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In conversation with… Cédric Villani

Early on a February morning, we’re standing outside one of the many trendy cafes in Fitzrovia. Down the street we spot a man striding our way, wearing a full suit, a hat, a giant spider brooch and hastily tying a cravat. It could only be superstar mathematician Cédric Villani.

Cédric is passing through London on his way back from the US, but this is no holiday. In his two days here, he is attending a scientific conference, giving a public lecture, and taking part in a political meeting. His packed schedule leaves the increasingly-busy Fields medallist just enough time to join us for breakfast.

Fields medal

One afternoon in early 2010, Cédric was in his office at the Henri Poincaré Institute in Paris, getting ready to pose for publicity photographs. The photographer, from a popular science magazine, was setting up his tripod when the office phone rang. Cedric leant over and picked it up. It was Lázló Lovász, president of the International Mathematical Union.

Fields medal ceremonies are held every four years, and six months before each ceremony, the winners are alerted by telephone about their success. During these six months, they are sworn to secrecy, but with the photographer in the room, Cédric suddenly realised that he might be in possession of the shortest-kept secret ever. By some miracle, the tripod had proved sufficiently interesting for the photographer, or perhaps he didn’t follow the English conversation, and the secret remained safe.

If you try too hard to win a Fields medal, you will fail.

Cédric had first realised that winning the Fields medal was a possibility at some point in 2004, when he was 31. Fields medals are only awarded to mathematicians under the age of 40, and until the phone call arrived, Cédric only placed his chances of winning at around 40%. “The prospect of winning the medal does put some pressure on you during your 30s. But everybody knows—it’s part of the common mythology—that if you try too hard to win it, you will fail.”

In August 2010, Cédric was officially awarded the medal at the International Congress of Mathematicians in front of 4000 mathematicians and journalists. Finally, he was allowed to celebrate: he did so by taking a dozen colleagues to a fancy restaurant in Germany, thereby relieving himself of half the CAN$15,000 prize money.

The Boltzmann equation and Landau damping

Kite

The Boltzmann equation can be applied up where the air is clear less dense. Image: public domain

While enjoying a hearty breakfast, Cédric explains his research to us. “In this room, we are surrounded by air. You can use the Navier–Stokes equations to describe this air. But at higher altitutes, where the atmosphere is more dilute, these equations do not work so well. Here, it is better to use the Boltzmann equation.” The Boltzmann equation describes the statistical behaviour of a gas, and Cédric has worked on two areas related to this equation: the influence of grazing collisions, where two particles pass very close to each other; and on the increase in entropy as time passes.

Cédric’s other work, completed with Clément Mouhot, looked at the mechanics of plasmas: high-energy soups of electrons and positively-charged ions which are formed by superheating gases. Roughly speaking, if a plasma is exposed to a brief electric field, then the electric field will become very small as time goes by. This decay effect is called Landau damping. In the 1940s, Lev Landau proved that this damping occurs for a linearised approximation of a plasma. Cédric and Clément proved this result for the full non-linear system of equations.

It was the work in these two areas that led to Cédric being awarded the Fields medal, although he has worked in other areas as well. Imagine you have a large pile of sand and a hole to fill (with the same volume as the sand). How should you go about moving the sand to fill the hole, while minimising the total work you have to do? This is an example of an optimal transport problem.

He used the ideas of entropy from his study of the Boltzmann equation and applied them to this problem, and used this to establish a link between the non-Euclidean curvature of a manifold and properties of the entropy. This led to a “whole bunch of research related to non-Euclidean geometry”.

Career choices

Academia is where my heart belongs.

If the young Cédric had had his way, his research life would be very different. “When I was a kid, I wanted to go into palaeontology. I recently had a great discussion with Jack Horner, the world’s most famous expert on the subject—‘Mr Dinosaur’, and it was like reconnecting with my youth.”

So is he happy in mathematics academia? “Academia is where my heart belongs. I like industry, and I sit on the advisory boards of several companies, but I’m an academic guy. My research has not had an application so far that I am aware of. But, with applications, when they come it will be much later.”

Traces of Cédric’s early passion can still be spotted though. He owns a cuddly toy dinosaur called Philibert, and leaves maths books open to keep him entertained. Years later, he found that Alan Turing, one of his greatest heroes, used to do the same with his teddy bear at university.

Grumpy Gauss

Grumpy Gauss, oil on canvas. Christian Albrecht Jensen (1840)

In fact, Turing is the hero in his recently-penned graphic novel,  Les Rêveurs Lunaires. Excited readers will be disappointed, however, as “even though England is everywhere in the book, English publishers have not yet been interested in making an translation.” This is a double-shame, as you will remember from Chalkdust issue 04 that comic books about maths are `hot’.

He is, however, less sure whether he would like to travel back in time to work with Turing or other mathematicians. “People like Gauss—so fascinating, so superhuman. But he was known for being rather grumpy; maybe it would not be so pleasant! Then take Riemann—a genius! But a bit depressive; maybe he was not so fun to work with. I’m not sure if he would want to see me.”

A day in the life of a Fields medallist

Life is rarely routine for Cédric. In a usual year he travels to 20–25 countries, and has roughly 30 different appointments each week. When he can, he enjoys a quiet family breakfast at home. The contents of this breakfast have not changed since he was a child, and include bread, jam and hot unpasteurised milk. For today, however, he makes do with a full English with scrambled eggs.

I never give fashion advice. I always tell people: “find your own way”, as I did find my own way.


Dairy products seem to feature heavily in Cédric’s day-to-day life. Impressively, he is able to visualise every shelf in his favourite cheese shop and name, in turn, every item on sale. This is very important to him, as otherwise he could return home from grocery shopping to find himself without one of his many favourite cheeses.

He is in London to give a lecture to the public, something that he spends a large amount of time doing these days, “much more so than to mathematicians. But both are good: different feelings, different preparations.” Overall, since winning the Fields medal and gaining fame, Cédric claims that his time for research has been “divided by hundreds”.

Indeed, the public lecture is not his only commitment in London. He is currently attending a meeting at the Royal Society about the numerical abilities of animals. This meeting included great revelations about the mathematical abilities of frogs—evidenced through their calls involving sounds of varying number and length—as well as fish, bees and chimpanzees. “One of the crazy things that emerged from this conference is that the tendency to put small numbers on the left and large numbers on the right is not merely a side effect of how we write numbers. You can also find this—in some sense—in newborn chicks and fish.”

When in France, Cédric is recognised everywhere he goes, and is (still) posing for photographs. He is regularly featured on the covers of science magazines, and is often confronted by giant billboards of his face. If you are planning on winning a Fields medal, do not panic: he assures us that you will quickly get used to this.

