU ARE
negative
sexy prime Pictures: Marmite: AZAdam, CC BY 2.0
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What is the point of intersection?* *not to be judgmental Elizabeth A Williams
S
months ago, one of my A-level students shared a puzzle with me. It’s a neat puzzle in its own right, but what came out of our conversation is even neater. My student made a fantastic insight into understanding what a logarithm was trying to say.
Puzzle Let y = bx where b > 0 and b ̸= 1. Consider the line which passes through the origin and is tangent to this graph at point T. What are the coordinates of T? y
T
x
This is great fun to slap into your graphing program of choice (all hail Desmos!). Look at y = 2x , y = 3x , maybe y = (1/2)x . Draw the line y = mx and twiddle its gradient until the line just meets the exponential. The program will give a good approximation to the coordinates of T, and voilà, you’re off and away to happy sandboxing and conjecturing. Pause the article here to have a play for yourself. 33
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chalkdust It’s a prey neat result. But—cracks knuckles—there is much satisfaction to be had in proving a conjecture. My student and I brandished our pencils, and this is what we wrote. Take y = 3x . The tangent line through the origin intersects the graph at point T with coordinates (t, 3t ). There are two ways to calculate the gradient of this tangent line. Because the line goes through (0, 0) and (t, 3t ), Δy/Δx = (3t − 0)/(t − 0) = 3t /t.
Also, as y = 3x , dy/dx = ln 3 × 3x . At point T, the gradient of the tangent is ln 3 × 3t .
Equating these two ( ) expressions and solving for t gives t = 1/ ln 3. Thus the coordinates of T 1 / ln 3 are 1/ ln 3, 3 . ( ) In general, for y = bx , the coordinates of T are 1/ ln b, b1/ ln b .
Ta-dah! We sat back in our chairs.
Except… what kind of number is 31/ ln 3 ⁈ Our conjecture had strongly hinted at what to expect, and this wasn’t quite it. I leaned forward again, scratching my pencil across the paper to find a simplification. My student remained still, regarding the expression 31/ ln 3 thoughtfully. Then he observed, “That’s e. It’s just e.” ! How did he recognise it on sight? One of my mathematical mantras is logs are powers (ommm). That is, logb c is the power to which you raise b in order √ to get c. The number logb c is named by the description of what it does, much like how writing 5 means the number such that when you square it, you get 5. We’re identifying a specific number by its property rather than by its explicit value. This mantra directly generates the delightful construction blogb c which now clearly shakes out as c. See? logb c is the power you raise b to, in order to get c. We’ve raised b to that very power, so what do we get? c. Hurrah! My student had taken this mantra to heart. And then he took it one step further. He reasoned as follows. We know, by the mantra, that eln 3 = 3. Also, the expression 31/ ln 3 means take the (ln 3)th root of 3. If we take the (ln 3)th root of both sides of eln 3 = 3, then on the LHS, we are le with just e. Hence, 31/ ln 3 equals e. I think this is a beautiful insight. It comes from viewing a power as a root and then seeing the power of that root, as it were. Until my student pointed it out, I hadn’t realised this elegance. Elizabeth A Williams Elizabeth is a freelance mathematician accompanied by Zeke, a free-range rat.
c realityminus3@gmail.com d electromagneticretaliation.wordpress.com a @RealityMinus3 chalkdustmagazine.com
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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
Power in numbers Talithia Williams The biographies of more than 30 female mathematicians are featured in this inspiring and really entertaining book. Although I would recommend it to everybody, it is definitely a must for those who are still deciding if a career in maths would be for them. For the more advanced audience, it might be missing some technical detail, but it is still a very enjoyable read.
ggggi The Theoretical Minimum
Electromagnetic Field
George Hrabovsky, Leonard Susskind and Art Friedman A really good series of popular science books with actual equations! Too bad they’re all about physics…
The most fun that it’s possible to fit into 3 days. Bring on 2020.
ggggg
ggggi
Statue of Lagrange in Turin “I’ve seen beer statues.”
Piled Higher and Deeper
ggiii
Jorge Cham Entertaining and hilarious. Jokes that any graduate student/academic will instantly relate to.
8 out of 10 maths A genuinely funny evening of maths-themed comedy improv.
ggggh
ggggi
Resistance is Futile Jenny T Colgan A seamless combination of romance and maths. Shall we call it mathsmance?
Scorpion Drake Not enough fs on the cover.
hiiii
ggghi
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Mathematics and art: the ELHP
Adam Atkinson
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people like to hear about mathematics being used to address real-life problems. I am going to claim that the problem I describe in this article is real-life because it arises from a conversation between two non-mathematicians.
Specifically, one of my sisters-in-law did an art degree, and as part of a project she did for this she visited Sardinia to interview the sculptor Pinuccio Sciola. At least some of his works are quite large, by which I mean maybe 3 m high or more, based on things I see on the web. During their conversation, he said something about wanting to install one of his sculptures on a named mountain somewhere in the Catania area, for the benefit of the residents. I don’t know his exact words, but my sister-in-law found this remarkable enough that she reported it to me and other members of the family. I’m not naming my sister-in-law here because she is not a public figure, and feels that this article is not the way she would choose to become one.
ELHP The logo of the ELHP
It may be important to note that my sister-in-law lives in a small town near Catania, and this may be what prompted him to say this. It seems entirely likely that he had not spent any real time looking into this idea. Indeed, I am told that when he later visited Catania he immediately realised that his idea was probably unrealistic. chalkdustmagazine.com
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chalkdust Let’s use maths, and some other disciplines, to consider his idea, pretending for the sake of discussion that he or someone else really does want to go ahead with it. You may notice that even though Sciola named the mountain, I have chosen not to do so. I shall do this later. For starters, let’s have a look at the mountain. I took this photograph from just outside Catania airport. Catania itself is mostly nearer to the mountain than the airport is, but if we’re going to build a sculpture or statue on a mountain to be seen from a city with an international airport, we might as well make it so that people arriving from overseas can appreciate the statue as soon as they arrive. Apart from anything else, when you’re in town, buildings are oen between you and the mountain.
