In this issue... Features 4
In conversation with Bernard Silverman
We chat to Bernard about maths in Westminster
10 Linear algebra with diagrams Paweł Sobociński gets graphical
28 Variations on Fermat
Robert Low and Thierry Platini explain a marvellous idea that this margin is too small to contain
34 Slide rules
Bernd Ulmann shows us the slide rule concealed in his pocket
42 The simplest difficult task Wojtek Wawrów sees how simple ideas lead to hard maths
48 Origami tesseracts
Debugging insect dynamics
L ulu Beatson folds some paper
50
Atheeta Ching gets stung by a bee
20
Regulars
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3 9 18 32
Page 3 model What's hot and what's not Dear Dirichlet Chalkdust comic by Tom Hockenhull
39 How to make... ...a slide rule
40 Which object are you? 50 Significant figures Florence Nightingale
55 On the cover 58 Prize crossnumber 61 Roots: Mary Somerville by Emma Bell
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64 Top ten: parts of a circle 1
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Editorial Director Rafael Prieto Curiel The team Rob Becke Hugo Castillo Sánchez Atheeta Ching Thuy Duong “TD” Dang Alex Doak Aryan Ghobadi Nikoleta Kalaydzhieva Antigoni Kleanthous Rudolf Kohulák Anna Lambert Tom Rivlin Mahew Scroggs Pietro Servini Belgin Seymenoğlu Yiannis Simillides Adam Townsend Cartoonist Tom Hockenhull chalkdustmagazine.com contact@chalkdustmagazine.com @chalkdustmag chalkdustmag chalkdustmag Chalkdust Magazine, Department of Mathematics, UCL, Gower Street, London WC1E 6BT, UK.
Even when working in mathematics, it has been impossible to avoid noticing what everyone has been talking about lately: tight election results, possibly unexpected outcomes and politics. But what can mathematicians offer to this global debate? In this issue, you can read our interview with Bernard Silverman, chief scientific advisor to the Home Office. He gives us an insight into the way the work of scientists and mathematicians feeds into politics and the difficulties of combining the human world of government with the rigorous world of science. Flicking further through the pages of our fih issue, you will also find two remarkable stories. Firstly, that of Florence Nightingale, who insisted on the use of better quantitative tools to make beer decisions and improve policies. Secondly, Mary Somerville, an activist, and one of the first female members of the Royal Astronomical Society, who demanded women’s suffrage. Both Nightingale and Somerville le their mark, not only in science but also in the political arena. I hope that you will enjoy the selection of articles that we have picked for our fih issue. Inside, you will rediscover linear algebra using diagrams, be tempted into buying a slide rule (or making your own), and learn about four-dimensional space using origami. The last of these was wrien by one of our youngest ever authors, an A-level student named Lulu, showing us that there are no age barriers when it comes to enjoying mathematics. Rafael Prieto Curiel Editorial Director
Acknowledgements We would like to thank our sponsors for their support, and for helping us keep Chalkdust free. Special thanks go to Robb McDonald and Luciano Rila, plus all the staff at the UCL Department of Mathematics. We would also like to give a special mention to Sam Brown for writing our LATEX templates: without your work, Chalkdust wouldn’t look anywhere near as good. We would also like to thank David Colquhoun and Hannah Fry, among many others, for their support. This issue would not exist without its excellent authors, who continue to send us top quality articles. Most of all, we would like to thank you, our readers, for your continued enthusiasm. We now acknowledge that you have 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 2017. All rights reserved. If you wish to reproduce any content, please contact us at Chalkdust Magazine, Dept of Mathematics, UCL, Gower Street, London WC1E 6BT, UK or email contact@chalkdustmagazine.com
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Bread is a staple of many diets. From delicious garlic bread to crunchy pizza, it’s enjoyed throughout the world. But have you ever wondered what mathematics lies just beneath the crust? No? Well, we’re going to tell you anyway.
Bread dough is initially a bubbly liquid, with bubbles connected to other bubbles in a ‘matrix’. These bubbles will collapse, provided that both the temperature and temperature gradient are high enough. To start with, the bubbles at the surface (which is hotter than the interior) reach a temperature at which they are likely to fracture. At this point, the temperature gradient is also high, with plenty of cooler liquid dough nearby. However, when the temperature of the interior has increased sufficiently to allow the bubbles inside to burst, the temperature gradient is much lower, the matrix has set, there is less liquid dough nearby, and so less collapse can take place.
But that’s not all! We can refine the model by considering the movement of the ‘crust boundary’ (where bubbles collapse) as the dough rises, as well as the vaporisation of moisture inside the bubbles. Both of these allow for the transfer of heat and affect the thermodynamics of the whole process.
So in the future, please try to remember all the maths that worked hard to ensure the crustiness of your bread! And, on that note, we’re off to get pizza...
References Jefferson DR, Lacey AA and Sadd PA (2007). Crust density in bread baking: Mathematical modelling and numerical solutions. Applied Mathematical Modelling 31 (2) 209–225. Jefferson DR, Lacey AA and Sadd PA (2007). Understanding crust formation during baking. Journal of Food Engineering 75 (4) 515–521.
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In conversation with...
Bernard Silverman Nikoleta Kalaydzhieva and TD Dang
I
’ been said that a degree in mathematics opens many doors, but to many this might seem a slight exaggeration. Bernard Silverman, however, is an excellent example of a mathematics graduate who has indeed done it all. Silverman is currently the chief scientific advisor to the Home Office, a statistician, and an Anglican priest. These are just a few examples of his many achievements, starting from the gold medal he won at the 1970 International Mathematics Olympiad—the only person to do so from the western side of the iron curtain—at the beginning of his mathematical career. He went on to read mathematics at university, and eventually obtained a PhD in data analysis in 1977. “I was always interested in maths, but as time went on I became keen on doing it in a way that has applications in different things, and that is what drew me to statistics.” He jokingly adds that he felt he was never good enough to be a pure mathematician. In the course of our conversation with him, he took us on a journey through the diverse areas in which he has applied his statistical approach.
Data analysis With a PhD in data analysis, we felt that Silverman was the perfect person to ask about how big data has changed the landscape of statistics. At the start of his career, “the towering figure was John Tukey, who wrote a book called Exploratory Data Analysis. At the time of its publication, it was an enormous advance because it had the idea of leing the data tell us the story, instead of fiing it to completely specified parametric models.” These ideas are still present in big data, chalkdustmagazine.com
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chalkdust he says, but “of course the challenge has gone from dealing with just a small set of numbers to enormously large data sets”. Silverman describes some of his work as seeking needles in haystacks or, in other words, “dealing with the problem of understanding when what we see is real, rather than just noise. All of this arises from handling these enormously large data sets: that is something that wasn’t there before.” During his academic career, Silverman has wrien several books. His work varies from the theoretical to more practical applications of statistics. His work, described in his book Functional Data Analysis, enabled advances in spatial statistics and image recognition. “In functional data analysis,” he says, “the data is not just numbers but curves and surfaces, and this will continue to be the case in the future, with our data consisting of a lot of pictures or sounds, which we need to make statistical sense of.” Understanding this more complicated statistical situation is something we should look forward to, along with machine learning. Silverman predicts a “closer melding of machine learning and big data. The data will set its own questions as well as answer them.”
Calculator development & the next technological breakthrough Although Silverman has always been interested in computational statistics, when he began his statistical journey during his master’s at Cambridge, there were no computers as we know them today. This meant that it would take him “weeks to write a simple programme to draw a density curve from some data”. His desire to develop the technology that would overcome this difficulty led him to move into industry, working for Sinclair Radionics on the Sinclair Cambridge Programmable calculator. First released in 1975, it was marketed as an affordable pocket proThe Sinclair Cambridge calculator, a close relative grammable calculator and Silverman proudly of Silverman’s calculator, as seen in the Science recalls that “you could keep 36 programme Museum steps in its memory”. At the other end of the spectrum, he expects the next technological development to be in quantum computing, which “if quantum computers ever get going—which is an ‘if’, we don’t know for sure—would be a complete game changer”. And, on a more immediately practical day-to-day level: “you just turn on your new computer and it recognises you and logs you in. This is again an application of functional data analysis, using your face as the data.”
Climate change and genetics Silverman’s aitude is to take a statistical approach to many different subjects, especially those of enormous global importance. For example, some of his collaborative work is in climate change with the Smith School of Enterprise and Environment and in human genetics with the Wellcome 5
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chalkdust Trust Centre for Human Genetics. Climate change modelling is different from standard statistical modelling: “it is not done by conducting experiments or taking measurements, because the models are all computer models. Rather, it is a question of making sense of all the different trajectories of temperature that you can possibly get under given scenarios.” In genetics, however, the challenge is different. In genomes and DNA, the idea is to “use data analysis to see what information there is in the very high-dimensional data that is generated”.
When people say that “all the evidence points in this direction”, you should be worried.
Home Office
Silverman leads a team of researchers at the Home Office. The Centre for Applied Science and Technology is a branch of the Home Office that focuses on serving a range of the government’s national interests: anything from crime prevention and community safety to security and identity assurance. Protective security is an example where maths, specifically finite element analysis, plays an important role. Finite element analysis is a method used to simulate complex physical systems: how explosions and earthquakes affect infrastructure, or the effect of wind on skyscrapers, among many other things. Take, for example, a model of an explosion in a building. We first consider the structure of the building as being fixed. The structure can be split up into units, and the blast will move back and forth through these units. When the explosion is triggered, waves propagate outwards, and initially the pressure of the system is very high, but as time goes on the range of pressures decreases. This work, carried out (some years ago, now) at the Centre for Applied Science and Technology thus involves modelling data, without actually carrying out experiments with physical resources—rather problematic in this example! It is not surprising that political decisions are not based solely on science, although it plays an important role: ultimately, a politician’s job is not the same as that of a scientist and priorities sometimes differ. However, on issues that really require scientific input and where evidence is needed (for example, when a policy is to be draed regarding driving under the influence of drugs), it is the role of scientists to determine how to carry out any experiments or surveys that may be reSteve Cadman, CC BY-SA 2.0 quired. And, of course, although policies may The Home Office, Marsham Street be based on politics, underneath it all is how they will actually be carried out: this is where science decides what will and will not work. Silverman emphasises the importance of “making sure that policymakers are aware of evidence and the objective”. Presenting this evidence is very different from presenting a research paper: simplicity is oen preferred over detail (although presenting detail in a simple manner is even beer!) and information needs to be incredibly clear. “It should empower your audience and make them feel comfortable about understanding it”. Aer that, though, “what they do is up to them”. chalkdustmagazine.com
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Modern slavery
There are between 10,000 and
Silverman is part of the prime minister’s task force 13,000 victims of modern slavery on tackling modern slavery. In particular, he manin the UK. aged to work out a way of using the techniques of multiple systems estimation and generalised mark recapture to estimate the size of a hidden population: the slaves of today’s society. He explains the basics with a simple example from his paper Modern slavery: an application of multiple systems estimation. Suppose that we want to estimate the number of fish in a pond. You first catch 100 fish, mark them and release them back into the pond. Aer leing the fish swim around for a bit, you catch another batch of 100 fish and count how many are marked. If, for example, 20 are marked (one fih of the second batch), the natural estimate for the whole population size would be 100 ÷ 1/5 = 500. This is the approach of mark recapture. Multiple systems estimation extends that idea to situations where there are more than two lists. You can use it to estimate the number of individuals that are not on any list, giving what is known as the ‘dark figure’. By using this approach, Silverman concluded that, for a confidence of 95%, there are between 10,000 and 13,000 victims of modern slavery in the UK. To underline the importance of research in the decision-making process of the Home Office, he adds that “the whole government strategy was built on this analysis. The paper was launched at the same time as the Modern Slavery Act was launched, but we are still trying to understand the scale and nature of this problem.”
Improving the image of maths and science As a man who has always been interested in science, Silverman believes that “people are very odd about it. On the one hand, everybody wants all the latest gadgets; but on the other, they are suspicious about scientists and don’t like them very much.” As someone who has had first-hand experience of science in both academia and public service, why does Silverman think that people perceive science in the way they do? He notes that science isn’t perfect, but the way it is taught at school does not suggest this. At school, we are only taught facts, ideas and methods that have been around for centuries, which have been perfected time and time again before they have reached our blackboards. Modern science, however, is experimental and, for those who have not yet understood this, “it feels either very boring or irrelevant”. The truth is that it is neither: “it is very exciting. Communicating the excitement is important.”
Women in mathematics In recent years, there has been an increased awareness of the lack of females in mathematics and science. We were interested to hear Silverman’s opinion on why that is so, and what we can do to reduce the gender imbalance. He admits that “it has been an enormous struggle to get girls interested in science. I have always wondered to myself why that is. We have to say what it is about the way we do science that women find off-puing. Because I don’t think women are any worse at it or men any beer.” He believes that the culture of science is “still very macho and very male-dominated”, which is why some women might find it off-puing. To move forward, our research into how to fix this needs to be more subtle than the current penchant for asking women in the system why they stayed in it. 7
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Priesthood What people may find surprising about Silverman is that aside from his extraordinary work in science, he also takes an interest in religion. He earned a bachelor’s degree in theology from the Southern Theological Education and Training Scheme, and is currently a practising Anglican priest. So what about his views regarding the relationship between science and religion? He says that although he is not always certain about his faith, it remains an influence on everything that he does. However, “any view of religion that doesn’t take into account modern science must be wrong, because the observations of modern science are incontrovertible. They are the world that we live in; they are the world as we have it.” He ends with the comment that “it is not a way of thinking I’ve always had, but it is what makes sense to me. It is part of your whole existence.” Something, we suppose, that could also be said about mathematics and statistics. Nikoleta Kalaydzhieva and TD Dang Niki is a PhD student at University College London, working in analytic number theory. TD is an undergraduate at UCL who actually understands the 1967 James Bond spoof Casino Royale, starring David Niven, Peter Sellers, Woody Allen, and Orson Welles as Le Chiffre. @televisionduck (TD)
My least favourite number Numbers are everywhere in maths. Some numbers are great, but many are simply awful. Throughout this issue, we share some of our least favourite numbers. We’d really love to hear about yours! contact@chalkdustmagazine.com, @chalkdustmag or chalkdustmag Send them to us at and you might just see them on our blog!