Politics

Cédric enjoying a popular maths magazine

Cédric enjoying a popular maths magazine. Image: Chalkdust

When we meet Cédric, the French election is in full flow. As part of his stay in London, he is attending a meeting for the candidate he describes as the “young, centrist guy”. He is one of seven scientists on a board that provides scientific policy advice to the European Commission. However, he doesn’t recommend becoming too involved in politics, as he thinks there is no way to find time to pursue both a serious research career and a serious political career.

“The current political climate is far from science in general. Science, as a field, is much more respected by society than politics. So there is reputation to be lost by going into politics. But the most popular politician in French history is Napoleon, and he was keen on mathematics, and a big protector of mathematicians and scientists. He was elected to the academy of sciences, attending when he could, and enjoyed discussions with many of the best mathematicians of his time. But he was always late…”

Keen not to be late himself, Cédric finishes his eggs and heads off to his next commitment. It would seem, however, that Cédric does not always listen to his own advice: in June he became an elected member of the French parliament, as a member of the young, centrist guy’s party.

TD, Scroggs and Yiannis The Undergrad enjoy Cédric's company

TD, Scroggs and Yiannis The Undergrad enjoy Cédric’s company. Image: Chalkdust

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Cardioids in coffee cups

Picture the scene. It’s 1am and you’re up late working on some long-winded calculations. The room around you is dark, a desk lamp the only source of light. Your eyelids start to droop. But the work must get done! Time to fall back on the saviour of many a mathematician: coffee.

But as you sit back down at your desk, you notice something weird. The light from your lamp is reflecting oddly from the edges of the cup, creating bright arcs—and it looks suspiciously like a cardioid curve! Time to investigate…

Work forgotten, you pull out a clean sheet of paper and—well, dear reader, you may have been more sensible than me and just gone to bed at this point, or finished the work you were meant to be doing. For me, though… well, let’s just say that sleep would be impossible until this mystery was resolved.

So. We have a cup. We have a light. We have an enigmatic looking curve. What’s going on?

This coffee is clearly demanding that we make theorems from it

Let’s shed some light on this

The paths of two collimated light rays and their different angles of reflection

To keep things simple, we’re going to model the base of the cup as a perfectly reflecting two-dimensional circle, and limit the incoming light to the plane that the circle is in. We can also set the radius of the cup to be 1 without loss of generality. Since the cup and the lamp are reasonably far apart, a decent assumption to make is that the light coming in is collimated, which means that all the light beams are parallel to each other, as if from a point source at ‘infinity’.

If we look at a single light ray coming in parallel to the $x$-axis, we know that the angle of incidence and the angle of reflection are equal, as measured from a line normal to the circle. If a second ray comes in parallel to the first, it will hit the surface at a different angle, so must reflect off at a different angle to the first, and so the reflected rays will no longer be parallel. Instead, they will overlap as shown above.

As we build up light rays, the shape of the envelope begins to emerge. The only difference between this diagram and the physical system is that here, the overlap makes a dark envelope, whereas in the cup the overlap makes a bright envelope

A rule of ray tracing tells us that all light rays parallel to the axis will go through the focal point of a curved surface. This is the point where the two light rays cross in the diagram above. We can therefore expect that to be the brightest point of the pattern that we see in the coffee, since it is hit by the most light.

But what about the rest of the curve? If we draw in a few more light rays, as in the diagram to the left, we start to see areas where many different rays overlap and can build up a picture of the curve. Treating the incident light rays as a family of curves, the bright pattern seen is their envelope. The envelope is a curve that at every point is tangent to one of the incoming rays, and so by moving along its length we move between the different members of the family. Due to the tangent property, it is also the boundary of the most dense area ‘swept out’ by the curves, so in many cases this corresponds to the curve you’d get by joining up all the points of intersection. If we can find the equation of this envelope, we can find out exactly what shape is being formed in the bottom of the cup.

Calculus to the rescue!

A hint about how to find the equation we need comes from the definition: we need to find a curve that is tangent to our family of curves at every point. So, perhaps unsurprisingly, a good way to do this is to use calculus. For a smooth family of curves, we first find a general equation for the curves by expressing them in terms of some parameter, say $a$. We can then find the equation of all their tangents by differentiating with respect to $a$. Since we need these two equations to match at every point along the envelope, to find the equation of the envelope we solve them simultaneously. For example, say we wanted to find the envelope of straight lines that enclose equal area between them and the axes—picture this as a ladder propped against a wall, but sliding down it.

The equation of a straight line in terms of both axis intercepts has the form

$$ \frac{x}{a} + \frac{y}{b} = 1, $$

where $a$ and $b$ are the intercept points. These are both parameters that describe the family of curves, so we use the fact that we want to keep the area $A= ab/2$ constant to eliminate $b$:

$$ \frac{x}{a} + \frac{ay}{2A}= 1. \qquad (1) $$

Differentiating this with respect to $a$ gives
$$ -\frac{x}{a\hspace{1pt}^2} + \frac{y}{2A} = 0. \qquad (2)$$

So equations (1) and (2) are what we want to solve simultaneously. In this case, it’s possible to do this by eliminating $a$ between the two equations, giving the equation of the envelope as

The curves and envelope formed for $A=1$

$$\sqrt{xy\hspace{1pt}} = \sqrt{\frac{A}{2}},$$
or, if we limit ourselves to the first quadrant,
$$xy = \frac{A}{2}.$$

This is the equation of a hyperbola. An example case for $A=1$ is shown to the right. In this case, we were able to eliminate the parameter from the equation to leave it only in terms of $x$ and $y$, but as we will see later, it may be easier to leave envelopes in their parametric form.

But what about the coffee?

We can use basic geometry to express our family of curves in terms of $\theta$

Now we know how to find the equation(s) for an envelope, we can apply this method to our cup scenario. We know the equations of the lines coming in, since they’re all just straight lines parallel to the axis, but we need to find out the equation of the reflected light beams. It turns out that this can be done by taking advantage of the fact that the cup is circular and throwing some trigonometry at it.

Consider a light beam coming from the right and striking the cup at a point $(x,y)$. The beam is at an angle $\theta\hspace{0.2mm}$ to the normal of the surface, so using

angle of incidence  =  angle of reflection

we know it will reflect at the same angle, as shown in the top right diagram. Using the fact that the red triangle is isosceles, the point $(x,y)$ is therefore at an angle $\theta$ from the negative $x$-axis, so we can parameterise the point as $(-\cos\theta, \sin\theta)$ as the radius of the cup is 1.