The view from Catania airport
I don’t know that Sciola said where on the mountain he wanted to place a sculpture, but it seems as though if you’re going to do this at all you might as well put the sculpture right on top of the mountain. Our first discipline might be geography: how far is the peak of this mountain from Catania airport? There is a ‘measure distance’ feature on Google Maps, and drawing a line from where I took the photo to a reasonable guess at where the peak is, I get 32 km. There’s also an issue of vertical separation, since the mountain is 3329 m tall. There are plenty of higher mountains in the world, but that’s not bad. The airport is very near the coast, and my eyes aren’t that far above the ground. Let’s pretend the photograph was taken at exactly sea level. We’re not going to be pretending anything we do has more√than about 1 significant figure anyway. Using Pythagoras we get the straight line distance as 33292 + 320002 = 32173 m. Let’s just say 32 km. Pretend the Earth is flat for now. Since we can see the mountain, we could at least in principle see things on it. But how big would they need to be? How are we going to define what we mean by how big a statue or any other object appears to be? We could use steradians. Imagine a transparent sphere centred on your eye, of radius 1 m. If you look at an object over 1 m away, rays from that object to your eye hit the surface of this pretend sphere in some places and not in others. The area of the sphere the object occupies is the ‘solid angle’ the object occupies centred on your eye. ‘The solid angle subtended at your eye’ or ‘the object subtends a solid angle at your eye’ is how I learned to say this sort of thing. Since the surface of a sphere of radius 1 is 4π square metres, if you sit inside a hollow spherical statue then the statue blocks your vision in all directions (even if you rotate your head/body) so the statue occupies a full 4π steradians. If you press your nose up against an infinite flat wall, that will occupy near enough 2π steradians. More plausible statues will subtend much smaller solid angles than that. Instead of using steradians we could define the full sphere to be 1 and measure fractions of a sphere 39
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chalkdust instead. Trying to use a human’s field of vision as our 100% value would run into problems with different people having different amounts of peripheral vision. But we’re not going to use this kind of approach at all. However, as an exercise, consider what solid angles various countries on the surface of the Earth subtend at the centre of the Earth, or what solid angle your computer screen subtends at your eye while you are using your computer. Instead of worrying about solid angles, let’s just use normal angles. When you look at the object, there will be two points on it that appear to be furthest apart. Perhaps the head and feet of a human, or the head and tail of a dragon. The ‘angular diameter’ of an object is the normal angle subtended at your eye by the (apparently) furthest separated points of the object. We could measure this angle in degrees, radians or as a fraction of a full circle. This is probably going to be a much easier value to estimate than the solid angle subtended by a statue. We could consider statues which were basically obelisks or nails or something else tall and thin. The angular diameter of a tall thin thing will depend on the angle you look at it from as well as the distance. Consider: Here is your eye gazing upon a tall thin cylindrical statue. Your eye is about level with the base of the statue. Lines to the top and boom of the statue from your eye are about 60 degrees apart. So the angular diameter of the statue is about 60 degrees. Here your eye is about the same distance from the centre of the statue, but the base of the statue is above you. The lines to the top and boom of the statue from your eye are now much closer together, so the angular diameter of the statue is smaller. Now you’re almost directly beneath the statue. The angular diameter of the statue could become almost zero assuming the radius of the cylinder were very small. We can avoid this issue by considering spherical statues! A spherical statue at a given distance will always have the same angular diameter. We know we can see very distant stars in the night sky, and they have extremely small angular diameters. It’s clear that there isn’t really a minimum size our sculpture could be to be visible at all: if it gives off enough light we’ll be able to see it no maer how small it is. If we want to calculate the angular diameters of spheres, can we just use their diameters and pretend we’re looking at a circle straight-on? Actually, no we can’t in general. The blue dot is your eye. You are admiring a spherical statue of radius r. Its centre is at distance R from your eye where r ⩽ R as otherwise your eye is inside the statue. The angular diameter of the statue is θ. The height/length/width/diameter of the statue is of course 2r.
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r θ
R
chalkdust In the first diagram to the right, we see that r = R sin(θ/2).
θ 2
In the second diagram, we see that 2r/R is 2 tan(θ/2)—pretending that the diameter of our sphere determines the angular diameter directly, which leads to an angle that is slightly too small. The length of the bold segment of a circle of radius R in the third diagram is Rθ, and this is about the same as 2r. We may call 2r ‘h’ for height if we’re not talking about spheres. We are going to claim that when θ is small 2r/R is good enough to calculate θ, and that going the other way, Rθ is close enough to 2r. To take a concrete example, half a degree is 0.00873 radians to 3 significant figures. The sin and tan of this are also 0.00873 to 3 significant figures.
r R
θ 2
r R
θ
R
If I put a spherical statue on top of a mountain, the centre of the statue will get further away as the statue gets bigger. I shall also ignore this. I’m also, for now, not worrying about the Earth being approximately an oblate spheroid. A sufficiently small but very bright statue would send photons to just one cell on our retina. Would a sculptor be satisfied with a sculpture that is for all practical purposes a single pixel? I’m not an artist, so don’t really know, but it seems unsatisfactory. I can see that an artist might choose the colour of a single pixel with great care. The intensity and colour could even vary with time if we wanted something more than a static dot. However, let us suppose that this is not what Sciola had in mind. At this point we need to know something about human physiology. About how big should the smallest details of the sculpture be for people to be able to see those? It appears that the finest paern the human eye can distinguish is about 120 lines per degree. So single dots are half a minute of arc. (A minute is one-sixtieth of a degree. And a second is one-sixtieth of a minute. Look up the definition of a parsec!) At 32 km, half a minute of arc works out to be about 4.7 m. Let’s say 5 m. We could start to think in terms of pixel art with 5 m pixels but perhaps this approach is wrong. Is a 2 × 2 block of pixels art? Et cetera. We could instead consider existing statues or other items which are viewed from some distance away, and where more than their mere existence can be perceived, and ask ourselves what their angular diameters are when viewed from those distances. If these statues or other objects are approximately spherical, so much the beer. We can of course consider things such as the Angel of the North, Christ the Redeemer and so on but the moon seems to be a very good candidate. It’s approximately spherical, its angular diameter varies very lile for most people and is about half a degree, and ‘about the same size as the moon’ would not be an outrageous starting point for what we might want a statue to be. If we want some kind of ‘moon unit’ like this to use in comparisons, we could call it the Zappa. Half a degree, at 32173 m, means a diameter of about 281 m. Other astronomical candidates might be Venus, Jupiter or Mars, but their angular diameters vary quite a bit. Their maximum angular diameters are about one minute for the first two and about half a minute for the third. If you were a sculptor, would your statue looking the same size as one of these 41
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chalkdust planets be acceptable? The sun, famously, has about the same angular diameter as the moon but you aren’t supposed to look straight at the sun. Consider also eye tests. To be able to distinguish one leer from another, we need to be able to see not merely the leers themselves, but smaller parts of them. If you wanted to put advertising slogans on the moon, how long could they be? A sheet of A4 paper is about 30 cm tall. For the height of a sheet of A4 paper to subtend half a degree at our eye, we would need 2πR/720 = 0.3 m so R has to be 34.4 m. If you’re looking at something on a shelf in your room that’s, say, 3 m away, how big must it be to subtend half a degree at your eye? 2π × 3/720 = h. So h is about 2.6 cm. If we want to limit ourselves to a 100 m statue, we have 2πR/720 = 100 m so for it to look the same size as the moon, we would only need to be 11.5 km away. But looking at Google Maps, we’d still be inside the park surrounding the mountain. What should our statue be of? One might think that the patron saint of Catania, St Agatha, would be a candidate, or the elephant, the symbol of Catania. The problem with these is that the top of the mountain can be seen from other, closer, communities who might feel that this was unreasonable. And St Agatha, as usually depicted, is not approximately spherical. I submit, then, that a solution which should be acceptable to all is to build a statue of my stuffed hedgehog, Herisson. Herisson is round enough that I think we can treat him or her as approximately spherical. And so we have the Extremely Large Herisson Project (ELHP): we wish to erect a statue of Herisson on top of a mountain 3.3 km tall, 32 km away, so that it appears to be about the same size as the moon. Let’s say that we want Herisson’s largest dimension to subtend the same angle that the moon does. Herisson
As seen above, this works out to be about 281 m. This is of course a lot. However, looking for large statues we find that the Spring Temple Buddha in China is 128 m tall, on a 25 m base. Our proposal is not fantastically larger than the largest statue in the world. Alternatively, we could sele for the Extremely Large Herisson being a lile less than one Zappa in size, perhaps. There are non-mathematical considerations, however. Merely doing calculations with similar triangles ignores the problem of haze. When in Catania recently trying to take a photograph of the mountain, 5 days out of 7 it was behind haze, fog, cloud or similar. Being able to see anything at all at 32 km, especially when it is above common altitudes at which clouds are found, is problematic. One is also driven to wonder how planning permission for this project would work. On Wikipedia we find this image, which purports to show the town boundaries in the Catania area. The mountain peak is the point in the top right where mulchalkdustmagazine.com
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Town boundaries around the proposed location of the ELHP
chalkdust tiple sectors meet. It might be necessary to obtain planning permission from all the towns whose areas meet at the summit. Might it be necessary to get two lots of planning permission from the town whose area meets the summit twice? Of course, since much of the mountain is a national park, there could be further complications. I may ask at least one of the town councils about this project at some point. I have in fact wrien to one town council, that of Nicolosi, asking about this. At the time of writing, in late June 2018, I have had no reply. Building a 281 m statue on top of any 3329 m mountain would be a challenge, but in this case there are extra difficulties: The mountain under discussion is in fact Mount Etna, one of the most active volcanoes in the world. Google can find plenty of images of spectacular events on and around Etna. One experiment we can do is to view normal-sized Herisson at suitable distances to see if one Zappa does indeed seem about right, and also try to see if when his or her angular diameter is half a minute (the single pixel option) he or she is indeed visible at all. (He or she is white, so against many backgrounds might stand out enough to be seen.) A very long tape measure would of course be needed. The picture to the le shows me on a field trip to Catania, holding up a copy of Chalkdust with Etna in the background. Ideally we should have made sure the distance to the camera and the size of the magazine were such that it looked the same size as the moon would on the mountain behind me. This photo serves to illustrate a way we could decide what a decent size of statue would be. A stepladder would clearly have made it possible to hold the magazine so that it seemed to be on top of Etna, but one was not immediately to hand. In this shot I am only 25 km from the top of Etna, so 2π × 25000/720 = h. A mere 218 m statue would suffice. On a later field trip it seemed useful to examine the proposed construction site itself. On the le I am in front of the main crater Annamaria Cucinotta with a normal-sized copy of Chalkdust. The photo on the right has some people in it to give a sense of the scale of the place. The main thing I learned on this trip was that I never want to do it again. Making it all the way up to the crater is quite hard. Transporting construction materials for the sculpture would be a major undertaking.