The golden ratio, ϕ = 1.618... Tom Rivlin
All the ancients revered it. It holds the secret to beauty and art itself. All of nature is based on it. Its proportions are perfection itself. It has magical properties…
Am I doing it right?
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…except none of that is true. It’s just half of one plus root five. It’s the solution to a quadratic equation. It’s the limiting ratio of a Fibonacci sequence; the growth factor of a logarithmic spiral. It appears in some places in nature. It appears nowhere in human anatomy. It was used by some classical artists. It has some neat mathematical properties. Whoever runs the golden ratio’s PR department is doing a great job, but the golden ratio needs to get over itself. The hype ruins it. 0/10
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& WHAT’S
WHAT’S
HOT NOT
HOT
Agree? Disagree? @chalkdustmag chalkdustmag
HOT
Schrödinger’s cat
Even better if they’re royalty-free images, because it’s not cool to steal other folks’ cartoon-based work
It definitely gets our vote. Stats can totally help with elections and referenda
NOT
NOT
Getting it wrong 2016 whoops…
NOT HOT Lowercase V Upsilon Not even its own
NOT
Used to be full of fun maths; now full of people declaring the end of the world. Sad!
HOT Psephology
Schrödinger’s cat
Maths memes
letter vntil 1386
HOT Pont Saint-Bénézet Satisfies the Euler characteristic despite being old and unfinis—
Maths is a fickle world. Stay à la mode with our guide to the latest trends.
υ
Rising star of the Greek alphabet (apart from **) and its capital looks like a diving whale
NOT Solids of constant width Roll over, 20p and 50p (while preserving your width), there’s a new quid on the block!
NOT Bridges of Königsberg
HOT Dodecagons
So many better ways of drawing a house these days
Because solids of constant width aren’t edgy enough
Pictures CC BY-SA 3.0 Brexit map: Wikimedia users Mirrorme22, Nilfanion, TUBS, Sting. Bridge: Wikimedia user Chiugoran. Fair use New pound coin: Royal Mint. ★ Because an asterisk—from the Greek asteriskos—is a rising star, right?
More free fashion advice online at chalkdustmagazine.com 9
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Linear algebra... with diagrams Vera de Kok, CC BY 2.0
Paweł Sobociński
A
succinct—if somewhat reductive—description of linear algebra is that it is the study of vector spaces over a field, and the associated structure-preserving maps known as linear transformations. These concepts are by now so standard that they are practically fossilised, appearing unchanged in textbooks for the best part of a century.
While modern mathematics has moved to more abstract pastures, the theorems of linear algebra are behind a surprising number of world-changing technologies: from quantum computing and quantum information, through control and systems theory, to big data and machine learning. All rely on various kinds of circuit diagrams, eg electrical circuits, quantum circuits or signal flow graphs. Circuits are geometric/topological entities, but have a vital connection to (linear) algebra, where the calculations are usually carried out. In this article, we cut out the middle man and rediscover linear algebra itself as an algebra of circuit diagrams. The result is called graphical linear algebra and, instead of using traditional definitions, we will draw lots of pictures. Mathematicians oen get nervous when given pictures, but relax: these ones are rigorous enough to replace formulas. chalkdustmagazine.com
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Diagrams Let’s start with a picture of a generic wire. This is our first diagram. (W) We say that (W) is of type (1, 1) because, as a diagram, it has one dangling end on the le, and one on the right. Diagrams can be stacked on top of one another; for example, the following is obtained by stacking (W) on top of itself, obtaining a (2, 2) diagram.
Wires can be be jumbled up, with the following (2, 2) diagram called a twist. (T) In addition to stacking, diagrams can also be connected in series if the numbers of wires agree on the connection boundary. The following results from connecting two twists.
It is useful to imagine that the wires are stretchy, like rubber bands. For example, we consider the following three (4, 4) diagrams to be equal.
We will need additional equations between diagrams. A nice value-added feature of the notation is that equations oen convey topological intuitions: eg we require that a twist followed by a twist can be “untangled” in the following sense. =
(S)
In some seings, eg knot theory, the above equation would not be imposed because knots—unsurprisingly—rely on the ability of wires to tangle. The final thing to say about twists is that we can “pull” diagrams across them, preserving equality. The following, known as the Yang–Baxter 11
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chalkdust equation, is an instance of this: we pull the top le twist across the third wire, which, for sake of legibility, is coloured red. (YB)
=
Given these insights, it is not too difficult to prove that the diagrams we’ve seen so far are in bijective correspondence with permutations. For example, those in (YB) correspond to the permutation ρ on the three element set {0, 1, 2} where ρ(x) = 2 − x. An optional note for category theory aficionados: our diagrams are the arrows of a strict symmetric monoidal category with objects the natural numbers. The arrows from m to n are (m, n) diagrams. All diagrams in this article are arrows of such categories, which are called PROPs. Thus “pulling” diagrams across twists, as in (YB), is none other than the naturality of the braiding structure and (S) ensures that the braiding is a symmetry.
Copying Let’s imagine that wires carry data from le to right. Whatever it is, assume that we can copy it. To copy, we get a special (1, 2) contraption, illustrated below, which takes data from the wire on the le and copies it to the two wires on the right.
A defining feature of copies is that one should not be able to distinguish between them; copying, then swapping, is the same as just copying. We thus add the following: =
If I copy twice, I end up with three copies. There are two ways to do this, but they are indistinguishable. The following equation ensures this: =
The ability to copy oen comes with the ability to throw away, otherwise photocopying rooms would quickly fill up with scrap. Throwing away is done by the (1, 0) diagram below, which takes data and returns nothing.
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chalkdust Finally, if I copy and throw away one copy, I haven’t achieved very much in the grand scheme of things. This is the intuition behind the following equation.
=
The above structure is otherwise known as a (cocommutative) comonoid. Copying is awkward to denote with standard formula syntax. For example, f(x, y) is a standard way of denoting an operation that takes two arguments and returns one result. But how can we denote an operation that takes one argument and returns two results? The kinds of diagrams that we have seen so far are a solution.
Adding Next, we imagine that data on the wires can be added. For the sake of concreteness, we may as well assume that wires carry integers. Then a (2, 1) addition operation takes two arguments and has a single result.
Adding comes with an identity element, and we introduce a (0, 1) gadget for this.
The intuition is that the diagram above outputs 0 on its result wire. Given that addition is commutative, associative and has an identity element, the following equations ought to be uncontroversial.
=
=
=
The above structure is otherwise known as a (commutative) monoid. An intriguing thing about the diagrammatic notation is that, ignoring the black/white colouring, the equations involved for monoids are mirror images of those for comonoids. 13
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Copying meets adding The fun really starts when diagrams combine adding and copying. From now on, we will draw diagrams that feature all of the following:
copy
throw away
add
output 0
subject to both the comonoid and monoid equations we considered before. But now the two structures can connect to each other, leading to some new and interesting situations. First, when we copy zero, we get two zeros. Similarly, when we discard the result of addition, it’s the same as discarding the arguments. This leads to the following:
=
=
One of the most interesting equations concerns copying the result of an addition: it is the same as if we copied the arguments and performed two additions, separately.
=
The last equation is about discarding zero. The effect is… nothing, the empty diagram. =
This is the point in the story where linear algebra starts bubbling up to the surface. Diagrams—with all of the equations we have considered so far—are now in bijective correspondence with matrices of natural numbers. Moreover, composing diagrams in series corresponds to matrix multiplication, and stacking two diagrams to a direct sum. At first sight, this may seem a lile bit magical: we drew diagrams by stacking and connecting basic components; we never even defined the natural numbers! Moreover, multiplying matrices involves ordinary addition and multiplication of integers. So where is all of this structure hiding in the diagrams? The best way to get the idea is via examples. Following the correspondence, the (1, 1) diagrams below ought to be 1 × 1 matrices. In fact, they correspond to (0), (1), (2) and (5). Notice the chalkdustmagazine.com
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chalkdust paern? To get the number, count the paths from the le to the right.
Can you figure out how to multiply and add numbers, as diagrams? If yes, you’ll enjoy proving that multiplication distributes over addition. This is the first example of a common phenomenon in graphical linear algebra: basic algebraic operations—oen considered primitive—are actually instances of the algebra of diagrams. One final example: the 2 × 3 matrix ( ) 1 0 2 0 1 1 corresponds to the (3, 2) diagram below. To get the (i, j)th entry, count the number of paths from the j th input to the i th output.
This algebraic structure is known as a (bicommutative) bimonoid, or bialgebra. Bimonoids are common all over mathematics, eg in algebra, combinatorics, topology and models of computation. Once familiar with this paern of interaction between a monoid and comonoid, you will see it everywhere. The most exciting part of the story comes when we confuse the “direction of flow” in diagrams. This means, roughly speaking, that copying and adding can now go both from le to right, and from right to le. Our diagrams will now feature all of
with all of the equations considered so far. The way to make sense of this is to stop considering the le side of the diagrams as “inputs” and the right side as “outputs”. Technically, it means not thinking of addition and copying as functions, but rather as relations, which, unlike functions, can always be reflected. For example, addition—as a relation—is the set of pairs (( ) ) x , x+y , y 15
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chalkdust while “backwards” addition is the relation ( ( )) x x + y, . y As before, this leads to several new situations to consider. Intriguingly, it turns out that the same equations describe the interaction of copying and adding with their mirror images.
=
=
The above are known as Frobenius equations, and are the other common way that monoids and comonoids interact. Just as for bimonoids, this paern of interaction can be found in many different places, all over mathematics and its related fields. Enumerating all the equations would be an overkill for this article, so let’s go straight to the punchline. Previously, our diagrams were a bijection with matrices of natural numbers, even if it took some mental yoga to see the correspondence. This time, the magic ramps up a few notches: diagrams are now in bijective correspondence with linear relations over the field Q of the rational numbers, AKA fractions. But let’s go through this step by step. First, where do fractions come from? Before, when everything flowed from le to right, there was a bijection between (1, 1) diagrams and natural numbers. But with mirror images of copying and adding around, direction of flow is confused, bringing additional expressivity. For example, the following (1, 1) diagram is the diagrammatic way of writing 32 : it connects a 2, going from le to right, with a 3 going from right to le. ( 32 )
Just as the multiplication and addition of natural numbers can be derived from the algebra of diagrams, so can the algebra of fractions that everyone learns in primary school. As it turns out, not all diagrams of type (1, 1) are fractions, which leads us to an interesting feature of graphical linear algebra. But first, a lile detour. A curious phenomenon of human languages is that some words are notoriously difficult to translate. Some have become quite well-known, almost cliché: Schadenfreude in German, furbo in Italian or hygge in Danish. This is a side effect of the subtle differences in expressivity between languages: a concept natural in one is sometimes clumsy to express in another. Differences in expressivity also show up in formal languages. With this in mind—and please don’t freak out—the algebra of diagrams has nothing to stop you from dividing by zero. This is because reflecting a (1, 1) diagram means taking the reciprocal: eg chalkdustmagazine.com
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chalkdust 3 is the diagram with three paths from le to right, and 13 is the diagram with three paths from right to le. Keeping this in mind, the following would seem to be the diagram for 10 . (∞) It’s natural to be a bit nervous, considering the anti division-by-zero propaganda bombarding us from a young age: as a result, most of us suffer from an acute dividebyzerophobia. But nothing explodes, and it can actually be useful to work in this extended algebra of fractions, featuring division by zero. To tie up the story, let’s complete the taxonomy of (1, 1) diagrams: there is a diagram for each ordinary fraction, the diagram (∞) above and just two additional diagrams, illustrated below, which can be understood as two different ways of translating 00 to the language of diagrams. (⊥ and ⊤) In general, (m, n) diagrams are in bijective correspondence with linear relations: those subsets of Qm × Qn that are closed under pointwise addition and Q multiplication, ie that are Q vector spaces. For example, the linear relation of (∞) is { (0, q) : q ∈ Q }, and those of (⊥ and ⊤) are, respectively, { (0, 0) } and { (p, q) : p, q ∈ Q }. The linear relation for ( 32 ) is { (3p, 2p) : p ∈ Q }. Other examples of such beasts include kernels and images of matrices with Q entries; all of which are thus expressible with the language of diagrams. Now, since all of the diagrammatic equations involve only adding and copying operations, we conclude with an insight that’s not so apparent given the usual way of presenting linear algebra: linear algebra is what happens when adding meets copying. Paweł Sobociński Paweł is a theoretical computer scientist at the University of Southampton, focusing on compositional modelling of systems, developing the underlying maths (usually category theory), and applying it to real-life problems such as verification. Since 2015, he has been working on the Graphical linear algebra blog, rediscovering linear algebra with string diagrams. graphicallinearalgebra.net
sobocinski@gmail.com
My least favourite number
NaN
Pietro Servini
My least favourite number is NaN, which can look something like the figure on the le. I very much disapprove of this number as it usually means that there’s yet another mistake in my code. 10/0
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Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the prof’s help? Contact deardirichlet@chalkdustmagazine.com
Dear Dirichlet,
ism and the eld belief in free-market capital g-h lon p, dee y ver a e hav ally I person I switched subjects ently shocked to discover when rec s wa I but rk, wo d har of ue val of Karl Marx! I’ve earch area are staunch followers that most people in my new res rgy slowly leagues on this. I can feel my ene col new my h wit s ent um arg had many Surrey ate. How do I resolve this? — Feeling blue, draining with every passing deb ■ DIRICHLET SAYS: Your problem is simply connected to the fact that you now work in a non-conservative field. It is quite typical in these fields for your energy to slowly decay away over time. Since you come from a conservative field, you understand that the value of work is absolute, but that’s just not the case in non-conservative fields: the value of your work can be different depending on the path you take through life! If you ever want to escape an argument, though, try curling up into a ball. Since you come from a conservative field, your curl will vanish.