Plot of equation (3) for a few different values of $\theta$. Note that these are just the reflected rays

The slope of the reflected ray is then $-\tan2\theta$ and the equation of the line, in terms of $\theta$, is
$$ y – \sin\theta = -\tan2\theta\,(x+\cos\theta),$$
or, after some identity jiggery-pokery,
$$x\hspace{1pt}\sin2\theta+y\cos2\theta = -\sin\theta. \qquad (3)$$
A plot of a few of these curves, with different values for $\theta$, is shown in the middle right diagram and looks similar to what we see in the cup. That’s a good sign!

Differentiating the above equation with respect to $\theta$ gives
$$2x\hspace{1pt}\cos2\theta -2y\hspace{1pt}\sin2\theta = -\cos\theta. \qquad (4)$$

A plot of equation (3) with the calculated envelope drawn on

Now, eliminating $\theta$ between these would be downright disgusting, so expressing (3) and (4) in matrix form and solving for $x$ and $y$ gives
$$
\begin{pmatrix} x \\ y \end{pmatrix}
=
\frac{1}{4}
\begin{pmatrix}
\cos3\theta- 3\cos\theta \\
3\sin\theta – \sin3\theta
\end{pmatrix}.$$
Plotting this, it matches up very nicely on one side of the cup. But, unlike the real pattern in the coffee cup, this one has an extra bit of curve! And this equation, alas, doesn’t describe the cardioid we’d hoped for—this is the equation of a nephroid. To add insult to injury, by the time I’d worked all this through, my coffee was ice cold. Yuck.

In fact, the extra bit of curve appears due to all values of $\theta$ being allowed. As the sides of the cup will block about half the light, this imposes a restriction on $\theta$, so we only see half the curve in our coffee. And a nephroid is, actually, correct—shapes like these that form when light reflects off a curved surface are called ‘caustics’, from the Greek word for ‘burnt’, as they can be used to focus sunlight to start fires. Both the nephroid and the cardioid belong to a larger family of curves called epicycloids, which are categorised according to the number of ‘cusps’ (sharp bits) that they have.

But I wanted a cardioid, dammit!

The angular setup for a point on the rim. To keep the parameterisation the same as the last case, the angles of incidence and reflection have been defined differently

If nephroids aren’t your thing, it’s possible to get a cardioid caustic in a cup if we change the setup slightly. Instead of having a point source at infinity, let’s put the point source on the rim of the cup and see what happens. The geometry of the incoming and outgoing rays is shown to the right.
This gives the equation of the line as
$$
y\hspace{1pt}(1+\cos3\theta\hspace{1pt})+x\hspace{1pt}\sin3\theta = \sin\theta- \sin2\theta,
$$
and differentiating, we get
$$
-3y\hspace{1pt}\sin3\theta + 3x\hspace{1pt}\cos3\theta = \cos\theta- 2\cos2\theta.
$$
Solving these simultaneously is a tad more fiddly than before, but working through gives the envelope as
$$
\begin{pmatrix} x \\ y \end{pmatrix}
=
\frac{1}{3}
\begin{pmatrix}
\cos2\theta- 2\cos\theta \\
2\sin\theta – \sin2\theta
\end{pmatrix}.
$$
This is a cardioid. Yay!

Reflected light beams from a point source located where the rim touches the positive $x$-axis

Additional complexities

What we see in a cup is unlikely to be exactly one of the two previous cases. If the caustic is bright enough to be visible, the light source is probably not far enough away to be at ‘infinity’, and people don’t tend to go around putting point sources on the rims of their cups. If we have a light source that’s a finite distance from the cup edge, the incident rays will be at an angle somewhere between the nephroid and cardioid cases, so the curve seen is somewhere between the two.

There is another large assumption we’ve made here that renders the situation somewhat unphysical. Our cup is a two-dimensional circle! And although that makes the maths nicer, it’s not great for holding coffee. The physical principles are the same in 3D, with an extra angle to worry about, so what you actually see in a coffee cup is the intersection of the surface in the diagram on the right with the bottom of your cup.

This surface is called the ‘cusp catastrophe’, and can be found using catastrophe theory, which, among other things, looks at the behaviour of manifolds with singularities in them.

The cusp catastrophe. Rich Morris (singsurg.org), CC BY-NC-SA 4.0

A change of focus

Since we can focus light using almost any process that changes its direction, reflection caustics (or catacaustics) such as the ones we have been considering here are not the only type possible. A common refraction caustic is the rippling pattern of light seen on the bottom of bodies of water, and a rainbow is a caustic caused by a combination of reflection and refraction. More exotically, gravitational forces bend space-time and therefore the light travelling through it, which means that gravitational lensing can give rise to caustics of astronomical scale. The shape of the caustic gives key information about the astronomical object, and this method has been used to identify and analyse exoplanets around distant stars.

Although these physical systems look completely different at first glance, they’re linked by a single phenomenon. The same flavour of physics that describes how the light in your morning cuppa behaves also describes the behaviour of light on ridiculously huge scales in the universe. And that’s pretty cool, don’t you think?

So, the next time you sit down to enjoy a hot beverage, take a moment to appreciate the awesome things happening, quite literally, right under our noses.

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Mathematics for the three-fingered mathematician

We’re all familiar with using a couple of different bases to represent integers. Base ten for almost all purposes when we do our own calculations, and base two, or binary, for getting computers to do them for us. But there’s nothing special about ten and two. We could equally well use any integer, $b$, greater than two, so that the string of digits

$$ d_n d_{n-1} d_{n-1} \ldots d_0, $$

where each $d_i$ is positive and less than $b$, represents the integer

$$ \sum_{i=0}^n d_i b^i.$$

Some bases are slightly more convenient than others for doing arithmetic. Bases eight and sixteen are both used in various computer applications, and there is an active society, the dozenal society, devoted to using and promoting the arithmetical advantages of base twelve. Much less common, but far more interesting, is base three.

With base three, the digits are all 0, 1 or 2. But I want to look at a variation on this. Instead of using 1 and 2, I’ll use 1 and -1; but it’s not convenient to have minus signs in the middle of our numbers, so because of this and for reasons of symmetry I’ll represent them with 1 (for 1) and 1 (for -1). Base three is ternary, and this variation of it is called balanced ternary.