Both photos by Gunther Schmidl
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chalkdust But why stop at Catania? The city of Messina is about 70 km away. And if we calculate the distance to the horizon from the top of Etna we discover that it’s about 206 km. 160 km gets us to Palermo. 203 km gets us to Catanzaro in southern Italy. Most of Sicily is within 206 km of the peak, but we can’t quite get to Trapani or Marsala on the west coast of Sicily. However, the sculpture would be so big by this point that its feet on the mountain would be invisible but some of the sculpture could still be seen. Still, I think we have to insist that the feet be visible: if we make a tall thin statue big enough of course the top can be seen from almost half the planet, and some part of a large enough spherical statue can be seen from almost anywhere on the planet (though the statue may need to be larger than the planet itself). Also, if your eyes are higher than sea level you can get some extra range from that. One photograph I have seen online of Etna from Catanzaro was taken from a hotel balcony. It seems possible that from a tall enough building in Trapani or Marsala, the top of Mount Etna might just be visible if there’s nothing in the way. Note that I am completely ignoring refraction here. In real life, refraction could make a difference, and as a wonderful article by David E H Jones (AKA Daedalus) in New Scientist observes, if we replaced the Earth’s atmosphere with sulphur dioxide or reduced the Earth’s radius we could in principle see as far as we wanted. But both of these options are well outside the scope of the ELHP. Puing 206 km instead of 32 km into our formulae, we get ‘pixels’ of about 30 m and statue size 1798 m. At this point we have a Prodigiously Large Herisson Project, and it seems quite impractical. And there’s the question of which way he or she should be facing. Which cities get to gaze upon Herisson’s gigantic backside? Gunther Schmidl (personal communication) suggests making the statue rotate about a vertical axis. This is clearly the way to go. Adam Atkinson Not only does Adam not know anything about art, he doesn’t even know what he likes. He’s one of the Brighton MathsJam organizers.
c ghira@mistral.co.uk My favourite function
Clarinet spectrum Carmen Cabrera Arnau
My favourite function is the frequency spectrum (intensity v frequencies) of a forte F3 played on a clarinet. As we can see, the intensities of the 1st, 3rd, 5th and 7th harmonics are strongly dominant. This is due to the closed pipe structure of a clarinet. The fact that their intensity is present in a similar proportion creates a very complex waveform that is responsible for the characteristic woody sound. chalkdustmagazine.com
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chalkdust
Have you been taking your weekly dose of chalkdust?
What makes a non-fairy cake fair?
What gets your heart beating?
Thinking of getting a tattoo?
Why is nature so mathematical?
Why knot?
Is there harmony in music and maths?
How do you calculate words?
Pub?
All these questions, and more, have been answered in our weekly online articles, covering everything from cosmology to sport, ďŹ&#x201A;uid dynamics to politics, as well as great puzzles and mediocre jokes. Read it every week and sign up to our monthly newsletter at
chalkdustmagazine.com Knot: Flickr user clickykbd, CC BY-NC-SA 2.0. Piano: Camera Eye Photography, CC BY 2.0. Beer: Neil916, CC BY 3.0.
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The Fields medals 2018
chalkdust
The winners and their work Albert Wood
M
has always been about pushing the boundaries of knowledge, but the 21st century has also looked inward, forging a synthesis of previously disparate subjects. Nowhere was this philosophy more apparent than in the 2018 Fields medal winners, several of whom have drawn from many different fields in their most revolutionary works. Here, we look at the work that won mathematics’ most famous prize.
Caucher Birkar Anyone who has messed about with graph-ploing soware has dabbled in algebraic geometry to some extent—for the subject is about algebraic curves, the name given to geometric objects that come from polynomial expressions. For example, a parabola comes from the expression y = x 2 . A major problem of algebraic geometry is classifying algebraic curves. If we use only two variables to form a polynomial equation, then the resulting curve in complex space will look like a 2-dimensional surface (as opposed to the 1-dimensional curve we get over the real numbers). The great mathematician Bernhard Riemann discovered that these surfaces will always look like doughnuts, but The Fields medal winners were anperhaps with more than one hole, or no holes at all. The nounced in Rio de Janeiro number of holes is called the genus of the surface; knowing the genus gives you a good idea of what the whole object looks like. Though the case of these ‘Riemann surfaces’ has long been dealt with, classifying higher-dimensional complex algebraic curves is a much harder problem. Over the 20th century, mathematicians aempted to simplify the situation by breaking down complicated curves into easier-to-understand pieces; this is known as the ‘minimal model program’. Although this worked in some low-dimensional cases, it was unsuccessful in general until, in 2010, Birkar proved the existence of minimal models for a swathe of algebraic curves, cracking the field wide open and providing geometers with many new tools.
Peter Scholze The German Fields medal winner Peter Scholze focuses on the applications of algebraic geometry to number theory, which is the branch of mathematics dedicated to studying the integers. Finding prime factors or lowest common multiples of numbers is a simple exercise of number theory. If geometry and number theory sound incompatible to you, consider the problem of finding Pythagorean triples—integers that satisfy x 2 + y 2 = z 2 . Though as stated this is a number theory problem, it is converted into geometry very easily. Dividing by z 2 , we get the equation p2 + q 2 = 1, where p = x/z and q = y/z. This is just the equation of the unit circle! So the problem is reduced to finding rational points on the unit circle—which can be done by drawing slopes with a rational gradient from the top of the circle. Scholze’s work continues to provide a geometric framework for solving number-theoretic problems, particularly those relating to curves known as ‘p-adic’. His work has built on the shoulders of chalkdustmagazine.com
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chalkdust the 20th century mathematical giants Grothendieck and Fontaine by creating an entirely new type of mathematical object, the fabulously named perfectoid spaces, which have helped algebraic geometers study p-adic curves in unforeseen ways.