Dear Dirichlet,
I am extremely proud that my hus band is a decorated major in the Army—when we met he was just a private! Now the kids have all grown up and moved out, we’re contemplating an early retirem ent to somewhere in the sun. But my husband heard rumours that a couple more pip s on his shoulders might be com ing his way soon. Without being awkward, how can we find out if he’s likely to be pro moted this year, or whether the whispers are jus t nothing?
— Fingers crossed, Redditch
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DIRICHLET SAYS: As a keen amateur military strategist myself, I see
the problem here: will your husband get rank... or nullity? For any linear map T:V->W on vector spaces V and W, dim(Im T) + dim(Ker T) = dim V. By conservation, therefore, I advise the major to look as shabby as possible and to destroy his image. Only this will increase his chances of making... colonel. chalkdustmagazine.com
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Dear Dirichlet,
Once a month, beneath the ligh t of the full moon, I feel inhuman strength, an animal instinct, and an overpowering bloodlust consume me, before blacking out and waking up in strange places. But there’s more: I also feel an ove rwhelming urge to solve exact integrals, genera te pictures of pop culture charac ter s using graphs, and perform useless calculatio ns with obscure trivia. At first I thought I was just becoming a werewolf, but now I’m not so sure. Can you help me ?
— A husky fellow, Hounslow
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DIRICHLET SAYS: Your symptoms have only one explanation: you are
not a werewolf, you are a wereWolfram. It’s a known side effect of using Wolfram Alpha too much. But worry not, the solution is trivial: simply perform an inverse Furrier transform on yourself and you’ll be back in normal space in no time.
Dear Dirichlet,
enever there’s an the local primary school, and wh in r ane cle a as gs rnin mo rk I wo es always leave e into the school to cast their vot election on, the people who com and find everything e a lot of effort to go through so much litter! It can often tak ier? on how to make the process eas they’ve left behind. Any advice t sweeper, Bedford
— Swing sea
DIRICHLET SAYS: Tell me about it! The public really are the worst. That’s why I now live exclusively with badgers. Anyway, a contour integration should easily pick up any residue left at the polls. It’s then up to you to find the right path around them all, but it’s really not that complex.
■
Dear Dirichlet,
I’m working on your divisor problem but have a few questions. Have you received my previous letters asking for help?
— M N Huxley, Cardiff
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DIRICHLET SAYS: Please cease your endless correspondence. More Dear Dirichlet, including two seasonal specials, online at chalkdustmagazine.com 19
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Debugging insect dynamics Todd Huffman, CC BY 2.0
Atheeta Ching
S
dynamics are complex and have evolved over many generations. One strategy that is used is that of altruism: the act of helping someone else at a cost to yourself. In some insects, this takes on an extreme case where workers sacrifice their own fertility to help raise the queen’s eggs instead. While this may seem to go against the idea that animals want to pass on as many of their genes as possible, we’ll see why this is actually a viable strategy. Game theory examines how the frequencies of different strategies played in a game change over time. Evolutionary game theory looks at the special case where players cannot change the strategy they play: they’re stuck with the strategy they’ve inherited from the previous generation. In each round, players are randomly matched up and play a game, leading to an outcome dependent on their strategies. This outcome is called a payoff, and it affects their fitness and thus the number of offspring they produce. In the 1970s, John Maynard Smith and George Price developed evolutionary game theory to investigate ritualistic fighting behaviour in animals. A good example is how male stags will compete for territory during the mating season. Physical contact is actually unlikely to occur and the stags can spend hours staring and roaring at each other to determine who is the strongest. If things do escalate, many species of stags have branched antlers allowing them to wrestle rather than impale each other. Species with straighter antlers will tend not to use them in fights, but will resort to biting and kicking, which is far less dangerous. This strategy of assessing your opponent first and picking your fights carefully is clearly beneficial for the species as a whole, but it wasn’t originally clear why an aggressive strategy (where you kill your opponent and pass on your genes) isn’t more common in the animal kingdom. chalkdustmagazine.com
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chalkdust Smith and Price sought to examine this and they devised the hawk–dove game. A population (of the same species) is split into two groups: hawks and doves. Hawks are aggressive and will play until they win or are seriously injured. On the other hand, a dove is a pacifist and will surrender if its opponent gets aggressive (so it will never get injured). In this game, two players are matched up and compete for a resource (eg food), and the outcome depends on the strategies they play. These dynamics are shown in the following payoff matrix: If hawk If dove
meets hawk (v − c)/2 0
meets dove v v/2
Here, v is the value of the resource, while c is the cost of injury (from a hawk losing to another hawk). Typically in nature, we find that c is much larger than v. To explain the entries of the payoff matrix: when a hawk meets another hawk, there is a 50% chance it will win, gaining v ; but a 50% chance it will lose, losing c. When a hawk meets a dove, it will always win, gaining v, while the dove always loses without injury, receiving 0. When a dove meets another dove, there is a 50% chance it will win, gaining v and, if it loses, it does not get injured but receives 0. These dynamics can be analysed by the replicator equations. The change in proportion of a strategy i (xi ) is given by the fitness of the strategy (fi ), minus the average fitness of the population, all multiplied by the proportion of strategy i: n ∑ dxi = xi fi (x) − xj fj (x) . dt j=1
The distribution of the population into the n strategies is given by the vector x = (x1 , . . . , xn ) which, since they are proportions, has entries summing to 1. Using the values from the payoff matrix above leads to a single differential equation (since x2 = 1 − x1 ), with a globally stable steady state where the proportion of hawks is v/c, which is closer to 0 than 1. While this can explain the lack of aggressive strategies seen in nature, an extension of this is the hawk–dove–assessor game. An assessor plays as a hawk if they are stronger than their opponent, and as a dove if they are weaker. This is precisely what we tend to see in ritualistic fights in nature. The payoff matrix is given below, which you can verify yourself. If hawk If dove If assessor
meets hawk (v − c)/2 0 v/2
meets dove v v/2 3v/4
meets assessor (v − c)/2 v/4 v/2
We find that the strategy of being an assessor is an evolutionary stable strategy. This means that if the entire population is playing as assessors, then any invasion by another strategy cannot succeed and spread. The assessors will always dominate eventually. 21
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chalkdust Many different strategies can be represented in game theory, including cooperation, spite, selfishness and altruism. Altruism is when a player does something beneficial to the recipient at a cost to itself. This can be seen in games with a memory or reputation system: if the cost is low but the benefit is high, then an individual may act altruistically in the hope of the other player returning the favour later on, leading to a net benefit for them both. Perhaps the most obvious reason for an altruistic act, though, is if the players are related, meaning that they have genes in common. This stems from the idea of inclusive fitness: inclusive fitness = individual fitness + (relatedness × relative’s individual fitness), ∑ Rij fj . ω i = fi + j̸=i
This means that when examining the dynamics of gene frequencies, the fitness of an individual’s family should also play a role since some of the genes will be shared. Relatedness is defined to be the probability that a gene picked randomly from each individual at the same locus (position) is identical by descent. This works out intuitively: 0.5 between you and a sibling/parent/son/daughter (since each parent gives half of their genes to their offspring), 0.25 between you and an uncle/aunt, 0.125 between you and a cousin. Another name for inclusive fitness theory is selfish gene theory, popularised by Richard Dawkins’ book The Selfish Gene, which was influenced by ideas from fellow biologist George Williams. The term ‘selfish’ refers to how some genes may prioritise their own survival (over many generations) over that of the individual or even species. This gene-centred view of evolution helps explain altruism. William Donald “Bill” Hamilton was an evolutionary biologist who completed his PhD at the London School of Economics and University College London. He claimed that altruistic acts are favoured (and as a strategy, can spread through the population) if the relatedness between the players is greater than the cost to benefit ratio of the act, ie C R> . B This became known as Hamilton’s rule and, while it can be hard to quantify and test, a simple example occurs in prairie dogs. When these rodents are above ground and spot a predator, an individual is more likely to sound an alarm call when relatives are close by: an action that is costly as the individual draws aention to itself.
Colony bee-hive-iour An extreme case of altruism is eusociality: the highest level of organisation in animal social structure. This structure is what you probably think of when you consider an insect colony: a queen laying eggs and thousands of workers maintaining the nest. Some other traits of this include cooperative brood care and overlapping generations; however, its most distinct trait is chalkdustmagazine.com
Gamekeeper, CC BY-SA 3.0
Aa cephalotes castes
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chalkdust the division of labour into castes. These castes are specialised in that individuals from one caste lose the ability to perform the tasks of individuals from another caste. These castes don’t just vary physically, they can express different behaviours and even instincts! In the figure on the previous page, we can see many types of workers on the le, a soldier in the middle and two queens on the right. The soldier is larger than the workers and has a stronger jaw—interestingly this is mainly used for foraging heavy objects and not to defend the nest: that job is up to the workers. Most eusocial animals can be found in the third largest order of insects, called Hymenoptera, which includes bees, wasps and ants. Hymenopterans are also haplodiploids, ie males have one set of chromosomes (haploids) and females have the usual two (diploids). This bizarre fact means that males actually hatch from unfertilised eggs and females from fertilised eggs. Some strange relationships can come from this. For example, males have no father or sons, but have a grandfather and can have grandsons! But the most intriguing fact is that the workers (who are all female and make up the vast majority of a colony) help raise new brothers and sisters produced by the queen instead of having their own offspring. This fact puzzled Charles Darwin, as it wasn’t clear how a trait that leads to an individual not reproducing and passing on their own genes can be so prevalent in a population. This would mean that their individual fitness is zero; however, going back to inclusive fitness theory, we can see that they can still benefit from their relatives. In fact, we’ll see that this benefit outweighs that of having their own offspring. A mated queen can produce fertilised and unfertilised eggs for the rest of her life. From the figure to the le, we can see that female workers are more related to their sisters than any other relative (including possible offspring). Explicitly, two female workers share exactly the same set of blue chromosomes (from their father), which is already a relatedness of 0.5. The other set (in red) can either be exactly the same or different, depending on which was inherited from the queen, giving on average another 0.25, and making their relatedness R = 0.75. Following Hamilton’s rule, it indeed makes sense to help raise new sisters instead of offspring, who only have a relatedness of 0.5. In most species of eusocial Hymenoptera, the queen is aggressive and releases pheromones to discourage workers from laying eggs and, in some cases, Family tree of haplodiploid insects from even ovulating. However, it seems that not all workers are happy with this arrangement, and this can lead to several different types of conflict. spring 2017
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Conflict: sex allocation The queen is equally related to male and female offspring (with a ratio of 1:1) but female workers prefer sisters to brothers (with a ratio of 3:1). Consequently, it is common for the female workers to eat or kill male larvae so that the colony’s resources are not used in raising them. This means that the queen has wasted energy producing and laying the male eggs. A common solution to this is to produce a brood of all the same sex, reducing the motivation for the males to be destroyed. In reality, the ratio is between the two ideals for the queen and female workers. For most of these species, males (AKA drones) do not help with work in the colony: their only role is to mate with a young queen—so, apart from the genetics, female workers may benefit even less from having brothers. For the queen, however, males are still important to help spread her genes even further: if the males are able to mate with a new queen who then starts her own colony, all of the new workers will carry one set of chromosomes from the original queen. In some species, it has been noticed that the queen will lay batches of male eggs when food supplies are low, as female workers’ fitnesses are more affected by food during their developmental stages. However, since males don’t even help with foraging for food, this strategy would only work in the short term.
Conflict: male rearing In honey bees, 7% of male eggs are from workers but only 0.1% of adult males are a worker’s son. Why would workers lay eggs in the first place? Well, they are more related to their own offspring (R = 0.5) than to any brothers (R = 0.25) that the queen produces. However, laying workers are less hard-working and the queen would rather spend resources raising new children than grandchildren (to whom she is less related), so she tries to prevent this using pheromones. These pheromones inhibit the workers’ ability to lay eggs but are less effective in large colonies, especially if the queen is old. Worker policing occurs through workers destroying other worker-laid Jessica Lawrence, CC BY 3.0 eggs. We can calculate from the family tree on the previous page that Apis mellifera: queen workers are more related to new sisters (R = 0.75) than any nephews and workers (R = 0.33). The calculation for the second relation works as follows: comparing a female worker to her nephew, there are three possible cases involving the three different sets of chromosomes in the family tree (the blue set of chromosomes from the queen’s mate and the two red sets of chromosomes from the queen herself). One case is where the nephew has the blue set of chromosomes, which his aunt will also have, meaning she shares 50% of her genes with him. If the nephew has a red set of chromosomes, they can either be the same set as his aunt’s (again, 50% of her genes will be shared), or they can be the ‘other’ red set originally from the queen (so his aunt will share none of her genes in this case). Thus on average, a worker has a relatedness of R = (0.5 + 0.5 + 0)/3 to her nephew. Workers can determine whom the eggs belong to through chemical markers. You may think that there is a selection pressure (in evolution) towards workers who can lay eggs that mimic the queen’s… and you’d be right! These are called ‘anarchic workers’ and in the Cape honey bee, this trait is affected by multiple genes; but, for various reasons, the cost of having these altered genes outweighs their benefit. Note that this conflict does not appear chalkdustmagazine.com
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chalkdust in every eusocial hymenopteran: in some species, the castes are so specialised that workers aren’t even capable of reproducing in the first place.