Continue reading

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The mathematics of Maryam Mirzakhani

Maryam Mirzakhani, the first woman Fields medallist and an explorer of abstract surfaces, left us in the prime of her life. Rightly, the world press mourned her passing, but what I hope to do here is to write about the beautiful and difficult mathematics she loved working on. As a pure mathematician, she was usually driven by in-depth understanding of the different complex structures on abstract surfaces, rather than the search for application. Nevertheless, her work has been used in solving real life problems. But what I personally find fascinating about her is the courage and creativity she had in attempting and solving long standing problems and the variety of areas within mathematics she worked on; from complex geometry and topology to dynamical systems. Here is a more intuitive exposition of some of her achievements.

Surfaces

A tasty surface. Image: Descubra Sorocaba, CC BY 2.0

I am sure that if I asked you to give me an example of a surface you would be able to do so straight away. You might say the surface of the Earth is obviously one, and you would be right. However, defining what we mean by a surface mathematically is a little bit trickier. Let’s give it a go. A geometrical object is called a surface if, when we zoom in very closely at the points on the shape, we can see overlapping patches of the plane. If we were to use mathematical language we would say that a surface is locally homeomorphic to the plane.

A genus 1 surface

You might not find this definition particularly helpful so let us consider a few more examples. Oranges, tomatoes, apples and, for more delicious alternatives, cakes, cupcakes, ring doughnuts and pretzels are all surfaces. Well, almost! In order for them to be surfaces we need to picture them hollow (or like a balloon), rather than solid. If we consider those objects geometrically, meaning that we differentiate between different angles and size lengths, we notice that there are infinitely many of them. This is why we consider them topologically. Using continuous deformations we can turn almost all of our examples into a sphere, except the ones that have holes in them. They are considered to be in a class of their own. Thus we can classify the surfaces up to deformations (topological equivalence) by the number of holes, which we call the genus. We can see that the sphere has genus 0, the torus (ring doughnut) genus 1 and the 3-fold torus (pretzel) genus 3, thus these are all inequivalent surfaces.

Mirzakhani’s work was on Riemann surfaces. To turn a surface into a Riemann surface we need to give it additional geometric structure. For example, we can give the surface geometric structure that allows us to measure angles, lengths and area. An example of such geometry is hyperbolic geometry. It is the first example of non-Euclidean geometry; the only way it differentiates from Euclidean geometry is that given a line $\ell$ and a point $P$, that is not on the line, we can draw at least 2 distinct lines through $P$ that are parallel to $\ell$.

Parallel lines and a triangle on a hyperbolic surface

One peculiar consequence of this new axiom is that rectangles do not exist in hyperbolic geometry. Moreover, the angle sum for a triangle is always less than $\pi$. Mirzakhani’s early work was on hyperbolic surfaces, which are Riemann surfaces with hyperbolic structure. The problem with hyperbolic surfaces is that we cannot really visualise them, because the hyperbolic structure on the Riemann surface can’t be embedded in $\mathbb{R}^3$. However, we can try and describe roughly how you put the structure on the surface. Imagine our surface is made out of rubber and we can bend it and fold it in all dimensions. Now we add the hyperbolic structure, but for that we need, according to John Nash, 17 dimensions. If we next dip it in cement it becomes solid, and we can no longer stuff it into 3 dimensions, hence we can no longer visualise it completely.

On these surfaces, Mirzakhani studied special objects called closed geodesics. Roughly speaking, a geodesic is a generalisation of the notion of a straight line that we have on the Euclidean plane. We can define a geodesic more rigorously as a path between two points on the surface, whose length cannot be shortened by deforming it. For example, on the sphere, the geodesics are called great circles. These are simply the intersections of a plane going through the origin and the sphere itself.

A sphere sliced in half along a great circle

A closed geodesic is a geodesic that starts and ends at the same point. The simplest example of a closed geodesic is a circle.We also allow intersections, that is, geodesics that look, for example, like a figure-of-eight and are much more complicated. Using the hyperbolic structure on the Riemann surface, we can compute the lengths of these closed geodesics. A natural question we can ask is how many such closed geodesics are there on any hyperbolic surface of length $\leq L$? The answer was established in the 1940s by Delsarte, Huber and Selberg and it was named the prime number theorem for hyperbolic surfaces, because of the resemblance to the prime number theorem. That is, the number of closed geodesics, denoted by $\pi$, satisfies\begin{align*} \pi(X, L)\sim \mathrm{e}^L/L,\end{align*}as $L\rightarrow \infty$. Roughly speaking, the number of closed geodesics on a hyperbolic surface $X$ of length $\leq L$ gets closer to $\mathrm{e}^{L}/L$ as $L$ becomes very big. We can see that their number grows exponentially, meaning very quickly, but more importantly we also see that the formula does not depend on the surface we are on.

The next question to consider is what would happen if we no longer allow our geodesics to intersect themselves? Would our formula change much? Will the growth rate be significantly different? That is, we wish to compute the number of simple closed geodesics (simple meaning no intersections are allowed) on a hyperbolic surface $X$ of length $\leq L$, denoted $\sigma(X, L)$. In 2004, Mirzakhani proved, as part of her PhD thesis, that \begin{align*} \sigma(X,L)\sim C_{X} L^{6g-6},\quad\text{as }L\to\infty,\end{align*}where $g$ denotes the genus of the surface $X$, and $C_{X}$ is some constant dependent on the geometry (hyperbolic structure) of the surface. It is important to make clear that surfaces of a given genus can be given many different hyperbolic structures. As a consequence the number grows much slower (polynomially) but it also depends on the surface we are on. The surface may be the same, but the different structure implies that we would have different geodesics and their lengths would also be different. Whilst she was computing $\sigma(X,L)$, she discovered formulae for the frequencies of different topological types of simple closed curves on $X$. The formulae are a bit too complicated to explain here, but let us consider an example: suppose $X$ is a surface of genus 2; there is a probability of 1/7 that a random simple closed geodesic will cut the surface into two genus 1 pieces. How cool is that?!

Flight paths follow geodesics on the Earth’s surface

Even though these results are for a given hyperbolic structure, Mirzakhani proved it by considering all structures at the same time. We know that we can continuously deform surfaces of the same genus $g$ and they will be topologically the same, however geometrically they may be different. These deformations depend on $6g-6$ parameters, which was known to Riemann. We call these parameters moduli and we can consider their space, the so-called moduli space of all hyperbolic structures on a given topological surface. By definition, a moduli space is a space of solutions of geometric classification problems up to deformations. This is a bit abstract, so let us illustrate it with a simple example. Suppose our geometric classification problem is to classify the circles in Euclidean space up to equivalence. We would say that two circles are equivalent if they have the same radius, no matter where their centre lies. That is, our modulus (parameter) is the radius $r$ of the circle, and we know that $r\in\mathbb{R}^{+}$. Hence the moduli space will be the positive real numbers. So what can we do with these new spaces? Greg McShane observed that you can add a new structure; a so-called symplectic structure which, roughly speaking, allows us to measure volumes on moduli spaces. Mirzakhani found a connection between volumes on moduli spaces and the number of simple closed geodesics on one surface. She computed some specific volumes on moduli spaces and her celebrated result followed.