Akshay Venkatesh Venkatesh’s work stitches together the theories of dynamical systems, topology and algebra to prove remarkable facts about number theory. For example, one of his most famous results regards polynomial expressions of degree 2 (examples are z 2 , or x 2 + 2xy + y 2 ; note each component of these expressions is a power of two, or two variables multiplied together). These are known as quadratic forms, and are the source of a long-standing open problem in mathematics—how can we tell whether a change of variables will transform one quadratic form into another? For example, the two expressions given above are in fact equivalent, as if we replace z with x + y, we get the same expression—we say that x 2 + 2xy + y 2 represents z 2 . This procedure can simplify a quadratic form; in this case it changes a form of two variables into a form with one! It was proven in 1978 that if P is a quadratic form of p variables, and Q is one of q variables, then Q represents P if q ⩾ 2p + 2, no maer what P and Q are. This has since been beaten—Akshay Venkatesh’s work exploits dynamical systems theory to prove that in fact, the same is true if q ⩾ p + 4 —an astonishing and unexpected improvement.
Alessio Figalli Alessio Figalli is an Italian mathematician, whose main interests lie in the theory of optimal transport, which starts by asking the question ‘what is the most efficient way to move a mass from point A to point B?’ For example, if there are several factories producing a product, and a number of locations where that product is to be delivered, what is the most efficient way (in terms of total distance travelled) to have the items delivered?
Some crystals look really cool
One of Figalli’s most impressive achievements is the application of this theory to understand the changing shape of crystals. Crystals try to maintain the most efficient shape possible with their current energy level—this is analogous to the way that a soap bubble aempts to minimise its surface area while enclosing a fixed amount of air, and so forms a sphere. If a crystal is heated, it will try to optimise its structure to match the new energy level, by changing shape. Figalli answered the question of how drastically this can happen by modelling the change in shape of the crystal as a transport problem, and showed that the molecules of the crystal move proportionally to the square root of the energy increase. This demonstrated that crystal shapes are stable, ie a huge change in shape could only result from a similarly huge change in energy. Albert Wood Albert Wood is a geometry PhD student at UCL currently working with geometric flows— physics-inspired motion of abstract shapes known as manifolds. He used to work as a teacher in west London, until he became jealous of his students and so went off to continue his own education.
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Topological tic-tac-toe Alexander Bolton
T
(also known as noughts and crosses) is a classic game known for its simplicity, and has been popular since ancient times. You and a friend (or enemy!) take turns to mark the squares of a 3×3 grid. The winner is the first to get three of their symbols in a row (horizontal, vertical or diagonal). A winning game for X is shown below right. It’s not difficult to work out how to play optimally on a standard board, where you’re guaranteed to at least draw with your opponent. However, what if you’re not restricted to the standard twodimensional square grid? How would you play then? To present a fresh challenge and to make tic-tac-toe exciting again, here is a collection of puzzles where you will be swapping your standard square board with one on various topological surfaces. Have fun!
× ⃝
×
× ⃝
⃝
×
A winning game for X
The cylinder The first new board to consider is the cylinder. To form a cylinder as in the diagram below, imagine that the board is wrapped around so that the le and right edges of the board are connected to each other, like a piece of paper that has been rolled into a tube. Matching edges are denoted with a ↑. 49
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chalkdust In all the subsequent puzzles, it is X’s turn to play, and it is possible for X to win in some number of moves, even if O plays optimally.
Puzzle 1: cylinders How does X win both of these games on cylindrical boards? It is possible to win the first game in one move.
⃝
×
×
⃝
×
⃝
The first game can be won in one move. A demonstration of this, along with the folded cylindrical board, is shown below.
The winning move for the first puzzle, shown in red. The diagram on the right shows the board folded into a cylinder, with the winning line marked in blue.
The Möbius strip How about a Möbius strip? Imagine that the right edge of the board is wrapped around the back of the board and given a half turn, so that it connects to the le edge of the board. The half turn means that top and boom become flipped as we move off the le or right edge of the board. The edges with a half turn are denoted by a ↑ and a ↓. (If you want to actually construct this board, draw each grid square as a wide rectangle so that the grid is wide enough to wrap around with a half twist.)
David Benbennick, CC BY-SA 3.0
A Möbius strip
The figure on the next page shows adjacent squares in the new board by making copies of the Möbius strip board, and puzzle 2 gives two puzzles on the Möbius strip board. chalkdustmagazine.com
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...
9 1 2 3 7 6 4 5 6 4 3 7 8 9 1
...
A figure showing which grid squares are adjacent on a Möbius strip board. For example, if you go one place right from the top-right square 3, you will go to the boom-le square 7.
Puzzle 2: Möbius strips How does X win both of these games on Möbius strip boards? It is possible to win the first game in one move.
×
×
×
⃝ ⃝
⃝
⃝
The torus There’s no need to limit ourselves to only connecting the le and right edges—we can also connect the top and boom edges. If we return to the cylinder formed by wrapping the right edge of the board round to meet the le edge, we can now connect the top and boom edges together to form a torus. Now we have two pairs of connected edges, denoted by → and ↑. For tic-tac-toe purposes, the cylindrical and toroidal boards are identical, and this holds even if we change the game to be ‘make a line of length n on an n × n board’ for any n. However, for puzzle 3, you need to find a line of length 3 on a 4 × 4 board.
Puzzle 3: torus puzzle What should X do to make three in a row on a torus?
⃝ ⃝
×
×⃝
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chalkdust On the torus, we can think of any row (or column) as being the central one, so it’s easier to prove facts about the torus than for other boards. Have a go at the following challenges.
Puzzle 4: more torus puzzles Can you show that making any line of length n on an n × n cylindrical board is also a line of length n on an n × n toroidal board and vice versa (so a game on a torus is equivalent to a game on a cylinder)? Are any starting positions beer than others on a 3 × 3 torus?
Is it possible for the game on a 3 × 3 toroidal board to end in a draw, with neither player geing 3 in a row (the previous question gives you a shortcut to solving this one)?
...
The Klein bottle Now we will think about playing on a Klein bole, a shape that cannot be constructed in three dimensions without it intersecting with itself. Fold the top edge of the board over to touch the boom edge, and connect the le and right edges with a half twist like for the Möbius strip. The figure to the right shows which squares are adjacent on the Klein bole board, and puzzle 5 gives two puzzles.
...
3 9 6 3 9
7 1 4 7 1
8 2 5 8 2 ..
9 3 6 9 3
1 7 4 1 7
...
Puzzle 5: Klein. bottle puzzles How does X win both of these games on Klein bole boards? It is possible to win the first game in one move.
× ⃝
⃝
× ⃝
The projective plane Another shape that it is not possible to construct in three dimensions without the shape intersecting itself is the projective plane. This is a shape where the top and boom edges are connected by a half twist, as are the le and right edges. To imagine how it would be made, think about connecting the single edge of the Möbius strip to itself. (You’ll have to be very dextrous to actually chalkdustmagazine.com
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chalkdust create this from paper!) Puzzle 6 gives two puzzles using the projective plane. Top tip: construct an adjacency map like the one for the Klein bole.
Puzzle 6: projective plane puzzles How does X win both of these games on projective plane boards?