Conflict: caste fate Why wouldn’t a worker choose to develop as a queen instead? Doing so would be selfish but it yields a much greater inclusive fitness. In many species, the determining factor as to whether a female larva develops into a queen or worker is the amount and type of food it receives. Too many queens could cause the colony to run inefficiently or break down, so to prevent this, some species raise Waugsberg, CC BY-SA 3.0 their larvae in space-restricted cells where the food Apis mellifera pupae supply is controlled by the workers. This generally prevents the larvae from developing into queens. In species where this does not occur, excess queens are killed immediately aer emerging. A queen can actually mate with several males to produce offspring that may not be full siblings. A higher relatedness between workers, however, can reduce the incentive for such selfish acts. Focusing only on the genetics, an already-developed worker would benefit more from raising new sister workers than raising a new sister queen, as this could lead to the raising of nephews/nieces.
Conflict: matricide There are three parties in favour of the queen’s death: laying workers, non-laying workers and, surprisingly, the queen herself. Going back to Hamilton’s rule, a model was developed by Bourke quantifying the cost–benefit ratio of the queen’s death. These are based on multiple variables including the number of offspring a laying worker can have between the queen’s hypothetical death and the death of the colony. The colony wouldn’t survive long without a queen, and raising new queens isn’t always successful due to the risky mating flight, where a virgin queen flies away from the safety of the nest to mate with males. Bourke’s model applies particularly well to annual colonies where eggs are laid together once a year, aer which the colony members all die (apart from young mated queens). Aer ordering the cost–benefit ratios of the queen’s death, it was found that laying workers have the most to gain, followed by non-laying workers, and then the queen herself. This is intuitive as, without the queen, laying workers can raise their own offspring. As we have seen in the figure on page 23, this shouldn’t happen in the case where the queen has only mated once. However, if she has mated multiple times, the new workers may not be full sisters with the previous workers, and the relatedness would be less than 0.75. Additionally, if the queen is old and laying fewer eggs (or unfertilised eggs due to food shortages), the benefit of a worker laying their own eggs increases. Once laying workers start aacking the queen, it creates a positive feedback loop: when the queen is injured, she will produce fewer eggs and the critical ratio for the non-laying workers can be surpassed, meaning that they start aacking the queen too. When the queen is seriously injured and no longer able to lay many eggs at all, she would gain a higher inclusive fitness from 25
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chalkdust allowing the workers to lay eggs, since they would all be her grandsons. Thus, when the queen’s own critical ratio is met, she allows herself to be killed by her workers. Surprisingly, eusociality is also observed in mammals: specifically, two species of mole rat. The naked mole rat is an odd species, famous for its hairless appearance and high resistance to cancer. In captivity, they can live up to 31 years—an astonishing amount of time for a rodent! They live in underground nests under the rule of an aggressive queen who releases pheromones to discourage the workers from reproducing. Unlike the insects, the queen naked mole rat wasn’t born into her position, she had to fight for it! Her reign is also unstable: she will have to defend her crown from female workers. Mole rats (along with alJedimentat44, CC BY 2.0 most all mammals) are diploids, so at first glance, it Female naked mole rat seems that they are just as related to their offspring as to their siblings. However, naked mole rats are infamous for high rates of inbreeding, causing a high level of relatedness between the workers. A few of the males will have the role of mating with the queen, while the others function as workers. It still isn’t certain whether a high level of relatedness is required for eusociality to evolve, or whether it is a consequence of it, but clearly the two are linked. Does altruism, the act of helping others at a cost to yourself, truly exist in the animal world? In the case of eusocial insects, sacrificing your own fertility to raise the queen’s eggs seems like a noble gesture, but we have seen that the workers only do this in order to pass on more of their genes. Once this benefit decreases, conflict occurs and the workers will continue to pursue their own optimal strategy—killing the queen to lay their own eggs. Can any of this theory be applied to humans? The majority of interactions occur between ‘unrelated’ individuals. For social dynamics, game theory would explain altruistic acts in terms of good karma, expecting others to return the favour later on (which is risky). Perhaps having this expectation can be selfish; however, a common assumption in game theory is that players act ‘rationally’ in the sense that they always try to maximise their payoff. I’d like to believe, though, that people don’t actually think like this and that acts of uncalculated kindness in humans occur more oen than the theory would suggest. Atheeta Ching Atheeta is a PhD student at University College London in the area of mathematical ecology. He also has interests in evolutionary game theory and social dynamics. He enjoys drinking cider and mixers and hates it when people like Pietro (see pages 50–54) mispronounce his name. @atheeta2
Did you know... …that Pythagoras’ theorem still works if the squares are replaced by regular pentagons? chalkdustmagazine.com
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Have you been taking your weekly dose of chalkdust?
The limit does not exist?!
How do snowflakes form?
How does maths catch criminals?
Is MEDUSA better than BODMAS?
Should you buy these socks?
Why are all maps wrong?
Is evolution just a game? A present: to buy or not to buy? All these questions and more, have been answered on our weekly blog, covering topics from cosmology to sport, fluid dynamics to sociology, history to politics, as well as great puzzles. Read it every week and sign up to our monthly newsletter at
chalkdustmagazine.com
Snowflakes: K. Libbrecht, by permission. Crime: Catalina Olavarria, CC BY 2.0. Maps: Charly W. Karl, CC BY-ND 2.0. Love: John Chandler, CC BY 2.0.
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Adapted from one of Henry Holiday’s original illustrations to The Hunting of the Snark (An Agony in 8 Fits) by Lewis Carroll
Robert J Low and Thierry Platini
F
’ Last Theorem has been a source of fascination and the motivation for an enormous amount of mathematics over the last few centuries, both in aempts (eventually successful) to prove it and as the inspiration for other related questions. This is the story of how an algebraic question inspired by Fermat’s Last Theorem morphed into an analytic question, which subsequently turned out to be expressed best as a geometric question, which could be answered using basic methods of plane geometry.
Fit the first: curiosity A common strategy in mathematical research is to consider something that is already known, and to try to generalise or vary it hoping for something interesting to appear in the process. Our initial question was motivated by Fermat’s Last Theorem: are there solutions to x ni + y ni = z ni where x, y, z and n are positive integers and i2 = −1? An obvious simplification is to replace x n , y n and z n by x, y and z: if there are no solutions to xi + yi = zi for (non-zero) integers x, y and z, then there are clearly none to the original problem, where x, y and z have to be nth powers. chalkdustmagazine.com
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chalkdust So let’s assume that x, y, z satisfy this relationship. Now, any complex number a+ib can be wrien in the form r (cos θ + i sin θ), where r is called the modulus and θ the argument of the complex number (see figure below). In this case, z i = ei ln z = cos(ln z) + i sin(ln z), so z i has modulus 1 and argument ln z. So now we know that (x i + y i )(x−i + y−i ) = z i z−i = 1 and expanding the brackets gives 2 + (x/y)i + (x/y)−i = 1, which rearranges to (x/y)i + (x/y)−i = −1.
The modulus, r, and argument, θ, of a complex number. Using properties of right angled triangles, it can be shown that r2 = a2 + b2 and tan θ = b/a
Using (x/y)i = cos(ln(x/y)) + i sin(ln(x/y)), we obtain 2 cos(ln(x/y)) = −1, ie cos(ln(x/y)) = − 12 . Therefore an integer k should exist such that ln(x/y) = 2πk ± 2π/3 or, in other words, x/y = e2πk ± 2π/3 .
So, remembering that x and y are integers, all we need to do now is to check whether e2πk ± 2π/3 can be rational. It certainly seems unlikely to be rational: however, there are unlikely truths in mathematics. This seemed like a good time to climb up onto the shoulders of giants, and we did a lile digging into transcendental number theory (using Transcendental Number Theory by A Baker). A number is rational if it is a zero of an expression of the form mx + n, where m and n are integers, ie it is a root of a linear polynomial with integer coefficients; it is algebraic if it is a root of any polynomial with integer coefficients; and it is transcendental if it is not algebraic. Our search showed us that eπ is called Gelfond’s constant, which is known (from the Gelfond–Schneider theorem) to be transcendental. Furthermore, any algebraic function of it is transcendental, and so we can now conclude that e2πk ± 2π/3 is transcendental for all integer k. It then follows that there are no integer solutions to the equation xi + yi = zi and so there are certainly none where x, y and z are all nth powers, ie there are no integer solutions to x ni + y ni = z ni for any positive integer values of n. 29
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Fit the second: despair We were initially happy to conclude this. We had found something interesting. Unfortunately, while looking up transcendental number theory, we found out that (by much the same argument) it was already well known from this theory that there are no solutions to x m+in + y m+in = z m+in for positive rational numbers x, y, z and exponents of the form m + in, where m, n ∈ Z with n ̸= 0. So we had managed to prove a special case of a known result—and not for the first time. For a while, we found this sufficiently off-puing that we stopped thinking about the problem. But finally we rallied, and did what mathematicians do. We changed the question.
Fit the third: enthusiasm We asked a new question instead: are there positive real solutions to xi + yi = zi and, if so, is there anything interesting to say about them? In fact, we already saw the clue as to how to solve this: for any positive real number x, the number x i = cos(ln x) + i sin(ln x) is a complex number of unit modulus. So now, instead of considering x, y, z, we realise that we are looking for three complex numbers a (= x i ), b (= y i ), c (= z i ), such that a + b = c and |a| = |b| = |c| = 1. At first sight, this has not helped a great deal. But interpreted geometrically, the problem suddenly becomes easy. Given a, since a+b = c, we immediately know that |a+b| = 1. |a + b| is the distance between a and −b, which is 1. Similarly, we can use a−c = −b to see that |a − c|, the distance from a to c, must also be 1. So we need to find a pair of points on the unit circle, centred at the origin, both a distance 1 from a, as shown in the figure to the right. We should note that −b and c both being a distance 1 from a is necessary; we must also check that it is sufficient. But it is now clear from the geometry that the segment connecting a to c is parallel to that connecting −b to 0, so we do indeed have a + b = c.
Algebra becomes geometry
We can now use this geometry to find b and c given a: for if we let α, β, γ be the arguments of a, b, c (or equivalently, the natural logarithms of x, y, z), the relationship between α, β and γ is easily seen. Since each of a, b, c has unit modulus, the triangle with vertices at 0, a and c is equilateral, and so all internal angles are π/3. It immediately follows that c has argument γ = α + π/3 and that b has argument α + 2π/3. chalkdustmagazine.com
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chalkdust From a similar picture, with the roles of b and c reversed, the other solution is given by γ = α −π/3 and β = α − 2π/3. Assembling this, we see that given an arbitrary positive real number x = eα , there is a pair of solutions to xi + yi = zi given by y± = eα ± 2π/3 = xe ± 2π/3 , z± = eα ± π/3 = xe ± π/3 . Given this, we can now see that these real solutions have the following properties, which we found interesting: 1. Any solution x, y, z satisfies the relationship xy = z 2 . 2. Since eπ is transcendental, it follows that the ratio of any two of x, y, z is transcendental, and so at most one of x, y, z is algebraic. 3. Furthermore, since there is a solution for any positive choice of x, and since there are only countably many algebraic numbers, for almost all solutions the three values x, y, z are all transcendental. 4. Finally, this tells us that not only are there no integer solutions, which is equivalent to there being no rational solutions, but that the real solutions are not only irrational, they are very irrational, in the sense that they are (almost all) transcendental.
Fit the fourth: the mathematical endeavour In closing, let’s think about how this small investigation fits into what mathematicians do all day. Some people think of mathematicians as people who try to solve problems using mathematics; others think of them as people who try to prove theorems. Very crudely speaking, we could think of these activities as applied and pure mathematics respectively. But both of these are really different approaches to a bigger objective: finding out something interesting. What we went through above is a small version of this, which shows the typical features. You start off with something that you want to understand beer, you find out about it—and sometimes what you find out is that somebody else has already understood it—and you refine your question until you have something that you can understand beer. Then you share it with other people who find it interesting. At least, I hope we’ve done the last part! Robert J Low and Thierry Platini Rob and Thierry both teach mathematics at Coventry University. mtx014@coventry.ac.uk (Rob)
@RobJLow (Rob)
ab3334@coventry.ac.uk (Thierry)
My least favourite number
ε
TD Dang
I hate ε, but only a small amount.