Dynamical systems

“Pot as many balls as you can.” Image: Curtis Perry, CC BY-NC-SA 2.0

In recent years, Mirzakhani focused her attention on dynamical systems on moduli spaces. A dynamical system is simply a system that evolves with time. Originally, dynamical systems arose in physics by looking at the movements of particles in a chamber or planets in the solar system. It was observed that these large systems are similar to smaller ones, and by studying toy models we might shed some light on the actual physical dynamical systems.

One such toy model is the dynamical system of billiard balls on a polygonal table (not necessarily rectangular). Bear in mind that in this version of billiards we only use one ball and it can travel forever on a path as long as it doesn’t reach a corner. The billiard balls will take the shortest paths, thus they travel via geodesics, and this is where Mirzakhani’s research come into play. As we know by now, she studied surfaces rather than polygons, but if you orient the edges of the table in pairs and glue them together then you can turn it into a surface.

Even though billiard dynamics might seem simple, there are difficult problems that are still unsolved. One might ask if there are any periodic billiard paths, and if so, would the answer change if we change the shape of the table $T$? This problem has been solved: it is known that there is always at least one periodic billiard path for a rational polygonal table (by a rational polygon, we mean a polygon whose angles are rational multiples of $\pi$). But what if we now ask what is the number of such periodic billiard paths of length $\leq L$ on a table $T$, denoted by $N(T, L)$? It is conjectured that the following asymptotic formula holds.

\begin{align*} N(T, L)\sim \frac{C_T L^2}{\pi \text{ Area}(T)},\end{align*}where $C_T$ is some constant depending on the table. Alongside Alex Eskin and Amir Mohammadi, Mirzakhani made some progresstowards this result. They showed that $\lim_{n\to\infty}N(T,L)/L^2$ exists and is non-zero. Mirzakhani’s work unfortunately does not provide a solution, however she brought progress by showing that this number satisfies the following asymptotic formulaMoreover, she showed that for the asymptotic formula to even exist in the form above, there exist only countably many numbers $C_T$. Recently, Mirzakhani and Eskin’s work on billiard paths was applied to the sight lines of security guards in complexes of mirrored rooms.

Another example of her impact on dynamical systems is her work on Thurston’s earthquake flow. Suppose we have a Riemann surface $X$ of genus $g$, a simple closed geodesic $\gamma$ on $X$ and a real number $t$. Then we obtain a new Riemann surface $X_t= \text{tw}_{\gamma}^t(X)$ by cutting $X$ along $\gamma$, twisting it to the right by $t$ and re-gluing. Then we can define the flow at time $t$ to be

\begin{align*} \text{tw}^{t}(\lambda, X\hspace{0.3mm})=(\lambda, \text{tw}_{\lambda}^t(X\hspace{0.3mm}))\end{align*}

where $\lambda$ is geodesic lamination. The definition of geodesic lamination is quite technical, so in this article we can simply think of it as a disjoint collection of simple geodesics on $X$. Intuitively, we have a dynamical system like the movement of planets in time $t$, but in our case the objects that move with time are moduli spaces.

We get some sort of periodicity, because if $\gamma$ has length $L$, then $X_{L+t}=X_t$. Mirzakhani showed something truly remarkable: The earthquake flow is ergodic.
This means that if we follow the laminations along we would be very close to any point on the surface with probability 1. This came as a surprise, because until then there was not a single known example showing that the earthquake paths are dense.

Maryam Mirzakhani by Mehrzad Hatami

She might not have always wanted to be a mathematician, aspiring to be a novelist when she was younger, but she left a big mark on mathematics. As the first Iranian and first woman to win the Fields medal, I believe she has been an inspiration to many young girls and women, including me, to go into research and be optimistic when solving problems because the “beauty of mathematics only shows itself to the more patient followers”.

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Geographic profiling

Imagine you’re a police officer working on a huge case of serial crime. You’ve been handed the list of suspects, but to your horror 268,000 names are on it! You need to come up with a way of working through this list as efficiently as possible to catch your criminal. Along with the thousands of names, you’re also given a map with the locations of where bodies have been found (the map above). Given these two pieces of intel, how exactly would you prioritise your list of suspects? Have a go! Where exactly would you search for the criminal? We will reveal the answer at the end of article!

Peter Sutcliffe, also known as the Yorkshire Ripper, was the name on a list of 268,000 suspects generated by this investigation in the late 1970s. But how were the team investigating these crimes meant to cope with such an overload of information? These are the fundamental problems that geographic profiling is trying to solve.

How exactly does geographic profiling work? This article will introduce you to the fundamental ideas behind the subject. We will also look at the various applications, just like the Yorkshire Ripper case, along the way. These examples aren’t just in criminology though. The applications span ecology and epidemiology too!

The first model

Geographic profiling uses the spatial relationship between crimes to try and find the most likely area in which a criminal is based; this can be a home, a work place or even a local pub. Collectively we refer to these as anchor points. The pioneer of the subject, Kim Rossmo, once a detective inspector but now director of geospatial intelligence/investigation at Texas State University, created the criminal geographic targeting model in his thesis in 1987. The criminal geographic targeting model aims to do exactly what we struggled with at the beginning of this article: prioritise a huge list of suspects.

A gridded-up map. Alistair Marshall, CC BY 2.0


It starts by breaking up your map, populated with crime, into a grid, much like on the left. We assume that each crime that occurs, does so independently from every other. We then score each grid cell; the one with the highest score is likeliest to contain the criminal’s potential anchor point.

How do we calculate this score? An important factor is the distance between crimes and anchor points. We choose to use the Manhattan metric as our measure of distance. In this metric, the distance between points $\boldsymbol{a}$ and $\boldsymbol{b}$ is the sum of the horizontal and vertical changes in distance. This is written as:
$$d(\boldsymbol{a},\boldsymbol{b}) = \lvert x_a-x_b \rvert + \lvert y_a-y_b\rvert, \qquad \boldsymbol{a} = (x_a, y_a), \quad \boldsymbol{b} = (x_b, y_b).$$

The Manhattan metric is so-called because it resembles the distance you have to travel to get between two points in a gridded city like Manhattan.