⃝
×
× ⃝
⃝
That brings us to the end of our foray into topological tic-tac-toe. I hope you enjoyed these mindbending puzzles! The inspiration for this article came from Across The Board by John Watkins, the most complete book on chessboard and other grid puzzles. I would recommend this book for some further interesting puzzles, eg how many queens are needed so that every square on a chessboard is targeted or occupied by one of the queens? The author gives an interesting history of chess problems and shows that they have inspired important advances in maths. Alexander Bolton Alexander Bolton completed a maths PhD at Imperial College London, applying Bayesian statistics to cyber security problems. He now works as a quantitative analyst at G-Research.
My favourite function
The error function Hugo Castillo Sánchez
The error function erf(η) and the complementary error function erfc(η) = 1 − erf(η) are special nonelementary functions of great importance in probability, statistics, partial differential equations and many branches of physics.
2 erf(η) = √ π
∫
η 0
2
e−ξ dξ
1.0
I like this function because it solves many problems in fluid mechanics, heat and mass transfer. -2 -1 1 2 For instance, in time t- and position x-dependent -0.5 problems (such as diffusion-convection), the velocity v(x, t), temperature T(x, t) and concentration -1.0 C(x, t) profiles adopt the form of the √ complementary error function erfc(η) = erfc(x/a t) (where a is a physical constant), which provide extremely useful information in process and industrial equipment design. 0.5
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1
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#8 Set by Humbug
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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 2 February 2019. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 16 February 2019. One randomly-selected correct answer will win a £100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Nerd DVD, a dodecaplex puzzle 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. Find out more at d mathsgear.co.uk
My least favourite function
The vertical line Belgin Seymenoğlu
I have liked ploing functions since I was twelve, when I got my first graphing calculator. But one function always spoiled the fun: I could never plot a vertical line, ie x = constant, because there is no way to express it as an explicit function, ie y = stuff depending on x only. If that’s not enough, its derivative is infinite! How are you supposed to do calculus on this function⁈ chalkdustmagazine.com
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chalkdust
Across
Down 1 The first two digits of 11D. 2 A palindrome. 3 A prime number that is 16 more than 3 times 4D. 4 A prime number that is 16 more than 3 times 22D. 5 A prime number that is 16 more than 3 times 6D. 6 A prime number. 7 Greater than 1A. 8 A prime number that is 16 more than 3 times 26D. 9 A prime number that is 16 more than 3 times 36A. 11 Less than 21D. 14 The sum of this numberâ&#x20AC;&#x2122;s factors* is three times this number. 16 The sum of this numberâ&#x20AC;&#x2122;s digits is 18A. 17 This number and 22A are coprime. 20 A prime number that is 16 more than 3 times 3D. 21 A factor of 3A. 22 A prime number that is 16 more than 3 times 8D. 24 A multiple of 27A. 25 A palindrome. 26 A prime number that is 16 more than 3 times 23A. 28 The largest prime factor of one less than the square of this number is 7. 29 The sum of two consecutive square numbers. Also the sum of three consecutive square numbers. 31 The largest prime factor of one less than the square of this number is 5. 33 A multiple of 30A. 35 An odd number that is not prime.
1 The largest prime factor of one less (2) than the square of this number is 3. 3 The sum of this numberâ&#x20AC;&#x2122;s digits is (8) 18A. 10 Every digit of this number is odd.
(5)
12 The sum of 2D, 25D and 19A.
(5)
13 A multiple of 1D whose first two dig- (7) its are the same. 15 Less than 1A.
(2)
16 A palindrome.
(7)
17 20 more than 17D.
(3)
18 A multiple of 3.
(2)
19 A palindrome.
(3)
20 This number and 17D are coprime.
(4)
21 A prime number of the form nn + 1 (3) for some integer n. 22 This number and 20A are coprime.
(3)
23 A prime number that is 16 more than (4) 3 times 5D. 25 A multiple of 10.
(3)
26 Less than 1A.
(2)
27 A factor of 24D.
(3)
28 A seven-digit number.
(7)
30 The highest common factor of 17A (2) and 20A. 31 A multiple of 32A.
(7)
32 The first digit of this number is the (5) first digit of 33D. 34 A number k such that k Ă&#x2014; 2n + 1 is not (5) prime for any integer n > 0. 36 A prime number that is 16 more than (8) 3 times 20D. 37 Not a palindrome.
(2)
(2) (5) (7) (6) (4) (3) (2) (5) (8) (3) (3) (8) (3) (7) (3) (6) (5) (3) (5) (4) (3)
(3) (2) (2)
*Factors including 1 and the number itself.
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.. a Mobius surprise You will need A3 paper, scissors, glue
Instructions
1
Cut your A3 paper into 4 strips.
Take two strips and glue them perpendicularly to make a cross.
3
Make a Mรถbius strip with each of the perpendicular strips.
Cut along the centre line of one Mรถbius strip until you get to where you started. Then cut along the other.
5
2
4
Be surprised.
Tube map platonic solids, Frรถbel stars and slide rules: more How to make at d chalkdustmagazine.com chalkdustmagazine.com
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chalkdust
Gerolamo Cardano Emma Bell
T
time of the Renaissance resonates with the names of individuals who have made an impact in the development of mathematics. One name, however, echoes more loudly than the others: Gerolamo Cardano.
We know so much about Cardano’s life and personal opinions because of his candid and forthright account in De Vita Propria Liber (The Book of My Life). The memoir gives us unparalleled insight into the mind of this polymath as well as detailed information regarding his accomplishments during his lifetime. An analysis of the chapter “Things of worth which I have achieved” certainly gives a great starting point for research. Gerolamo Cardano was born in what would later become Northern Italy in September 1501. It was a lively period in the development of Europe and the world: the time of the Borgias, Copernicus, Michelangelo and Christopher Columbus. Cardano was the illegitimate son of Fazio, a lawyer, and Chiara Micheri, who suffered terribly during a three-day labour before 57
Wellcome Images, CC BY-SA 4.0
Portrait of Gerolamo Cardano autumn 2018
chalkdust the boy was born. The baby Cardano had to be “revived in a bath of warm wine” shortly aer birth—the whole story of the incident laid bare in the second chapter of Cardano’s autobiography. Cardano aributes his survival to the alignment of the planets, describing his birth horoscope in intricate detail. Fazio employed his son as a page for much of the boy’s childhood and ensured that Cardano had a good grounding in mathematics. Fazio was an accomplished mathematical thinker who had been cited in Codex Atlanticus, a collection of papers and drawings produced by Leonardo da Vinci between 1478 and 1519. Cardano talks of his education from his father, which began with basic arithmetic and moved on to geometry:
Aer I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself. Cardano studied at the University of Pavia, which had been founded in 1361, a school for philosophical and legal thinking. He transferred to the University of Milan when Pavia’s establishment was forced to close due to sieges from the French. It was there that he graduated as a physician. Cardano spent time in prison, during the prominence of the Inquisition, for casting the horoscope of Jesus. Horoscopes were held in high regard by Cardano, who explains the alignment of stars and planets at many momentous points in his life. This obsession meant that Cardano also used the zodiac to foretell the exact date of his death. He was correct in his prediction, although it is hypothesised that he took maers into his own hands to ensure that his forecast came to pass. Giovanni Dall’Orto
Thanks to the memoir, we know some very personal details of Cardano’s life. We know what he looked like, what he enjoyed eating, the disappointment that he had in his sons, and with surprising intimacy, the fact that he was impotent for much of his young adulthood. Cardano dedicates a whole chapter to “a meditation of the perpetuation of my name”—he was determined to be remembered. This column will focus in on three areas of his mathematical legacy. The University of Pavia
Algebra In 16th century Italy, a popular activity of the top thinkers of the time was to take part in equation solving competitions. Public contests would be held where those competing would challenge each other to solve increasingly difficult equations. The mathematicians were highly secretive, not wanting to give away their methods of working for fear of giving others an advantage. The solving of cubic equations was classed as an ultimate challenge. One mathematician, Niccolò Fontana (also known as Tartaglia the Stammerer) figured out an algorithm for solving third order polynomials, winning many competitions where the other competitors could only find approximate chalkdustmagazine.com
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chalkdust solutions. Cardano, somehow, managed to convince Tartaglia to tell him his method, and promised not to share it. Cardano kept his word and did not tell of Tartaglia’s method. However, in 1543 on a visit to Bologna University, Cardano and his student Ferrari met with the son-in-law of Scipione del Ferro, Hannival Nave. Del Ferro, who had died in 1526, was the lecturer for arithmetic and geometry at Bologna University, and Nave had taken over the role and inherited his father-in-law’s notebooks. It was in these notebooks that Ferrari and Cardano found a version of Tartaglia’s method which predated the Stammerer’s work. With this discovery, Cardano felt it was fair to publish the technique himself. ‘Cardano’s Rule’ is still used today to solve cubic polynomials. It was published in Cardano’s Ars Magna in 1545 along with the work of Ferrari, which reduced quartics down to cubics. This meant that, for the first time, equations which had no basis in the ‘real world’ could be solved. Ars Magna is an exceptional legacy. The work on cubics and quartics stimulated further research by others: abstract algebra had been born. This legacy came at a price however. Tartaglia was very unhappy with the publication and the pair were locked in an acrimonious dispute for decades.