ε/N 31
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SLIDE RULES: the early calculators Duke, CC BY-SA 2.5
Bernd Ulmann
B
it or not, I never leave home without my trusty slide rule in my pocket. In the years before 1970, this would have been totally normal for any engineer, with the slide rule being the archetypal symbol of the engineering profession, much like how the stethoscope remains that of the medical profession. However, with the advent of the pocket calculator, the slide rule has completely vanished from public view and its demise is a wonderful example of a paradigm shi, as described by Thomas S Kuhn in The Structure of Scientific Revolutions. I oen get asked “What are you doing with that thing?” when I grab my slide rule to convert miles to kilometres or, once I’ve finished refuelling, to calculate my fuel economy. Most of my students have never seen a slide rule and are at first quite incredulous when I show them my Faber–Castell 62/82N. Not that I’ve ever managed to convince one of them to switch to a slide rule, but at least I normally manage to instil some interest in them for these mathematical instruments. At the heart of the typical slide rule are logarithms. The first logarithm table was published in 1614 by John Napier of Merchiston (1550–1617) in Mirifici logarithmorum canonis descriptio. Other mathematicians, such as Jost Bürgi, also worked on logarithms and published tables of them in the following years. Logarithms are extremely useful when it comes to the multiplication or division of numbers as they allow you to replace the multiplication of two values, x and y, by the sum of their respective logarithms, xy = blogb (x)+logb (y) , chalkdustmagazine.com
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(∗)
chalkdust where b represents the base of the logarithm. Today, the most common bases are b = 10 (the socalled decadic or Briggsian logarithm, named aer Henry Briggs, who championed its use); b = e, where e is Euler’s number (yielding the natural logarithm); and b = 2 (the logarithmus dualis), which is especially useful in computer science. For our purposes, the base itself is not particularly important and, even if it were, the logarithm to some given base b can easily be determined from that to a different base b′ using the formula logb′ (x) logb (x) = . logb′ (b)
The early days of slide rules Performing multiplication or division using a logarithm table was, however, still quite time consuming and prone to errors. Firstly, no table was completely error-free as these were calculated and typeset manually. Secondly, looking up values in hundreds of densely typed pages added further complexity. The following picture shows part of a logarithm table from 1889:
As an example, we will try to multiply 6305.7 by 6310 using this table. Firstly, one has to determine the logarithms of these numbers. To do this, these values must be wrien in the form mbE , where b is the base we are working with, E is the exponent and m is the normalised mantissa, satisfying 1 ⩽ m < b. Working in base 10, this yields 6.3057×103 and 6.31×103 . If we were to work in binary instead of in the decimal system, this process is exactly the way in which floating point numbers are represented in modern digital computers. The table is then used to look up the logarithm of the mantissa. For the mantissa corresponding to 6305.7, we go to the entries associated with the 6300 numbers, read down until we reach ‘05’ and then across to the column headed ‘7’, which is the decimal part. We find the entry to be ‘7333’, to which we add the prefix ‘799’ (given in the top le hand entry of the table), to obtain the answer log10 (6.3057) ≈ 0.7997333. In a similar fashion, we get log10 (6.31) ≈ 0.8000294. So, taking into account that log10 (103 ) = 3 and recalling that logarithms turn multiplication into addition, we have log10 (6.3057 × 103 ) = log10 (6.3057) + log10 (103 ) ≈ 3.7997333, log10 (6.31 × 103 ) ≈ 3.8000294. 35
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chalkdust Again, referring back to equation (∗): if we are multiplying two numbers, we want to add their logarithms, and so the sum of these values is 7.5997627 and our answer is 107.5997627 , which we must now convert to something useful. The leading seven is the exponent, so we instantly know that the result must be of the order 107 . What we need now is the antilogarithm of 0.5997627, which the logarithm table (not shown here) tells us is approximately 3.9789. So the final result is 6305.7 × 6310 ≈ 3.9789 × 107 = 39789000, which is prey close to the exact solution 39,788,967.
Slide rules: rise and fall As you can tell, this technique is prey laborious and gave rise to various mechanisations that finally led to the slide rule. In 1624, Edmund Gunter (1581–1626) published the brilliant idea of dividing a ruler logarithmically. In the simplest case such a ruler had a scale running from 1 to 10 so that one could work directly with normalised mantissae. To multiply two numbers, their exponents would be determined and added, and then the logarithms of the two normalised mantissae were added by means of dividers, which were used to mark out the lengths on the ruler corresponding to them. The use of dividers meant that the procedure was still a bit cumbersome. It was greatly simplified by the idea of using two identical rulers, divided logarithmically, which could slide along next to each other. There is still some debate among historians about who actually invented this early slide rule, but most experts give that honour to William Oughtred (1574–1660). The following picture shows a simple slide rule from the early 20th century with only a few scales. The two boom scales (quite atypically denoted by E and V), are shown calculating the product 2 × 3.
The scale E is moved across so that it starts at the point 2 on V. The length 3 on E corresponds to the point 6 on V, which is the answer. What we have done is added the length of log10 (2) (on V) to the length log10 (3) (on E) to obtain the correct result. As all operations are done with normalised mantissae, a slide rule allows operations with numbers of arbitrary size—one only has to take care of the exponents manually. For multiplication, the exponents are added, while they are subtracted for division. Obviously, to divide two numbers one just has to subtract the lengths of their respective logarithms, so the picture above could also be seen as the calculation 36 = 2. One neat feature of a slide rule is that it not only yields a single result but a complete table of results. In the example above we not only got the single answer 6 but a table showing all multiples chalkdustmagazine.com
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chalkdust of two up to 10 (the range being restricted by the length of the slide rule). This is extremely handy when one has to convert miles to kilometres—as I oen have to do since my car was imported into Germany and still measures distances in miles. Obviously, a slide rule can’t help with the addition and subtraction of numbers, which has to be done manually, but it can help with much more difficult calculations than just multiplication and division. The following picture shows a more complex slide rule than the one used just. Here, both front and back are filled with scales, hence the name duplex slide rule:
The slide rule below, a Faber–Castell 2/83N, is one of the last and most powerful models ever built. Its front and back feature 30 scales: simple scales for multiplication to more complicated ones like cube scales, several trigonometric scales and double logarithmic scales, which can be used to √ compute x y and y x . Typically, a slide rule also has a movable slider with at least one hairline covering the whole vertical space between the topand boommost scale to facilitate readout. This slider oen features additional hairlines that are spaced at carefully selected distances to the le and right of the main hairline and allow the instant multiA Faber–Castell 2/83N plication of a value by fixed constants. Engineering slide rules, for example, typically have a special-purpose hairline to convert between horsepower and kilowas. The 2/83N slide rule shown above has many such hairlines, including one allowing you to compute log2 (x) using the double logarithmic scales: prey handy in computer science, but a bit ironic given that computer science more or less killed the slide rule. The idea of computing by adding or subtracting lengths extends readily from basic mathematical operations to more complex tasks. The picture on the far le shows a highly specialised slide rule still used in avionics to calculate things like fuel burn, ground speed and to update the estimated time of arrival: the E6B flight computer. And, finally, the picture on the near le shows one of the rarest and strangest slide rules: a circular slide rule, dating back to the days of the cold war and roughly estimating the number of casualties and catastrophic effects of a nuclear strike. Flight computer (le); nuclear slide rule (right)
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chalkdust It is a pity that these marvels have now become largely forgoen. They remain very useful tools— not only for the automobile enthusiast like me, who wishes to convert between miles and kilometres or quickly calculate fuel consumption data—but also for every student who seeks an intuitive understanding of logarithms. So if you ever find a slide rule at a flea market, buy it! You will see that it is a thing of beauty and a joy to use. Bernd Ulmann Bernd is professor of business informatics at the FOM University of Applied Sciences for Economics and Management in Frankfurt-am-Main, Germany. His primary interest is analogue computing in the 21st century. If you would like to know more about analogue computing, you can visit his website. analogmuseum.org
Send us your articles! Got an exciting idea? Want to share it with us? We publish new articles every week online, as well as every six months in our printed magazine. So send us your articles! All submissions are welcome, and if you’re new to writing, we’re happy to work with you to hone it into something special. Who will your article be rubbing shoulders with in the next issue of Chalkdust? Get in touch with us at
contact@chalkdustmagazine.com
My least favourite number
Googolplex
Belgin Seymenoğlu
Ever heard of googol (10100 )? You have? What about googolplex (10googol )? No? Well, while a googol is a one followed by a hundred zeroes, a googolplex is a one followed by googol zeroes. So it can be wrien as 100 1010 , which is not nice at all. Most calculators already can’t handle googol, which is bad enough, but googolplex takes it even further. One morning I asked my computer to print googolplex; it never got back to me. (log10 log10 10)/10
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chalkdust
a slide rule You will need: One copy of Chalkdust, scissors
Instructions
1 2
Cut out the scales on the left and right of this page.
3
Get a new copy of Chalkdust, because this page is now cut up so you can’t read what’s on the other side.
Do lots of fun calculations.
Tube map platonic solids and Fröbel stars: more How to make at chalkdustmagazine.com 39
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Which mathematical object are you?
START AS
HUMAN What matters? Whatâ&#x20AC;&#x2122;s inside or outside?
OUTSIDE YOU ARE NOW
a 2-manifold in â&#x201E;?3
â&#x201E;?2
How do you feel locally?
YOU ARE NOW
a set of points in â&#x201E;?3
SATISFYING CAUCHYâ&#x20AC;&#x201C; RIEMANN EQNS
ITâ&#x20AC;&#x2122;S MORE COMPLEX â&#x201E;&#x201A; THAN IDEAL & THAT IN MY Do you feel PRIME more than a set of points? A SET OF Do you CONNECTED daydream? Iâ&#x20AC;&#x2122;M MORE POINTS, AB STRACT IN FACT TH AN THAT YOU ARE YOU ARE NOW
How much make-up do you use?
NOT MUCH
NO
an at most 2-differentiable 2-manifold in â&#x201E;?3
Re (â&#x201E;&#x201A;)
BECOME A GROUP!
YOU ARE NOW
YOU ARE NOW
Im (â&#x201E;&#x201A;)
â&#x201E;¤P
â&#x2030;&#x2026;
Ď&#x20AC;1(YOU)
YOU ARE NOW
â&#x2030;&#x2026;
an infinitely differentiable 2-manifold in â&#x201E;?3
â&#x201E;¤(P)
Ď&#x20AC;1(YOU)
YOU ARE NOW
NOW
Ď&#x20AC;1(YOU)
YOU ARE NOW
YOU ARE NOW
a 3D topological space
Ď&#x20AC;1(YOU)
YOU ARE NOW
YES
LOTS
ITâ&#x20AC;&#x2122;S MORE COMPLEX THAN THAT
BOTH
LIKE
INSIDE
YOU ARE NOW
a group CHALKDUST FUNCTOR
a set of obnoxious points in
YOU ARE NOW
PROJECT ONTO ORIGIN
a đ?&#x201C;&#x2019;ategory of points in â&#x201E;?3 YOU ARE
YOU ARE
(0, 0, 0)
the group of order one
chalkdustmagazine.com
YOU ARE NOW
â&#x201E;?3
(â&#x201E;?, +)
TRIVIAL HOMOMORPHISM
HELL YEAH
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Flowchart explanation flowchart
START AS NON-UNDERSTANDING HUMAN
.
Do you understand manifolds? NO YES Have you seen the C-R eqns? NO YES Happy with categories?
NO YES
NO Know your ℤs?
YES Think of potatoes. If you zoom in enough, the surface looks like ℝ2.
Do you want to understand the flowchart? Get it?
NO YES
NO Functions that are differentiable on ℂ satisfy the Cauchy–Riemann equations.
OK?
YES
NO
These are collections of objects and ‘morphisms’. For example, NO if your objects are sets, your morphisms are functions. Think of functors as morphisms between categories. Yes? ℤP is the integers modulo p (see pages 42–47). ℤ(P) is the set of fractions where the denominator is not divisible by p. It’s called the ‘localisation of ℤ at prime ideal (p)’.
YES
ℤℤℤ?
YES
NO
YES
? π1(?)
YES
Called the ‘fundamental group’, it’s the set of loops starting/ending at a point in a given topological space. We can ‘add’ loops together by going around one, and then going around the other. Then finally we say that loops are equivalent if you can squash one into another without breaking it. Then it’s a group! π1(
NO YES
)?
The heart-warming message of this flowchart is that at the end of the day, we’re all the same: isomorphic to the group of order one! Pictures: Wikipedia logo: Wikimedia Foundation, CC BY-SA 3.0
My least favourite number
–1/12
Matthew Scroggs
What is 1 + 2 + 3 + 4 + 5 + · · · ? Obviously, you’ll get to +∞. If anyone ever tells you that the answer is –1/12, they are being silly. –12/10
Did you know... …that your article could appear in issue 6? More info at 41
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chalkdust
The simplest difficult task Wojtek Wawrów
M
of us have heard of the RSA algorithm and how it’s very useful for cryptography. In order to crack it we need to be able to factor large numbers, but experience has told us that the problem of factorisation, while very simple to describe, is very difficult to do in practice. Yet there exists a problem that, though it might sound even simpler, is just as difficult.
Basic arithmetic We all know how to add or multiply two numbers. In particular, we can multiply a number by itself, which we call ‘squaring’. In other words, if I give you a number, say 127, then I’m sure you will be able to tell me, possibly aer some time, that its square is 16,129. Piece of cake. But consider now the inverse problem: suppose I gave you the number 33,124. Can you tell me what number this is the square of? In other words, what is the square root of 33,124? The answer, √ √ of course, is 33,124 (or − 33,124, but we’ll get back to this point later). However, this feels a bit like cheating, so let’s suppose that I want a more explicit answer, without the square √ root. This might not make much sense in general—for example, what answer more explicit than 2 could I possibly expect? So let’s make a deal: if I ask you for a square root of a number, I will make sure that the answer is an integer n. Now, how to approach this problem? A rather naive method would be to just take the numbers below, say, 1000 and square them all to look for a hit. This will, of course, take a horrendous amount of time, especially with larger chalkdustmagazine.com
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chalkdust numbers. But consider the following method: we first try taking some number that we expect to be about the size of n. Let’s start with 200. We find that 2002 is larger than 33,124. So this is a miss—our square root n is not 200. But we get to know a bit more: n is strictly less than 200. So let’s try a smaller number, say 170. Then 1702 < 33,124, so n is larger than 170. We can now try some number between 170 and 200 to find some tighter bounds for n. Aer five or six more such guesses, we will finally find out that n = 182. Maybe this isn’t the most effective method, but it’s still beer than trying all 181 smaller numbers! At this point, let’s note a very important thing: to find the solution we made use of the structure of the integers, which was not required in the statement of the problem. To be more precise, we used the ordering of the integers in order to learn something about n given only n 2 . This suggests that the problem might become harder if we consider it in a system that doesn’t have this structure.