This is the most suitable metric for our work, but it’s worth noting there are more that can be used (depending on the system you’re studying). Now we could just start searching at the spatial mean of our crimes and work radially outward from that point, however one rogue crime occurring far away from the rest could easily throw a spanner in the works. Instead we use something called a buffer/distance decay function.

$$ f(d) =
\begin{cases}
\dfrac{k}{d^{h}}, & d > B \\
\dfrac{kB^{g-h}}{(2B-d)^g}, & d\leq B\\
\end{cases}$$

A criminal isn’t likely to commit a crime close to an anchor point, out of fear of being recognised, so we place a buffer around it. In addition, to commit a crime far away from home is a lot of hassle, so the chance of a crime decays as we move away from the anchor point. This is why our buffer/decay function looks a bit like a cross-section of a volcano. The explicit function, $f(d)$, is written on the right, where $k$ is constant, $B$ is the buffer zone radius and $g$ and $h$ are values describing the criminal’s attributes, eg what mode of travel they use. With our distance metric, $d$, and buffer/decay function, $f$, we are now able to compute a score for each grid cell.

For $n$ crimes, the score we give to cell $\boldsymbol{p}$ is
$$ S(\boldsymbol{p}) = \sum_{i=1}^{n}f(d(\boldsymbol{p}, \boldsymbol{c}_i)), $$
where $\boldsymbol{c}_i$ is the location of crime $i$. So finally we have a score for each grid cell and we can prioritise our list!

An example of the geographic profile created using the criminal geographic targeting model

Plotting these scores on the $z$-axis produces a mountain range of values, like on the right. We can now prioritise by checking residencies at the peak of this mountain range and working our way down. Notice the collection of peaks around a particular area: this gives us an indication that perhaps the criminal uses more than one anchor point.

An important question: how can we be sure this even works? Does it really identify anchor points efficiently? What do we even mean by “efficient”? This is answered with a quantity called the hit score. This is

$$\text{hit score} = \frac{\text{number of grid cells searched before finding the criminal}}{\text{total number of grid cells}}. $$

So ironically, the lower our hit score, the better our model performs. This is sensible, since we want to search as little space as possible to catch our criminal.

The Gestapo case

Otto and Elise Hampel distributed hundreds of anti-Nazi postcards during the second world war. The Gestapo’s intuition on where the Hampel duo might live was based on themes almost exactly the same as geographic profiling. Inspired by a classic German novel, Alone in Berlin, our group revisited the Gestapo investigation and published our findings in a journal that is so highly classified we are not able to read it.

By analysing the drop-sites of the postcards and letters we were able to show that geographic profiling successfully prioritises the area where the Hampels lived in Berlin. Crucially, this study actually showed the importance of analysing minor terrorism related or subversive acts to identify terrorist bases before more serious crimes occur.

A statistical approach

The criminal geographic targeting model is an incredibly useful tool and is used to this day by the CIA, the Metropolitan Police and even the Canadian Mounted Police. Mike O’Leary, professor at Towson University, Maryland asked why the criminal geographic targeting model only produces a score, when we require a probability. So he developed a way of using geographic profiling under the laws of Bayesian probability.

Bayes’ rule is better in neon. Image: Wikimedia Commons user Mattbuck, CC BY-SA 3.0

O’Leary uses Bayes’ rule as seen on the right. How do we apply it to criminology? We want to know: what is the probability that an offender is based at an anchor point given the crimes they have committed? Using Bayes’ rule, instead we pretend we know where the anchor point is and ask; what is the probability of the crimes occurring given our anchor point? We use the formulation
$$\Pr(\boldsymbol{c}_1, \boldsymbol{c}_2, \boldsymbol{c}_3, \boldsymbol{c}_4\text{…}\;|\;\boldsymbol{p})\; = \;\prod_{i=1}^{n}\Pr(\boldsymbol{c}_i\;|\;\boldsymbol{p}),$$
where the equality derives from the assumption of independent crimes.

Below, we can see a comparison between Rossmo’s criminal geographic targeting model and O’Leary’s simple Bayesian model. The problem with O’Leary’s model is he assumes that a criminal only has one anchor point. Unfortunately this is rarely the case. As we mentioned earlier, an anchor point could be a home, a workplace, a local pub or even all of the above. So we obtain a probability surface, but we only consider one anchor point. The criminal geographic targeting model entertains the idea that multiple anchor points exist, but doesn’t give us an explicit probability. What we really need is a way of combining both methods. Does such a method exist?

(a) The criminal geographic targetting model

(b) The simple Bayesian model

Examples of the geographic profiles created using the criminal geographic targeting and simple Bayesian models

The elusive tarsiers

Image: Callum Pearson

South-east Asia, specifically Sulawesi, houses a huge number of endemic species. Often habitat assessments of cryptic and elusive animals such as the tarsier (right) are overlooked, primarily due to the difficulties of locating them in challenging habitats. Traditional assessment techniques are often limited by time constraints, costs and challenging logistics of certain habitats such as dense rainforest.

Using only the GPS location of tarsier vocalisations as input into the geographic profiling model we were able to identify the location of tarsier sleeping trees. The model found 10 of the 26 known sleeping sites by searching less than 5% of the total area (3.4 km$^2$). In addition, the model located all but one of the sleeping sites by searching less than 15% of the area. The results strongly suggest that this technique can be successfully applied to locating nests, dens or roosts of elusive animals, and as such be further used within ecological research.

The best of both worlds

The Dirichlet process mixture model is the best of both the criminal geographic targeting and the simple Bayesian models. So far we’ve only stated that we’re either working with one anchor point, or many. The beauty of the Dirichlet process mixture model is that we don’t need to specify the number of anchor points we are searching for. Instead, there is always some non-zero probability that each crime comes from a separate anchor point. So multiple anchor points can be identified while using a probabilistic framework. Introducing multiple anchor points is challenging since we need to know:

  1. How are all the crimes clustered together?
  2. In each cluster of crimes, where is the anchor point?

Actually, what would be really useful is if we knew the answer to just one of these questions. If we knew how the crimes were clustered, finding the anchor points is easy (we use the simple Bayesian model to find the source in each cluster). But also, if we knew where the anchor points were, allocating crimes to clusters is easy (and of course we know where our criminal lives!). The solution to this problem is to use something called a Gibbs sampler. We use a Gibbs sampler in cases where we want to sample a set of events that are conditional on one another. In our case, anchor point locations depend on the clustering of crimes, but the clustering of crimes also depends on the anchor point locations. The steps the Gibbs sampler will take are:

  1. Randomly assign each crime an anchor point (even though we don’t yet know where the anchor points are).
  2. Find each anchor point by using the simple Bayesian model on each assignment.
  3. Throw out the assignments of crimes to anchor points and now re-assign crimes but using the locations found in previous step. Throw out the old anchor point locations and find new ones using this new assignment.
  4. Repeat steps 3 and 4 many, many times.