Imaginary numbers Ars Magna, and the solving of cubics and quadratics lead to another ‘discovery’ – that of imaginary numbers. Cardano posed this problem in his book: “divide 10 into 2 parts so that that product of those parts is 30 or 40”. Cardano states that this problem is seemingly impossible, but then shows how he can solve it with a bit of imagination…
Cardano archive
Or,
5+ 5−
√
√
−15 −15
25 − −15 = 40. Cardano did not take this discovery further, only using it as a “folly”, but it appears to be the first time that the square root of a negative number is used in such a way. This use of an imaginary number was eventually given prominence by Leonhard Euler and Carl Friedrich Gauss in the eighteenth century. Complex numbers, the mix of imaginary and real numbers, have applications today in areas such as fluid mechanics and electrical engineering. 59
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Probability A century before the correspondence between Pascal and Fermat, considered by many to be the origin of probability theory, Cardano wrote Liber de Ludo Aleae (The Book on Games of Chance) in which he calculated the different possibilities and outcomes of the gambling games of which he was so fond. He introduced the notion that the probability of an event could be regarded as a value between 0 and 1. Cardano tells us, “I was inordinately addicted to the chessboard and the dicing table” in his autobiography, partaking in some form of being every day, which is even more enlightening when we discover that one of the chapters in the book is dedicated to methods of cheating! Unfortunately, although the work was completed in 1564, it was not published until 1663. As a result, it did not have the impact of Pascal’s and Fermat’s leers. Five of the leers between the pair covered the same games of chance that Cardano had examined in his book. Cardano also investigated a way to distribute the winnings of a game of chance if the game was interrupted and not able to be completed—known as the ‘problem of points’. His answer was based on the points already won, while Pascal and Fermat based their solution on the probability that each would win if the game had properly concluded.
Dice players in the 16th century
Of all his accomplishments, Cardano seems most proud of the reputation that he forged for himself. Parts of his autobiography are solely concerned with the “testimony of illustrious men”, which gives numerous quotations about Cardano’s character, as well as an extensive list of other publications which have cited his work. Two passages stand out:
Julius Caesar Scaliger ascribed more titles to me than I should have thought of arrogating to myself, calling me ‘ingenium profundissimum, felicissimum et incomparabile’ [A man of most profound, most favoured, and incomparable genius] Of problems solved or investigated I shall leave something like forty thousand, and of minutiae two hundred thousand, and for this that great light of our country used to call me ‘the man of discoveries’. Cardano was determined, throughout his life, to be discovered himself. It seems appropriate to now rediscover his legacy, and to herald his achievements for all to hear. Emma Bell Emma is the maths enhancement manager at Grimsby Institute of Further and Higher Education.
c belle@grimsby.ac.uk a @El_Timbre Emma Bell
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All the drama happening in your favourite soaps this season
SOAP UPDATE COMBINA
e l a d a m Lem
T I ON S T .
Norris moves to Permutation Street to get his accounts in order.
A dilemma for one of the Dingles when they miscount the number of fenceposts required to build a fence.
∇×E Watts runs continuously in circles around the Red Rec.
Jack & Vera think about stats on the way back from the Ellipse and invent the Duckworth-DuckworthDuckworth-Lewis method.
Another Dingle realises a fox has attacked when they count more pigeonholes than pigeons.
Galois challenges Les Dennis to a duel over Gail Platt’s affections. Our survey said… eh-uhh.
The Dingles realise they’re the only ones anyone’s heard of. Galois gets run over by a tractor.
DecreasedEnders
s r u o b h g i e N N e ar e s t
Dot Cotton gets angry and becomes Cross Cotton.
Galois gets eaten by a shark. What a drongo.
Ian’s family start singing together, forming a Bealean group.
Strewth! Karl Kennedy has a barbie and invites six neighbours. He realises either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances, thereby inventing Ramsay Street theory.
Albert Square realises he’s a road, not a character. Galois stays up all night and is shot on the stairs. But by whom? Dun. Dun. Dun dun dun d-d-d-d-
Images: Farm, David Wright CC BY-SA 2.0. Coronation Street, Lewis Clarke, CC BY-SA 2.0.
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he t
To o g Tr to b ood uc e
chalkdust
Colin Beveridge
I
is altogether too hot, it is altogether too full of people, and it is altogether too lunchtime for what feels like 8pm. Fortunately, the organisers have fed mathematicians before, and have thoughtfully provided plain paper tablecloths and pens with which to postulate, puzzle and prove while we eat. We’re in Atlanta, Georgia for the 13th Gathering 4 Gardner, or G4G: every couple of years, mathematicians, magicians, sceptics, jugglers and assorted others gather to honour the work and memory of popular science writer Martin Gardner with a week-long conference. An idea from an earlier talk had lodged in my head. Cindy Lawrence of MoMath—New York’s Museum of Mathematics, where one can ride a square-wheeled tricycle or explore the inside of a Möbius strip, the kind of thing that Gardner would certainly have wrien about—had raved about Truchet tiles. What are they? Well, start with a square, coloured either black or white. Pick two diagonallyopposite corners and shade them with a quarter-circle of the other colour, as shown. Make several. Then place them however appeals to you! The way they’re set up, it’s practically impossible not to start making paerns: blobs and whorls that seem almost alive. Of course, you don’t need the tiles themselves. You can just as easily doodle them on a convenient sheet of paper, such as a tablecloth. And you can just as easily start asking yourself questions like, what kinds of tiles can you get if you remove the restriction of squareness? What happens if you chalkdustmagazine.com
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chalkdust move into three dimensions? And is there any prey maths underlying the prey paerns?