Modular arithmetic A simple example of a system that has multiplication (and addition, for that maer) is the set of integers modulo some natural number N > 1. In modular arithmetic, we say that every two integers a, b are the same if they differ by a multiple of N. More precisely, we say that a and b are congruent modulo N, wrien a ≡ b (mod N ), if and only if the number a − b is a (possibly negative) multiple of N. We can identify numbers modulo N with the possible remainders when dividing by N, ie with the numbers {0, 1, . . . , N − 1}. Now we can state the problem we wish to consider: given the integers y and N, find an integer x such that x 2 ≡ y (mod N ).
Barry Mangham, CC BY-SA 3.0
For example, let’s say that y = 2 and N = 7. Then you can see that 32 ≡ 2 (mod 7), so 3 is a valid, though not unique, answer to our problem. On the other hand, if I gave you y = 2 and N = 5, you will eventually realise that there is no solution. Like before, I won’t give you such numbers. Although the question of determining the existence of square roots is very interesting in itself, here we will restrict ourselves to finding the square roots under the assumption that they exist.
Modular arithmetic is sometimes called clock arithmetic because clocks count modulo 12
As with many things in mathematics, this problem simplifies quite a bit when we consider prime numbers, so let’s assume for now that N is a prime number p. If p = 2, then every number is its own square, which is a rather boring case, so we will suppose that p is odd. We only know that y ≡ x 2 modulo p, and we somehow want to extract the value of x from it, so we want to get rid of the square. We can try exponentiating y further. Since we don’t know how to take arbitrary powers of numbers modulo p, the exponent has to be an integer. This wouldn’t work if we were solving y = x 2 in the integers: y k will always be bigger than x for k > 1. But in modular arithmetic we have at least one tool that makes this idea feasible: Fermat’s lile theorem, which states that x p ≡ x (mod p). We would therefore like to try and raise y to the power p/2, 43
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chalkdust but sadly p/2 is not an integer. The closest integers are (p − 1)/2 and (p + 1)/2, but taking y to these powers gives us y(p−1)/2 ≡ (x 2 )(p−1)/2 ≡ x p−1 ≡ 1 (mod p), y(p+1)/2 ≡ (x 2 )(p+1)/2 ≡ x p+1 ≡ x p−1 · x 2 ≡ x 2 ≡ y (mod p). This doesn’t seem to be of much help… but wait! Since y(p+1)/2 is y, then y(p+1)/4 squared will be y—provided that (p + 1)/4 is an integer, which is equivalent to saying that p + 1 ≡ 0 (mod 4), or p ≡ 3 (mod 4). Moreover, this gives rise to a rather fast algorithm—the (possibly large) exponent (p+1)/4 might make you think that we are bound to make (p+1)/4 multiplications, but thankfully modular exponentiation can be made very effective using so-called repeated squaring. So we have figured out a way to find square roots in, essentially, half of the cases! Is there a way to find square roots in the remaining half: modulo primes p ≡ 1 (mod 4)? Indeed there is! A beautiful algorithm was discovTrial and error is the best nonranered sort-of independently by Alberto Tonelli in 1891 dom algorithm we have. and Daniel Shanks in 1973, with the explanation that Shanks’ “tardiness in learning of these historical references was because [he] had lent Volume 1 of Dickson’s History [of the Theory of Numbers] to a friend and it was never returned”. The algorithm lets us find square roots modulo primes p that are equal to 1 modulo 4 almost as quickly as the one described above. It has one drawback though: in order to use it, we need to find a number that is not a perfect square modulo p. It might sound easy enough: it can be shown that half of the numbers indivisible by p are not squares, which suggests that a randomised search could be efficient (testing whether a number is a square modulo a prime can be done quickly using a concept called Euler’s criterion). Trial and error is the best nonrandom (ie deterministic) algorithm we have, so its computation time depends on the size of the smallest non-square. Very surprisingly, the only useful bounds on it have to be proven under the assumption of very complicated number-theoretic conjectures like the generalised Riemann hypothesis! Given that, we won’t pursue this problem further, but we highly recommend that interested readers look up the Tonelli–Shanks algorithm.
Factorisation There is certainly a lot more to be said about square roots modulo prime numbers, but given that we have already found some rather efficient algorithms, we will now move on to finding square roots modulo composite (non-prime) numbers. First, let’s try working with a prime power, say N = pe . There is a very useful result known as Hensel’s lemma which lets us “li” a root of a polynomial in pe to a root in pe+1 . The idea is the following: if we know that x 2 ≡ y (mod pe ), we can write x 2 = y + ape . Now we consider numbers of the form x + bpe and we hope that one of them will give us a root in pe+1 . If we square them, we get x 2 + 2bxpe + b 2 p2e ≡ y + (a + 2bx)pe (mod pe+1 ). Now it’s a maer of choosing b such that a + 2bx is divisible by p. This will give us a desired solution as long as p does not divide y and p ̸= 2. It is a rather instructive exercise to consider what happens in the remaining cases. Things get a lot more interesting when N has more prime factors, say N = pe11 ...pekk . First, using the above method, we can find square roots xi of y modulo pei i for each i. A theorem evocatively chalkdustmagazine.com
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chalkdust known as the Chinese remainder theorem then states that there is a number x satisfying, for all i, x ≡ xi (mod pei i ). If we look at x 2 , we find, for each i, x 2 ≡ xi2 ≡ y (mod pei i ), from which it follows that x 2 ≡ y (mod N ). This looks nice and all, but note that we have used, in a very essential way, the factorisation of N, which, as we know, is a very difficult problem. Maybe there is a way to find square roots modulo N without needing to factor N ? And here is the point we’ve been building towards: finding square roots modulo composite numbers is just as difficult as factorisation. To see why this is true, we need to find a way of finding nontrivial factors of N provided an algorithm for finding square roots modulo N. First, we may assume that N is odd and is not a perfect power, otherwise factoring it is moderately easy. If this holds, then we can write N as a product PQ of two relatively prime odd numbers (two numbers that have no common factors). Now suppose that y ̸≡ 0 (mod N ) has a square root x (we don’t care what this square root is, we just assume it exists). If y and N have a common factor, then this common factor will be a nontrivial divisor of N and we have factored N. Now assume that y, N are relatively prime. Clearly, −x is also a square root of y, but the cool bit is that there will be more square roots. How can we use such additional square roots to factor N ? Suppose x1 , x2 are two roots of y modulo N such that x1 ̸≡ x2 , x1 ̸≡ −x2 (mod N ). Since x21 ≡ y ≡ x22 (mod N ), N must divide x21 − x22 = (x1 − x2 )(x1 + x2 ). But, from our definition of x1 and x2 , N divides neither factor on the right, so N and x1 − x2 (as well as N and x1 + x2 ) must have their highest common factor strictly between 1 and N, which lets us find a nontrivial factor of N using the Euclidean algorithm! This idea underlies all modern factorisation algorithms, including Shor’s quantum version. Only one question remains: how can we use our square root finding algorithm to generate two distinct roots of a number? The answer is incredibly simple: take a number y that we already know a square root of, and hope that the algorithm gives you a different root. We knew which square root we squared to get y, but the algorithm doesn’t know which of the (at least) four square roots we used. So by choosing a number randomly, squaring it and asking our algorithm for a square root, we have at least a 50% chance of finding another square root, thereby leing us factor N.
Cryptography This discussion tells us that finding the square of a number is easy, but finding a square root modulo N Alice and a dodo that may or may not be is hard, unless we know the factorisation of N. These called Bob observations were made by Rabin in 1979, when he realised that they could be used to construct an extremely simple public-key cryptographic system, nowadays known as the Rabin cryptosystem. 45
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chalkdust As is usual in cryptography, we will describe this technique using a scenario in which Alice wants to send a secret message to Bob, but this message has to pass through a public channel. First Bob chooses two large primes p, q that are congruent to 3 (mod 4). In practice, these numbers will be tens of digits long. He will keep these primes secret, but he will make N = pq public. Aerwards, Alice takes her secret message and encodes it as a number m, which we assume is smaller than N and relatively prime to it, and computes the remainder of m2 when dividing by N, calling it M. M is the encrypted message that can now be sent to Bob. Since Bob knows p and q, he can find square roots of M modulo these primes by computing M(p+1)/4 (mod p), M(q+1)/4 (mod q) (this is why we assumed that p, q are 3 (mod 4)), and finally, using the Chinese remainder theorem, we can combine these square roots to get a square root modulo pq = N. At this point we are almost done. The only remaining question is, how do we know that this square root is m? And here is the issue: we don’t.
Ambiguity Earlier, we established that M will have at least four distinct roots modulo N. Actually, we can show that it has precisely four roots. If m were a message that could reasonably be distinguished from a random string of characters, eg if it were a sentence wrien in English and then encoded as Ascii, there would be a high chance that Bob will be able to figure out which of the roots is Alice’s message. However, in general, there is a very small, but unfortunately nonzero, chance that two of these roots will encode equally valid messages, for example if we are trying to send a large numerical value. Of course, Alice hasn’t provided enough information to let Bob unambiguously find out what the message is. But maybe Alice can provide some additional information that will enable Bob to identify the message? If so, can she do this in a way that will not make it easier to break the algorithm? Let’s write the four roots of M as m, N − m, x and N − x, where x ≡ −m (mod p) and x ≡ m (mod q). We can easily remove half of the ambiguity: note that precisely one of m and p − m is odd; and precisely one of x and p − x is odd number. Therefore if Alice specifies “my message is odd”, Bob will be able to remove half of all the possibilities, and at the same time Alice won’t make it any easier to find the square roots. Specifying whether the message lies in {m, p−m}(or) {x, p−x} is more difficult. The simplest way to solve this problem is to use the Jacobi symbol ba , where a and b are integers, and b is positive ( ) and odd. We won’t need its most general definition, but we need to use the following facts: ba is ±1 if a, b are relatively prime; for primes p ≡ 3 (mod 4), () ( a ) (a)(a) ( ) ( ′) () ( ) = − pa and pq = p q ; if a ≡ a′ (mod b) then ba = ab ; and finally, ba can we have −a p be efficiently calculated even if we can’t factor a or b, thanks to so-called quadratic reciprocity. In particular, we find that ( ) ( )( ) ( ( )) ( ( )) ( )( ) ( ) ( ) m m m −m −m −m −m −m N−m = = − − = = = N p q p q p q N N ( x ) (N−x) and similarly N = N ; but ( ) ( )( ) ( )( ) ( )( ) ( ) x x x −m m m m m = = =− =− . N p q p q p q N chalkdustmagazine.com
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chalkdust So Alice can specify in which ( )of {m, p − m} or {x, p − x} her message lies using the value of it being different to that of Nx ! And again, this won’t help anyone find the square roots.
(m ) N
,
Therefore, we can uniquely specify which is the correct square root out of four possibilities using precisely two bits of information.
Comparison with RSA At the beginning, I intentionally made a not-so-precise statement that in order to crack RSA, we need to factor large numbers. This isn’t false, but it isn’t known to be true either—what we actually need is to be able to find the e-th roots modulo N; but in the case of RSA, the root is known to be unique. This is known as the RSA problem. As opposed to the above square root finding problem, the RSA problem is not known to be equivalent to factorising, so it is consistent with our current knowledge that the RSA problem might be solved without an efficient factorisation algorithm. Rabin’s cryptosystem is known to be secure, provided we believe factorisation is hard. Beyond that, apart from the disambiguation of No RSA-breaking algorithm is roots, Rabin’s cryptosystem is conceptually simpler more efficient than factorisation. to the RSA cryptosystem, since it only requires us to square numbers and find square roots. Also, encoding messages is about as simple as is conceivable, and the generation of public and private keys (N and p, q respectively) is very simple as well. In RSA, the generation of keys requires the further step of finding suitable exponents that we later use, and the encryption exponent is bound to be larger than 2. As for decryption, in RSA the decryption exponent can be prey much any number between 3 and (p − 1)(q − 1), so it is usually much larger than the exponents used in Rabin’s system. The fact that we have to do two exponentiations and then later use the Chinese remainder theorem makes the decryption take roughly the same amount of time in the two systems. However, the disambiguation issue is what has decided which of the two algorithms became dominant for so many years. The need to compute the Jacobi systems introduces an additional level of complexity to the algorithms, which isn’t worth the implementation effort. Although finding square roots is provably as hard as factorisation, in practice what maers is the algorithms we know, and no RSA-breaking algorithm is more efficient than factorisation. Regardless, both of these algorithms are quickly becoming obsolete: elliptic curve cryptography provides us with more secure systems using smaller private and public keys. But that’s a story for another time… Wojtek Wawrów Wojtek is an undergraduate mathematics student at Adam Mickiewicz University in Poznań, Poland. He is fond of number theory and algebra, but no field of maths is odious to him. wojowu.students.wmi.amu.edu.pl
Did you know... …that it is impossible to comb the hair on a hairy ball flat without leaving a tu? 47
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L
Ma Parker’s drinking straws held together by pipe cleaners, this is an easy way to explore the symmetry of the four-dimensional companion of the cube without fancy modelling soware or a rapid prototyping machine. The wireframe oblique projection uses 24 sheets of A4 paper to create two classic origami cubes placed in parallel and joins their vertices with slanted beams. In a real tesseract these connections are made through a fourth axis but unfortunately I didn’t have the time to rip a hole into another dimension. Nevertheless, the model is a simple way in which to begin to understand the shape as a continuation of the dimensions we already experience. A point is to a line as a line is to a square, a square is to a cube as a cube is to a cubic prism.
The image on the right, which appears frequently in pop culture, is the perspective projection. The cube is projected away into 4D space and diminishes in volume just as the rear face of a cube appears to have a smaller area. However, the oblique projection remains my favourite since it somewhat resembles a rhombic dodecahedron, which is also the vertexfirst projection of this hypercube. In other words, it is what we would see if one were to fall corner-first through our three dimensional ‘plane’. This particular way of falling exposes the maximum volume, just as a particular plane slice of a cube can expose a maximum area. Coincidentally, the number of vertices of the rhombic dodecahedron at a distance of 0, 1, 2, 3, 4 from any given vertex respectively, are 1, 4, 6, 4, 1—just as in the fourth row of Pascal’s triangle.