This produces a new profile like on the right below. We can now compare this to our other two models on the left. We can see the Dirichlet process mixture model displays fewer peaks than the criminal geographic targeting model, but that these peaks are tighter. This in turn will reduce the hit score of our search.

(a) The criminal geographic targetting model

(b) The simple Bayesian model

(c) The Dirichlet process mixture model

A comparison of the three main geographic profiling models

The malaria case

Water bodies with mosquito larvae. Image: © OpenStreetMap contributors. Cartography CC BY-SA 2.0

Throughout history, infectious diseases have been a major cause of death, with three in particular (malaria, HIV and tuberculosis) accounting for 3.9 million deaths a year. Targeted interventions are crucial in the fight against infectious diseases as they are more efficient and, importantly, more cost effective. They are even more crucial when the transmission rate is strongly dependent on particular locations. For example, we were tasked with finding the source(s) of malaria outbreaks in Cairo by considering the breeding site locations of mosquitos.

All accessible water bodies within the study area were recorded between April and September 2005, and 59 of these were harbouring at least one mosquito larva. Of these 59 sites, seven tested positive for An. sergentii, well-established as the most dangerous malaria vector in Egypt. Using only the spatial locations of 139 disease case locations as input into the model, we were able to rank six of these seven sites in the top 2% of the geoprofile.

Applying the method

The geoprofile associated with the Yorkshire Ripper body dump sites (black dots). The anchor points of Peter Sutcliffe are labelled as red squares. Image: © OpenStreetMap contributors. Cartography CC BY-SA 2.0

We’ve done it! We now have a robust method for searching for our criminal. A list of 268,000 suspects is no longer so intimidating. Without this technique in 1975-1981, however, there was a lot more work for the team investigating the Yorkshire Ripper case. On top of a huge list of suspects, 27,000 houses were visited and 31,000 statements were taken during the investigation.

If we apply our model to the crime sites we were given at the start of this article, we produce the contour map on the right. In this case the areas in white describe the highest points on our probability surface, whilst areas in red describe the lowest. In addition to the contours, we also see two red squares right at the top of the map. These are the two homes Peter Sutcliffe resided at during the period of his crimes. The hit scores for his two residences are 24% and 5% respectively. So by searching only 24% of our total search area, we’ve managed to find both residences. This is far better than a random search which would find them after searching, on average, 50% of our area.

Peter Sutcliffe’s homes are clearly marked on this map but we must remember an important point about geographic profiling: that it is not an ‘X marks the spot’ kind of model, but rather a method of prioritisation.

Investigating an old case

Dramatic scenes covered the newspaper front pages, such as this from 1888. Image: The Illustrated Police News

We can’t talk about the Yorkshire Ripper without mentioning the notorious 1888 London serial killer, Jack the Ripper. The five locations around Whitechapel where bodies were dumped were studied using geographic profiling to try and gain a better idea of where Jack the Ripper may have lived.

The map overleaf shows us the associated geoprofile, with Jack’s suspected anchor point obtaining a hit score between 10-20%, much better than a non-prioritised search!

This is just one example of many cases where we can utilise our new model to study cases from the past where such tools were not available.

Geographic profiling began in criminology, but now spans ecology (catching invasive species) and epidemiology (identifying sources of infectious disease) too. This means saving a hefty chunk of time and money, as well as developing prevention strategies to minimise any negative impacts these problems may cause.

The geoprofile associated with the body dump sites (black dots) of Jack the Ripper. Jack’s anchor point (the red square) is suspected to be around Flower and Dean Street. Image: © OpenStreetMap contributors. Cartography CC BY-SA 2.0

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Roots: Blaise Pascal

The influence of Blaise Pascal is most keenly felt in his work on probability and the binomial theorem, illustrated by the famous Pascal’s triangle. It cannot be denied that Pascal’s triangle is a thing of mathematical beauty. However, this array of numbers was not discovered by its namesake, rather its applications and importance were highlighted by Pascal in his work, akin to Pythagoras’ theorem, which was certainly not invented by Pythagoras himself. However, Pascal’s legacy to mathematics goes further than the instantly recognisable triangle …   Continue reading

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Pretty pictures in the complex plane

Some of the greatest works of art in history have been produced by mathematicians. One fascinating source of mathematical artwork is fractals: infinitely complex shapes, with similar patterns at different scales. Fractal geometry has dramatically altered how we see the world. Technology has many uses for fractals, one of which is the production of beautiful computer graphics. These pretty pictures are used to present a large amount of information about a function in a clear and comprehensible manner,  and the simplicity of the maths involved in producing these pictures is fascinating.

Pretty pictures in the $z$-plane are widely used as computer graphics, book covers and even sold as works of art.

Modern art studies have often been dismissive of the power of beauty in mathematics, with the idea that “beauty is not in itself sufficient to create a work of art”. Mathematics produces rigid and inflexible answers, whereas art is free-moving and open to interpretation. However, it is undeniable that these pretty pictures demonstrate true beauty, not only in the images but also in the mathematics behind them.

The mathematics behind pretty pictures

Extremely simple functions can be used to produce these pictures. For example, let’s consider the quadratic function $f\hspace{0.4mm}(x\hspace{0.3mm})=x\hspace{0.3mm}^2+c$, for some constant $c$. An iterative method is applied to the function. First, a seed (let’s call it $x_0$) is selected to be the initial value for iteration. The solution of the function is then subsequently recycled as the new input value, $x$. In this way:

\begin{align*}
x\hspace{0.3mm}_1&=x\hspace{0.3mm}_{0}^2 + c,\\
x\hspace{0.3mm}_2&=x\hspace{0.3mm}_{1}^2 + c = (x\hspace{0.3mm}_{0}^2 + c)^2 +c,\\
x\hspace{0.3mm}_3&=x\hspace{0.3mm}_{2}^2 + c = \cdots\\
\text{and in general, }x\hspace{0.3mm}_n&=x\hspace{0.3mm}_{n-1}^2+c.
\end{align*}

We continue until the iteration either converges to a fixed point or cycle, or diverges to infinity. The orbit is the sequence of numbers generated during the process of iteration: $x_0,x_1,x_2,x_3,\ldots,x_n$. If we only apply real numbers to the quadratic function we limit the graphical representation of the iterations to a line. To produce pictures in the plane, we use complex numbers instead.