Sébastian Truchet and his tiles The tiles Lawrence talked about are not, strictly speaking, due to Truchet. In 1704, Sébastien Truchet published A Memoir On Combinations, in which he discusses squares split diagonally into triangles (pictured right), giving four possible orientations for each tile. It’s an interesting read. Truchet methodically looks at the number of ways you can place two such tiles next to each other, edge-toedge. (He says that he’s started work on three tiles, but isn’t happy with it yet. I’ve seen nothing to make me believe he ever was happy with it.) In the paper, Truchet carefully reasons that there are:
B A
D C Orientations A, B, C and D
• four possible orientations for a single tile; • four positions to place a second tile next to a first (north, south, east, or west); and • four orientations for each of the second tiles … making a total of 64 possible arrangements. He then notes that some of the arrangements are indistinguishable from others: placing a tile in orientation A to the le of a tile in orientation B is the same as placing B to the right of A, reducing the number to 32. Furthermore, some arrangements are rotations of others—for example, arrangement AA (pictured below) is a rotation of arrangement CC (not pictured below unless you’re reading Chalkdust upside down). In all, he reduces the 64 original possibilities to 10 (six appear in eight configurations each, and four—those with ‘stripes’ across the middle—appear four times apiece). It turns out to be worthwhile to consider the overlying structure of these arrangements. Of the ten possibilities, eight appear in pairs: swapping the colours of one gives the other. The other two are self-inverse: swapping the colours gives a rotation of the original arrangement. And that’s where Truchet tiles remained until the 1980s, when CS Smith, writing in Leonardo, took a deeper dive into the topology of Truchet tiles.
Arrangement AA
Arrangement BB
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Arrangement AB
Arrangement CD autumn 2018
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Arrangement AC
Arrangement AD
Arrangement BD
Arrangement CA
Arrangement CB
Arrangement DB
Among other things, he suggests several possible changes to the design—for example, removing colour from the equation altogether and simply tiling with diagonal lines, or—rather nicely—by a pair of arcs in opposite corners. This is, in terms of symmetries, just the same as a diagonal, but placing large numbers of them together makes for much more appealing paerns—which you should totally play around with, but aer you’ve finished reading this article. Smith also suggests extending these tiles to include colours: if you make the cut-off corners one colour, and the remaining strip of the tile the other, you have exactly the blobby-cornered tiles MoMath brought to G4G. But why limit yourself to squares?
Extending tiles to 2n-gons Don’t get me wrong, I have nothing against squares. They’re certainly in my top ten favourite shapes. Just… even if the tessellation paerns are neat, the configurations of the tiles themselves are not all that interesting. When you go beyond four sides using blobby corners and two colours, though, fascinating things begin to emerge, without even having to put the tiles next to each other. (From here on, you can assume that all tiles are blobby-cornered and two-coloured.) Indeed, any regular polygon with an even number of sides can be turned into blobby Truchet tiles (although hexagons are the only ones that would tile the plane alone). Odd numbers of sides don’t work with this kind of colouring, because the vertices need to alternate between two colours. How many hexagonal tiles are there? Well, it depends how you count. I choose to count rotations of the tile as the same tile—so there exactly two square tiles, one with two isolated white corners and one with two isolated black corners. With hexagons, it’s simple enough to do the counting: let’s start by considering the three white corners and how they connect (or don’t connect) to each other. Either each corner is on its own, all three are connected, or a pair of corners is connected and the other isolated. (Note that if any member of a group of corners connects to another corner, all members of that group must chalkdustmagazine.com
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chalkdust connect—we can’t have a case where the first corner connects to the second and the second to the third unless the first and third are also connected.) These correspond to the figures to the right—three isolated white corners and three interconnected white corners, followed by—more interestingly - an isolated white corner, followed by a black band that ends halfway across the hexagon, followed by a white band and an isolated black corner. I’ve paired them like this for a very good reason: inverting the colours on the black dots tile gives the white dots, and vice versa; inverting the colours on the split hexagon gives… a rotation of itself! I count the hexagons as having three possible tiles, one of which is self-inverse.
White dots
Split hexagon
Black dots
What possible patterns are there for an octagon? Obviously, octagons don’t tile the plane on their own (although there’s nothing to stop you filling in the gaps with square tiles!) Independent of how we’re going to arrange them, we can still consider the viable paerns. Again, it’s good to start by considering the four white corners and how they connect. When none of them connect, we have another white dots pattern; when all four connect, we have a corresponding black dots paern. These two are inverses.
White dots
Black pants
Black stripe
If two neighbouring corners are joined, and the other two le unaached, the resulting shape looks like a pair of black pants. Its inverse pattern, the pair of white pants, arises from any three white vertices being connected.
Black dots
White pants
White stripe
We could also connect a pair of diagonallyopposite corners to give a white stripe; the black stripe comes from connecting one pair of adjacent vertices, then connecting the remaining pair. The octagons therefore have six possible tiles, none of which is self-inverse. (Or do they? Have I counted correctly? How do you know?) Decagons are where (for me) it gets interesting: but perhaps you want to try working out how many decagon tiles there are yourself first. Once you’ve had a go, turn the page… 65
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White dots
White link
White bikini
Blackpool tower Stripes
Black dots
Black link
Black bikini
Pair of pairs of pants
Whitepool tower
There are ten decagon tilings, up to rotation—of which two are self-inverse. That’s a structure we’ve seen before: it’s the same as the structure behind Truchet’s pairs of adjacent tiles. Don’t you think that’s neat? Well, hold my coffee.
Truchet tiles in three dimensions The following day, I pick a table with the MoMath people, who have a set of the square blobby tiles out to play with, black with white corners one side, white with black corners on the other. “How many ways can you make a cube?” asks Tom. We pick up the tiles and try to arrange them, one per face. Let’s adopt the reasonable rule that the corners of each tile that meet a vertex must be the same colour. Each face then has two diagonally-opposite black vertices, and two diagonally-opposite white vertices, and there are only two ways to place a blobby tile to satisfy that: either the black vertices are joined by the stripe, or the white ones are. We can therefore think of each face as either black (if its black vertices are connected), or white (if the white pair is joined). The puzzle reduces to finding how many ways there are to colour the faces of a cube using only two colours. If all six faces are white, there’s only one possibility. The same goes for no white faces. If five are white, there is again only one possibility (up to rotational symmetry); this is also true for one white face.
All white
One black
Adjacent black Opposite black
With four white faces, there are two possibilities: either the black All black One white Adjacent white Opposite white squares are on opposite sides, or they are on adjacent sides. As you might expect, the same goes for two white faces. chalkdustmagazine.com
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chalkdust With three white faces, there are again two possibilities: either the three faces meet at a vertex, or they wrap around like a tennis ball.
Mixed corner
Mixed tennis ball
Now, simply listing the possibilities isn’t all that interesting. However, the structure behind the list is: of the ten arrangements, eight have a natural colour inverse, and the other two are self-inverse switching the colours gives you a rotation of the same cube.
That’s precisely the same as the structure of the decagonal tile arrangements—and Truchet’s original pairs.
Coincidence? It’s not clear to me whether this repeated structure is a coincidence, or some deep property of Truchet tiles. It’s not unnatural for a ten-element set to have that same structure—two elements that are their own inverses, and eight that form inverse-pairs. Indeed, if you consider the group of integers from 0 to 9 under addition modulo 10, you also get that structure (0 and 5 are self-inverse). However, to consider the tiles or cubes as a group, we’d need a way to combine them in pairs, and if there’s an simple operator for any of the Truchet sets, it’s not obvious to me. So, I’m opening it up to you, knowing that the readership of Chalkdust has, collectively, far more insight than I do: is this common structure a coincidence, or something deeper? While you’re working it out, draw some Truchet tiles of your own. You’ll be glad you did. See you later, tessellator! Colin Beveridge Colin is the author of Cracking Mathematics and The Maths Behind, wrien to prove that he has nothing to prove (by contradiction).
a @icecolbeveridge d colinbeveridge.co.uk My favourite function
The empty function Sam Porritt
∅:∅→∅
Mildly amusing to set theorists but about as useful as an ash tray on a motorcycle.