A 3D perspective projection
Schläfli symbols used by mathematicians are a compact, recursive description of regular polytopes and tessellations. In two dimensions, a triangle is simply {3}, a square is {4} and all regular p-sided polygons are {p}. chalkdustmagazine.com
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In three dimensions, polyhedra with q regular p-sided polygons around each vertex are represented as {p, q}; so cubes can be described as {4, 3}, for example. A tesseract, {4, 3, 3}, has 3 cubes around an edge. The Dali cross, Corpus Hypercubus, a net of the hypercube, uses eight cube cells, giving rise to the 8-cell as another name for the shape. Just as in the net of a cube, the tesseract’s faces must be lied one dimension higher in order to fit them together. Once constructed, each vertex of the tesseract has four edges, giving it the vertex figure of a regular tetrahedron. With the corners removed to expose new faces you would be le with its dual polytope, a 16-cell {3, 3, 4}.
A cubic network
A tesseract, and its parallel projection
A tesseract network
Hypercubes sound like the avant-garde invention of a mad mathemagician, but their geometry is being used in parallel computing. Processor networks like IBM’s Blue Gene are able to execute many calculations simultaneously from the bit to the task level, reaching a number of floating point operations per second of the 15th order of magnitude. Hypercube networks also have the advantage of being decentralised, meaning that there is no single node which could cause them to fail. A hypercube of N dimensions has 2N vertices, each of degree N. The nodes are conventionally labelled in binary with N digits, where directly adjacent nodes differ by one binary digit. The longest path between two nodes in such a network is of length N. You can find out more about shapes in higher dimensions from the Numberphile channel on YouTube, the MathWorld website and Ma Parker’s book Things to Make and Do in the Fourth Dimension. Lulu Beatson Lulu is an A-level student and a wannabe mathemagician. passionfruit-studies.tumblr.com
lulu.beatson@gmail.com
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Significant figures Florence Nightingale, statistician Pietro Servini
Pietro Servini
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her copy of Thomas à Kempis’ fieenth century The Imitation of Christ is the inscription, in her own writing, “I only wish to be forgoen”. Although remaining a very private figure throughout her lifetime, this was not a desire that could ever be fulfilled. Since her death on 13 August 1910, at the age of 90, Florence Nightingale’s fame and the legend of the lady with the lamp has failed to dim. That her name has become a synonym for nursing is, however, misleading; and the public perception of her as a nurse does a great disservice to her life as a pioneering statistician.
The lady with the lamp The legend of Florence Nightingale as the lady with the lamp was born in a filthy hospital in the midst of the Crimean war, which was fought between the Russians on the one side and a coalition of Oomans, British, French and Sardinians on the other. Beginning in October 1853, it trundled on until March 1856 with, in the words of the historian Alexis Troubetzkoy, few other wars rivalling it in “greater confusion of purpose”. On 21 October 1854—aer reports in newspapers of the terrible conditions in which the British wounded were being kept led to a public outcry—Florence was put in charge of a team of nurses, many coming from religious houses, and sent to Turkey, where she was to be responsible for the nursing at the military hospital of Scutari, a former army barracks that had been built over an enormous cesspit. The idea was revolutionary: the army establishment was vehemently opposed to the presence of female nurses in its hospitals. However, Nightingale chalkdustmagazine.com
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chalkdust had the support of Sydney Herbert, at the time secretary of state for war and a staunch ally of Florence until his death in 1861. On 24 February 1855, the Illustrated London News published an engraving of Florence with her lamp, the only glow of light and hope in a gloomy room, filled with wounded men. In the same month, John MacDonald, in an article for the Times, wrote that “when all the medical officers have retired for the night and sickness and darkness have seled down upon the miles of prostrate sick, she may be observed alone, with a lile lamp in her hand, making her solitary rounds”. And so the image stuck.
Wellcome Library, CC BY 4.0
A coloured version of the engraving in the Illustrated London News
But despite this public perception of Florence, she did very lile actual nursing while in the Crimea and, indeed, throughout her whole life. In fact, the words of her sister, Parthenope, apply as much now as they did then: “what Florence does and what she is, is most faintly conceived of in England […]. The public there generally imagine her by the soldier’s bedside, where doubtless she is oen to be found, but as she herself said, how satisfactory, how easy if that were all. The quantity of writing, the quantity of talking is the weary work, the dealing with the selfish, the mean, the incompetent.” Much of her job involved the day-to-day running of the hospital, the management of the nurses (some of whom fell somewhere on the scale of hopelessness), the sweet-talking of the sometimes recalcitrant doctors and the constant bale against bureaucracy and obstructionism in the offices of the purveyor and holders of the purse-strings back in London. The legend of Florence Nightingale was undoubtedly born in the misery of Crimea’s hospitals. But so too was her burning desire for social change, beginning with the medical treatment of Britain’s soldiers. To fight her bales, she turned to the relatively new science of statistics.
Florence Nightingale: before the legend In an age when girls were rarely educated, Florence was lucky to have had a father, William Nightingale, who believed that everyone had a right to an education. Her early learning consisted of a wide range of subjects— chemistry, geography, physics, history, languages (classical and modern)— but only very basic mathematics. This changed when she managed to get hold of a copy of Euclid’s Elements, which once upon a time was virtually a prerequisite for serious mathematical study. It was her aunt Mai (a prominent character in the story of Florence’s life) who petitioned Fanny, Florence’s mother, to hire a mathematics tutor for her daughter. As with Wellcome Library, CC BY 4.0 Mary Somerville (see pages 61–63), whose On the Connexion of the PhysiFlorence in 1856 cal Sciences led Florence to analyse the moon’s path in detail, this request met with a considerable amount of initial opposition, until a tutor was finally obtained. There is a possibility that one of Florence’s tutors was James Joseph Sylvester (1814–1897), who made significant contributions to the theory of matrices, among other things, but no documentary evidence of this exists. 51
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chalkdust The Nightingale parents’ approach to their daughters’ education was, however, one of education for education’s sake. A relatively wealthy family, there was no need for Parthenope and Florence to apply their education in any meaningful way, or for them to break away from the passive roles that were assigned to women by societal convention. For Florence, however, this was unacceptable; her desire for action is perhaps best reflected in her aitude to God: although highly religious, she believed that humanity had a duty to act to sort out its own problems, rather than relying on petitions to a supreme being—a belief perhaps influenced by an experiment she carried out at the age of six to determine whether or not God answered her prayers. She found one of her first sources of inspiration in the work of Adolphe etelet (1796–1874), who had founded the Royal Observatory of Belgium but was more famous for developing his concept of l’homme moyen, the average man, whose behaviour could be characterised by the mathematical laws of probability. He carried out extensive surveys of mortality, suicide and crime rates: his 1835 book Sur l’homme et le développement de ses facultés, ou Essai de physique sociale (translated into English in 1842 as Treatise on Man) challenged the conventional consensus that individuals had a freedom of choice, showing that societal factors like education predisposed people to certain actions. His work was a groundbreaking—and controversial—embodiment of the statistical revolution that was underway.
Statistics in the nineteenth century Statistics was a relatively new science at the turn of the century, its increased use triggered by William Playfair’s (1759–1823) invention of graphical methods of representing data. States and ruling bodies had long collected information about the people who lived within their jurisdiction—the Domesday Book of 1086 was England’s first formal, surviving census—but its popularity grew throughout the course of the nineAn early bar chart by William Playfair teenth century and its use evolved from that of determining taxation to describing the welfare of society. Decennial censuses were introduced in 1801, with the first four being supervised by John Rickman, a high-ranking clerk in the House of Commons and a friend of the Nightingale family. In 1841, the census was expanded and took a form similar to the one used today: a record of all individuals present in a household on a certain night. Throughout government, various offices and departments began collecting statistics; and in 1837, the civil registration of deaths, births and marriages became mandatory. The wealth of new data that was becoming available and the opportunities it opened up is perhaps comparable to the current excitement generated by the promise of big data. It was now possible to apply the science of statistics to the fields of economy, politics, social problems, the condition of the poor and, of course, medicine. chalkdustmagazine.com
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Florence Nightingale, statistician Hospitals for much of the nineteenth century did not play much of a role in healthcare. The rich were tended to at home, the poor by their family, while the so-called ‘undeserving’ poor (those who were condemned as being too lazy to work) were sent to workhouses where they were forced to carry out menial labour. When they fell sick, they were treated in the workhouse infirmary, where conditions were more conducive to dying than recovering. Any hospitals that did exist were mainly for the ‘deserving’ poor—who had to present themselves with a leer of recommendation— and did not treat fever, serious conditions such as tuberculosis, cancer or smallpox, and did not deliver babies. Nurses themselves were largely religious sisters or domestic servants filling time between private posts. Either way, no training or experience was required; tales of drunkenness and sexual shenanigans between the hospital bedsheets were rife. However, this was beginning to change: industrialisation was leading to an influx of people in major cities and towns from more rural areas, increasing the pressure on the workhouses and hospitals. It was against this background that Florence Nightingale began her bale for an improvement in the provision of medical care. By the time of her death, government was well on the way along the path of reform that would eventually lead to the National Health Service (NHS) Act of 1946 and the establishment of the NHS in 1948. Her immediate legacy, however, was the creation of the first school for nurses to be connected directly to a hospital, St Thomas’. In her fight, Nightingale was ably assisted by the statistician William Farr (1807–1883), an authority on epidemiology, who had set up a system for recording causes of death. It was he who encouraged Florence to analyse the mortality rates of soldiers following the Crimean campaign, which led her to the finding that the death rate of soldiers, even in peacetime, was twice that of the civilian population and, even more shockingly, that the mortality of all British troops in the Crimea was two-thirds that of those who remained at home. The dilemma turned to how best to present this information. The wounded at Balaklava, by William Simpson
The Nightingale rose diagram Florence Nightingale understood the need to present her findings in a way that the public could instantly understand or, in her own words, to “affect thro’ the eyes what we may fail to convey to the brains of the public through their word-proof ears”. This led her to the creation of the Nightingale rose diagram, a collection of which was termed the coxcomb. Perhaps inspired by Farr’s use of circular diagrams in his work on deaths caused by cholera in London, the Nightingale rose diagram is an example of a polar-area graph, invented by the French statistician André-Michel Guerry (1802–1886) in 1829, which differs from Playfair’s pie chart in that sectors all have the same angle but differ in their radius, which varies according to the data. The coxcomb appeared in the report of the Royal Commission on the Health of the Army (1857), whose chairman was Florence’s ally Sydney Herbert and which reported on the medical fiasco of 53
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chalkdust the Crimean war and made recommendations for future reform. On Florence’s insistence, one of these recommendations was that a proper method of statistical accounting be introduced; a desire that was achieved with the appointment of a sub-commission in April 1861 tasked with the establishment of a statistics branch of the Army Medical Department.
The rose diagram, as it appeared in Martineau’s book
Florence’s wish for a wholesale reform of the War Office, however, didn’t come to fruition, despite her nearconstant pressure on the various people who filled the post of secretary of state for war during her lifetime. She published numerous pamphlets that included her diagrams depicting the needless death of soldiers from insanitary conditions and neglect. This included helping Harriet Martineau in the writing of her book, England and her Soldiers, which was published in 1859 and contained her coxcombs, bringing them to a public audience for the first time.
Her work to improve the conditions of the army led Florence to turn her aention to its health in India, where the death rate was particularly high. Her concern, however, then expanded to include the sanitary condition of the people and she applied her rigorous statistical approach to link it with the mortality rates of both Indians and British soldiers. Her lobbying led to the establishment of a Royal Commission on the Sanitary State of the Army in India, whose report (which again included some of Nightingale’s findings) was published in 1864.
The lady with the lamp, again Florence Nightingale wasn’t forgoen. She stands on a plinth in Pall Mall next to her good friend Sydney Herbert and in front of the Guards Crimean war memorial: three Guardsmen overlooked by Honour, arms outstretched, cast in bronze from cannons captured at Sevastopol. een Victoria, on opening the new St Thomas’ hospital on 21 June 1871, referred to Florence as “the lady whose name will always remain associated with the care of the wounded and sick”. Prophetic words. But perhaps we should review the legend of the lady with the lamp: in the misery of Scutari, it may have been her lamp, immortalised in her statue, that provided hope for the wounded; but for most of her work, it was statistics that was the lamp that lit the way to effective reform. It was her groundbreaking use of statistics, as well as her desire to present it in an accessible form, that should form a fundamental part of her lasting legacy. Pietro Servini Pietro is interested in history and sport. He also happens to be doing a PhD in fluid dynamics at University College London. If he can combine any two of them it makes him a happy man. pietroservini.com
Did you know... …that it is impossible to find four consecutive integers that are all the sum of two squares? chalkdustmagazine.com
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On the cover
Dragon curves Matthew Scroggs
T
a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper n times, then the edge will make an order n dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of this issue of Chalkdust, and in the header box above, is an order 10 dragon curve.