Did you know... …that sixteen has seven leers; seven has five leers; five has four leers; four has four leers… No maer what number you start with, you will always end up at four. 67
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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.
Hey Chalkdust what about mathematical nails? Want to inspire some new nail equations? :)
Fabián Silva Grifé,
Locky,
Chalkdust flies Buddha Air.
Feels somehow wrong to use Chalkdust as a ruler for working on a blackboard.
a @Locky17
>eek< >eek< >eek< >eek< >eek< it is four ante meridiem and my rm3 has failed to go to bed >eek< >eek< >eek< I blame Chalkdust >eek< Zeke, Cardiff a @RealityMinus3
A bit tricky this [crossnumber]. Only started it last night, but finished in time to submit it today. PRL, Berkshire a @UsrBinPRL
chalkdustmagazine.com
a @afsg77
Shambolic Librarian,
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a @BucksLibrarian1
chalkdust
On the cover
Hydrogen orbitals
Tom Rivlin
Q
mechanics has a reputation. It’s notorious for being obtuse, difficult, confusing, and unintuitive. That reputation is… entirely deserved. I work on quantum systems full time for my job and I feel like I’ve barely scratched the surface of the mysteries it contains. But one other feature of quantum mechanics that’s oen overlooked is how beautiful it can be. So, for the cover of this issue, I wanted to share one aspect of quantum mechanics that I think is stunning. It’s a certain set of solutions to a differential equation: the orbitals of an electron in a hydrogen atom. In school, you’re taught that electrons orbit the nucleus of an atom like a planet orbiting a star. This is mostly wrong. The main problem is that electrons, protons and neutrons aren’t lile billiard balls, they exist as ‘clouds’ of probability. To understand what a hydrogen atom really looks like, imagine a cloud of something whizzing around a single proton. The proton’s positive charge aracts and traps the negatively-charged something in what we call the proton’s potential well. Imagine that cloud is denser in some places and sparser in others. That cloud of something can be just one electron whose position has been smeared out. The density of the cloud at a point represents the probability of finding the electron at that point in space. The electron’s position may be smeared out over all space, but it has different odds of being found at different points in space. In fact, it’s usually exponentially less likely to be found outside the small, confined volume of the potential well. 69
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chalkdust The mathematical explanation for this is that our system is obeying the Schrödinger equation. For our case, it looks like this: ( −h̄ 2 2m
) ∇2 + V(r ) ψ(r ) = Eψ(r ).
The Schrödinger equation is the foundational equation of quantum mechanics. It’s used to determine the wavefunction, ψ(r ), and the energy, E, of the components of the system. In this case the wavefunction represents the electron (with mass m) trapped in the electric potential well of the proton, which is represented by V(r ). The reduced Planck constant, h̄, (oen called “h-bar”) is a fundamental physical constant, and ∇2 is the Laplacian operator, which sums second derivatives over all the coordinates. The modulus squared of the wavefunction, |ψ(r )|2 , tells you what the density of that probability cloud is like: where are you more likely to find the electron? Most people who do quantum mechanics for a living spend their time solving this equation and its variants, myself included. The problem is that this is really, really hard. The Schrödinger equation for a hydrogen atom has analytic solutions you can write down, but with almost all other physical systems, you aren’t so lucky. Once you have more than one electron, the complexity skyrockets. Understanding the analytic solutions form an important part of a physics undergraduate’s introduction to quantum mechanics, especially in my field of research. I work on finding approximate solutions to the Schrödinger equation for more complex systems. To solve the Schrödinger equation, you can separate the wavefunction to get a radial part which is a function of the distance from the nucleus, r, and an angular part which is a function of the angles (θ, φ). Both parts have multiple solutions, and it turns out that you need three labels to identify these solutions. We call these labels quantum numbers. Here, the three are called n, l, and m. Puing these two concepts together, we can say: Eψnlm (r ) = Rn (r)Ylm (θ, φ). There’s lots of constraints on the allowed values of n, l, and m, but the most important one is that each number take whole number values only. This is where the ‘quantum’ in quantum mechanics comes from! The quantum numbers each have physical interpretations: they loosely correspond to the three spatial coordinates. Here, n corresponds to energy. Higher values of n mean the electron has a larger amount of energy, which, due to how electric fields work, also exactly corresponds to a larger distance chalkdustmagazine.com
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Plots of the solution with n = 0, l = 5 and m = 0 (top) to m = 5 (boom)
chalkdust from the nucleus. That means n is associated with the radial coordinate: the higher n is, the further from the nucleus the electron can be. Meanwhile, l and m correspond to angular momentum, and so they are associated with the angular coordinates. Roughly speaking, higher values of l correspond to the electron ‘orbiting’ around the nucleus with greater energy (in a weird, quantum mechanical way that doesn’t really look like a planet orbiting a star). Changing m means changing exactly how it orbits for a given value of l. What this all means in practice is that by varying the three quantum numbers you get a huge variety of electron distributions. For instance, n = 1, l = 0, m = 0 means that the electron isn’t orbiting the nucleus at all, so it’s most likely to be found right on top of the nucleus – opposite charges aract! When n, l, and m are all large you get things like concentric sets of lobes of varying shapes and sizes. Bringing it back to the cover, the pictures were all generated by making a 2D slice through the full 3D distribution at y = 0. The brighter a given point is shaded, the higher the value of |ψ(r )|2 is there—the higher the odds of finding the electron there are. The full 3D versions look like spheres, balloons, lobes, and other wild shapes. The 2D slices have a different sort of haunting beauty to them. The distributions can be concentric rings, orange slices, weird lobes, insect-like segments, and more. The front cover is the 2D slice of the solution for n = 9, l = 4, m = 1. The back cover contains all the allowed solutions from n = 1, l = 0, m = 0 up to n = 9, l = 7, m = 7. These orbitals are beautiful by themselves as pieces of abstract maths, but they also provide profound insights into the strange quantum nature of our reality. They’re a testament to the amazing power physics and mathematics can have when they work together to help us understand our universe. Tom Rivlin Tom is a PhD student in the UCL physics department, simulating atomic collisions. He likes to think that what he does ‘technically counts as maths’.
a @TomRivlin My favourite function
Dirac delta function Atheeta Ching
This function, denoted δ(x), is zero everywhere apart from at the point 0 and it satisfies the following constraint: ∫ ∞ δ(x) dx = 1 −∞
Loosely, this means the function is equal to ‘infinity’ at the point 0. This function appears quite oen in measure theory but also in modelling impulses. 71
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This issue features the top ten units of measurement. To vote on the top ten Chalkdust regulars go to d chalkdustmagazine.com At 9, and selling one tenth of the number of copies that number 6 sold: it’s a millimetre.
At 10, it’s the Zappa (see pages 38–44).
At 8, and not receiving much radio play due to being far longer than the rest of the top ten: it’s a furlong.
Following warm reviews from critics, degrees Celsius enters the top ten at 7.
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The new single by no-one’s favourite rapper 50 Centimetre is at 6.
Following the release of its 51st anniversary deluxe edition, The Velvet Underground and Picometre is at 4.
Forced up two places from last issue, it’s the Newton.
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At 3, and selling 273.15 more copies than this issue’s number 7: it’s Kelvin Harris.
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Ra =
Pictures Kelvin thermometer: Wikimedia commons user Martinvl, CC BY-SA 3.0.
chalkdustmagazine.com
At 5, it’s the Yardbirds tribute act whose members are all 8.5 cm taller than the originals: the Metrebirds.
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L l
Topping the pops this issue, it’s dimensionless constants.