Le: Folding a strip of paper in half four times leads to an order four dragon curve (aer rounding the corners). Right: A level 10 dragon curve resembling a dragon
The dragon curves on the cover show that it is possible to tile the entire plane with copies of dragon curves of the same order. If any readers are looking for an excellent way to tile a bathroom, I recommend geing some dragon curve-shaped tiles made. 55
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chalkdust An order n dragon curve can be made by joining two order n − 1 dragon curves with a 90° angle between their tails. Therefore, by taking the cover’s tiling of the plane with order 10 dragon curves, we may join them into pairs to get a tiling with order 11 dragon curves. We could repeat this to get tilings with order 12, 13, and so on… If we were to repeat this ad infinitum we would arrive at the conclusion that an order ∞ dragon curve will cover the entire plane without crossing itself. In other words, an order ∞ dragon curve is a space-filling curve. Like so many other interesting bits of recreational maths, dragon curves were popularised by Martin Gardner in one of his Mathematical Games columns in Scientific American. In this column, it was noted that the endpoints of dragon curves of different orders (all starting at the same point) lie on a logarithmic spiral. This can be seen in the diagram to the right. Sadly, not all logarithmic spirals are this cool (see page 8). The endpoints of dragon curves of order 1 to 10 Although many of their properties have been with a logarithmic spiral passing through them known for a long time and are well studied, dragon curves continue to appear in new and interesting places. At last year’s Maths Jam conference, Paul Taylor gave a talk about my favourite surprise occurrence of a dragon.
Normally when we write numbers, we write them in base ten, with the digits in the number representing (from right to le) ones, tens, hundreds, thousands, etc. Many readers will be familiar with binary numbers (base two), where the powers of two are used in the place of powers of ten, so the digits represent ones, twos, fours, eights, etc. In his talk, Paul suggested looking at numbers in base −1 + i (where i is the square root of −1; for more adventures of i, see pages 28–31) using the digits 0 and 1. From right to le, the columns of numbers in this base have values 1, −1 + i, −2i, 2 + 2i, −4, etc. The first 11 numbers in this base are shown below. Number in base −1 + i 0 1 10 11 100 101 110 111 1000 1001 1010 chalkdustmagazine.com
Complex number 0 1 −1 + i (−1 + i) + (1) = i −2i (−2i) + (1) = 1 − 2i (−2i) + (−1 + i) = −1 − i (−2i) + (−1 + i) + (1) = −i 2 + 2i (2 + 2i) + (1) = 3 + 2i (2 + 2i) + (−1 + i) = 1 + 3i 56
chalkdust Complex numbers are oen drawn on an Argand diagram: the real part of the number is ploed on the horizontal axis and the imaginary part on the vertical axis. The diagram to the le shows the numbers of ten digits or less in base −1 + i on an Argand diagram. The points form an order 10 dragon curve! In fact, ploing numbers of n digits or less will draw an order n dragon curve. Brilliantly, we may now use known properties of dragon curves to discover properties of base −1+i. A level ∞ dragon curve covers the entire plane without intersecting itself: therefore every GausNumbers in base −1 + i of ten digits or less sian integer (a number of the form a + ib where a ploed on an Argand diagram and b are integers) has a unique representation in base −1 + i. The endpoints of dragon curves lie on a logarithmic spiral: therefore numbers of the form (−1 + i)n , where n is an integer, lie on a logarithmic spiral in the complex plane. If you’d like to play with some dragon curves, you can download the Python code used to make the chalkdustmagazine.com pictures in this article at Matthew Scroggs Mahew is a PhD student at University College London working on boundary element methods. In his spare time, he works on a lile-known mathematics magazine called Chalkdust. @mscroggs
mscroggs
mscroggs.co.uk
My least favourite number
2016
Aryan Ghobadi
I mean, it’s just a horrible number! Worst prime factorisation ever: 5 powers of 2 and just 1 power of 7⁈? It’s insanity! 7/2016
Largest odd factors Pick a number. Call it n. Write down all the numbers from n+1 to 2n (inclusive). For example, if you picked 7, you would write: 8 9 10 11 12 13 14 Below each number, write down its largest odd factor. Add these factors up. What is the result? Why? Source: Daniel Griller (puzzlecritic.wordpress.com) Answers at chalkdustmagazine.com
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#5 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 ( chalkdustmagazine.com) by 22 July 2017. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 30 July 2017. One randomly-selected correct answer will win a ÂŁ100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Nerd DVD, a utilities puzzle mug and much, much more. Three randomly-selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping. Find out more at mathsgear.co.uk chalkdustmagazine.com
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Across
Down
1 The last five digits of the square of this (5) number are equal to this number.
1 This number is equal to the last four (4) digits of its square.
5 The square of this number is one more (3) than the product of four consecutive integers.
2 A number whose square and cube (2) between them use each digit 0 to 9 exactly once.
7 This number is equal to the sum of the (3) factorials of its digits. 9 A number a such that the equation (2) 3x2 + ax + 75 has a repeated root. 10 The geometric mean of 27D and 6D.
(3)
12 The 2nd, 4th, 6th, 8th, and 10th digits (11) of this number are the highest common factors of the digits either side of them.
3 This number can be wrien as the (4) sum of the fourth powers of two rational numbers, but it cannot be wrien as the sum of the fourth powers of two integers. 4 The geometric mean of 13A and 15A. (6) 5 The geometric mean of 15A and 18A. (8)
(5)
6 The geometric mean of 10A and 13A. (4)
14 This number contains each of the digits (9) 1 to 9 exactly once. The number formed by the first n digits of this number is divisible by n (for all n).
7 This number is a multiple of 3D but (12) not a multiple of 4. Its digits add up to 27D.
13 The geometric mean of 6D and 4D.
15 The geometric mean of 4D and 5D.
(7)
17 Each digit of this number (aer the first (7) two) is a digit in the product of the previous two digits of this number. This number’s second digit is 9. 18 An integer.
(9)
20 A number, n, such that n, n + 1, n + 2, (5) n+3 and n+4 all have the same number of factors. 21 A multiple of 5,318,008.
(11)
24 A multiple of 9 whose digits are all even. (3) 26 The number of the across clue whose (2) answer is 26. 28 In a base other than 10, this number can (3) be wrien as 1,000,000. 29 A number n such that n, n + 1 and n + 2 (3) are all the sum of two distinct square numbers. 30 The sum of this number’s factors is (5) 100,000. 59
8 This number can be made by con- (10) catenating two other answers in this crossnumber. 11 The number of squares (of any size) (12) on a 13,178-by-13,178 chessboard. 13 A multiple of both 6D and 20A.
(10)
16 The sum of this number’s first five (8) digits is one more than the sum of its last three digits. 19 The square of this number is a palin- (6) drome. 22 The number of 41-dimensional sides (4) on a 43-dimensional hypercube. 23 Each digit of this number is either a (4) factor or a multiple of the previous digit. 25 A number of the form n2 + 2n .
(4)
27 The total number of zeros, threes, (2) sixes, and nines that appear in the completed crossnumber. spring 2017
We love it when our readers write to us. Here are some of the best emails and tweets we’ve been sent. Send your comments by email to contact@chalkdustmagazine.com, on Twitter @chalkdustmag, or by post to Chalkdust Magazine, Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
We’ve got Chalkdust mail! Enjoyed the puzzle. All Christmas cards should come with maths puzzles! (Pascal’s on spoiler duty.)
Dominika Vasilkova,
Just discovered—and very much enjoying—Chalkdust good work Kyle D Evans, Hants
@kyledevans
Received T-shirt. Lad loves it. I think he has more maths expressions in his wardrobe than his fact book. Adam P, Shropshire
@ajtpartridge
@Dragon_Dodo
Dear Chalkdust,
My student spontaneously constructed her own Venn with snakes, women, and Medusas. I guess it’s Medusa week. Elizabeth & Zeke, Cardiff
@RealityMinus3
Happy birthday Chalkdust, the most advanced two year-old I know! Lisa Chalmers, London
@MsChalmersMaths
My Chalkdusts arrived today. Thanks. Still, crikey. Second class post, eh? I had no idea it could be this slow. Adam Atkinson, Horsham chalkdustmagazine.com
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Emma Bell
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2017, a new polymer £10 note will enter circulation in Scotland, featuring the portrait of a Scoish mathematician and scientist: the honour chosen by public vote.
At a time when democracy and equality are being fervently debated, this issue’s Roots column focuses on the legacy of Mary Somerville. In 1704, an annual publication was launched: the Ladies’ Diary, or Woman’s Almanack. The subtitle of the periodical explained its remit: Royal Bank of Scotland
Containing new improvements in arts and sciences, and many entertaining particulars: designed for the use and diversion of the fair sex. The Ladies’ Diary carried the portrait of an important woman on its cover, and contained typical almanac sections of the time with moon phases and eclipse tables, as well as fashion stories and recipes, alongside a section including riddles or enigmas. This section, in particular, proved to be extremely popular and the Ladies’ Diary went on to gain a reputation for mathematical puzzles. Readers of the publication would send in solutions, oen using pseudonyms, that were published in the following year’s edition. The mathematical difficulty of the problems varied widely and contributors to the almanac were both male and female, eminent and unheard of. Women’s participation in the sciences carried a social stigma, and barriers were put up to stop young women who may have shown an aptitude 61
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chalkdust for study. Aer all, it was believed that it was no good for their health and would be too much of a burden for them. An example puzzle from the 1798 edition was the following:
It was problems such as these that captured the interest of a young Mary Somerville. Born Mary Fairfax in 1780 in Jedburgh, a town on the Scoish Borders, she was the daughter of a high-ranking naval officer, Vice-Admiral Sir William George Fairfax. She grew up in Fife, and in keeping with the norms of the time, was not given an education other than being taught to read by her mother. Sir William returned from sea and was very disappointed in Mary’s intellectual progress. In Mary’s words—from her memoirs published posthumously by her daughter, Martha— “he was shocked to find me such a savage”. As a result, Mary was sent to boarding school for twelve months: a period of her life she compared to being caged, forced to recite pages verbatim from Johnson’s dictionary while wearing steel-rodded clothes in order to encourage good deportment. Mary felt that she had learnt nothing. She found the teaching methods inefficient and her family were disappointed in her progress, especially in relation to the school fees they had paid. She became aware of algebra while aending a tea party with her mother, an event that bored her “exceedingly”. A younger lady, Miss Ogilvie, showed her a periodical filled with “coloured plates of ladies’ dresses, charades and puzzles”. She thought that one of the puzzles was a simple arithmetic teaser, but when she continued reading she found “strange looking lines, mixed with leers, chiefly xs and ys”. She asked her companion about it, and was introduced to the word ‘algebra’, although Miss Ogilvie did admit that she could tell Mary “nothing about it”. She oen overheard her brother being taught at home and writes about her brother’s tutor:
I ventured to ask him about algebra and geometry, and begged him, the first time he went to Edinburgh, to buy me something elementary on these subjects, so he soon brought me Euclid and Bonnycastle’s Algebra, which were the books used in the schools at that time. Now I had got what I so long and earnestly desired. I asked Mr Craw to hear me demonstrate a few problems in the first book of Euclid and then I continued the study alone with courage and assiduity, knowing I was on the right road. Unfortunately, her family aempted to extinguish rather than quench this thirst for knowledge:
I had to take part in the household affairs, and to make and mend my own clothes. I rose early, played on the piano, and painted during the time I could spare in the daylight hours, but I sat up very late reading Euclid. The servants, however, told my chalkdustmagazine.com
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mother “It was no wonder the stock of candles was soon exhausted, for Miss Mary sat up reading till a very late hour”; whereupon an order was given to take away my candle as soon as I was in bed. […] My father came home for a short time, and, somehow or other, finding out what I was about, said to my mother, “Peg, we must put a stop to this, or we shall have Mary in a straitjacket one of these days”.
Mary Somerville
Mary’s first marriage to Captain Samuel Greig continued this theme, and her husband’s prejudices towards educated women stifled her academic progress. Widowed aer three years of marriage, Mary’s inheritance from Greig allowed her to pursue her interests fervently. She won a medal for solving a prize problem in the journal Mathematical Repository; but it was only aer her second marriage to Dr William Somerville in 1812 that her efforts began to be truly supported. From this point onwards, Mary never looked back: she published widely, became one of the first women members of the Royal Astronomical Society, and encouraged her friend, Ada Lovelace, to study mathematics.
In 1866, Mary sent her signature from her home in Italy to be included on the first petition to parliament on women’s suffrage, organised by Barbara Bodichon and delivered by John Stuart Mill. Mary Somerville’s tenacity and refusal to have her enthusiasm curtailed serves as an example for us all. She is an inspiration. Emma Bell Emma is a teacher from Grimsby. @El_Timbre
emmalbell@outlook.com
My least favourite number
27,000
Yiannis Simillides
My least favourite number is 27,000, because that is the amount of money I now owe the government. £27,000/my pocket
Square factorials Multiply together the first 100 factorials: 1! x 2! x 3! x ... x 100! Find a number, n, such that dividing this product by n! produces a square number. Source: Tom Button (via Woody Lewenstein at Maths Jam) Answers at chalkdustmagazine.com
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This issue features the top ten parts of a circle. To vote on the top ten geometry instruments go to chalkdustmagazine.com . At 10, and dan gerously close to having a "(si c)" add ed after it: the cen ter.
B eatin g its Am eric version to nu m ber an cover Stuck in th e Cen tr 9 th is week: e with You by Stea lers Wheel.
At 7, it's the Mysterons' favourite part of the circle: the circumferen ce.
At 8, and spend ing yet anoth er week inside the top ten: the interior.
At 6, everyone' s favourite pop twi ns: no, not Jed wa rd, it's two ang les subten ded from the sam e arc.
At 5 this week, the debut solo single from one half of the duo Circum ference: it's the radius. At 4, with such a Monumental cover of the song Baker Street that it wou ld even amu se Queen Victoria: it's the Circle line.
Ma kin g it to nu m ber 3 th is week after being ba nn ed by all th e UK's m aths depa rtm en t ra di o stations (except Loug hborou gh ): ta u. k, an d At nu m ber 2 th is wee ni ng ai provin g th at even contt m ake th ree squa res doesn' p ten: e to you too un cool for th uation. eq n sia th e ci rcle's Ca rte
Despite tau's repeated attempts to take it down with diss tracks, pi remains at number 1 for a record 3.14159 2 6535 years in a row, proving that it really is the only circle constant you need. 8 97
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