The abundant beauty in the plots is somehow increased when the simplicity of the mathematics is understood.

Through the process of iteration, each seed will either converge or diverge, and so for a given function we can divide the plane into an escaping set $E_c =\{ z_0 : |z_n| \rightarrow \infty \, \mbox{as} \, n \rightarrow \infty \}$ (that is, all the seeds that end up at infinity) and prisoner set, where the iteration tends to a point or becomes periodic.

The Julia set of a function

To go from the iterative procedure described above to the vivid images to the right, we need to introduce the idea of the Julia set of a function, named after the French mathematician Gaston Julia. Julia was an extraordinary man, who tragically lost his nose while fighting in the first world war. Despite the substantial injury, he made immense progress in the field of complex iteration and published the book Mémoire sur l’itération des fonctions rationnelles in 1918, which began the study of what we now call a Julia set.

The filled-in Julia set is the collection of points in the complex plane that form the prisoner set of a function, while the Julia set itself is the boundary of this region. The points within the filled-in Julia set remain bounded under the iteration since their orbits converge to an attracting point or cycle.

Connected and unconnected Julia sets of the quadratic function for different values of $c$

Conventionally, when pictures of the Julia set are shown, the filled-in Julia set is shaded black and varying colours are used to show the rate at which the escaping set diverges to infinity. The Julia set is therefore the edge of the black region. Maps 1–7 above show the Julia sets of the quadratic function for different values of $c$, with the escaping set colour-coded as follows: red areas represent points that slowly escape from the set, while blue areas signify points that quickly escape to infinity. The value of the complex constant $c$ influences the shape of the Julia set.

Maps 1, 4 and 5 all have black centres, which indicate that the Julia set is connected, while maps 3, 6 and 7 demonstrate unconnected sets. For these images, the Julia sets have no black regions and instead the pictures are just flurries of colour. It is not always easy to spot whether a Julia set is connected, however. In map 2, there is no obvious black region, but neither are there colourful individual flurries and instead we see a spiky line. In fact the set is connected, it is just that the filled in Julia set is so slender that the black line points are not visible in the image.

During the initial study of these sets, a fascinating criterion for connectivity was discovered concerning the critical point, $z_0=0$. If the critical point is used as the seed, we produce the critical orbit, which is bounded if and only if the Julia set is connected.

Fractal patterns appear in all plots, apart from when $c=0$ or $-2$. The picture below displays examples of magnified sections of the fractals, for $c=-0.7$ (maps 9–12), $c=-0.12 + 0.75 \,\mathrm{i}$ (maps 13–16), $c=0.1 + 0.7 \, \mathrm{i} $ (maps 17–20) and $c=-0.1 + 1 \, \mathrm{i} $ (maps 21–24). Each enhancement of a section produces what appears to be copies of the whole section, not just in overall shape but also with smaller embellishments on every “limb”. For connected plots, these fractals appear as loopy ovals and circles or thin, almost stick-like, sections. For disconnected plots, however, the fractals are grouped together in intricate floral patterns, revealing the same shape and pattern with each level of magnification.

Magnified sections of fractals for different values of $c$

Prior to computer technology, Julia had to rely on his imagination and manually carry out the iterations by hand. Fifty years later, another mathematician applied modern computing power to plot these pretty pictures, finally showing the sets in all their beauty…

The Mandelbrot set

The Mandelbrot set is named after the Polish mathematician Benoit B Mandelbrot, known for being the founder of fractal geometry. The word fractal is derived from the Latin fractus, which means broken, and describes the shape of a stone after it has been smashed.

Mandelbrot discovered that fractals appear not only in mathematics but also in nature, through crystal formation, the growth of plants and landscapes, as well as in the structure of the human body. In 1945, Mandelbrot read Julia’s 1918 book. He was fascinated and, with the aid of computer graphics, was able to show that Julia’s work contained some of the most beautiful fractals known today.

To create the Mandelbrot set, each complex value of $c$ is used as the constant term in the quadratic function $f\hspace{0.3mm}(z\hspace{0.2mm})=z\hspace{0.3mm}^2+c$ and iterated with the critical point $z_0=0$ as the seed. If the orbit escapes to infinity, the number of iterations taken for the modulus of the function to exceed a specified value is used to decide on the colour of the map at that point, $c$. Otherwise, when the orbit converges, the point is coloured black. The Mandelbrot set is the set of black points.

For example, if we let $c=-0.15+0.3 \, \mathrm{i}$ then we have the complex quadratic function $f\hspace{0.3mm}(z\hspace{0.2mm})=z\hspace{0.3mm}^2-0.15+0.3 \, \mathrm{i}$. We start with $z_0=0$ as the seed and the sequence of iteration (to 5 significant figures) is as follows:

\begin{align*}
z_1&={0}^2 -0.15 +0.3 \, \mathrm{i} &&\Rightarrow &z_1&= -0.15 +0.3 \, \mathrm{i},\\
z_2&=(-0.15+0.3 \, \mathrm{i})^2-0.15+0.3 \, \mathrm{i} &&\Rightarrow &z_2&= 0.2175 +0.21 \, \mathrm{i},\\
&&&&z_3&=-0.14679+0.20865 \, \mathrm{i},\\
&&&&z_4&=-0.17199+0.23874 \, \mathrm{i},\\
&&&&z_5&=-0.17742+0.21788 \, \mathrm{i}.
\end{align*}

Continuing to 30 iterations, the orbit has not escaped to infinity and instead converges to the point   $z=-0.17082+0.22361\, \mathrm{i}$ (again to 5 significant figures). Therefore, $c=-0.15+0.3 \, \mathrm{i}$ is within the Mandelbrot set and is coloured black.

On the other hand, if we take $c=-1.85+1.2 \, \mathrm{i}$, and hence the complex quadratic function $f\hspace{0.3mm}(z\hspace{0.2mm})=z\hspace{0.4mm}^2-1.85+1.2 \, \mathrm{i}$, then the sequence of iterations (to 5 sf) is as follows:

\begin{align*}
z_1&={0}^2 -1.85 +1.2 \, \mathrm{i} &&\Rightarrow &z_1&= -1.85 +1.2 \, \mathrm{i},\\
z_2&=(-1.85 +1.2 \, \mathrm{i})^2-1.85 +1.2 \, \mathrm{i} &&\Rightarrow &z_2&= 0.1325 -3.24 \, \mathrm{i},\\
&&&&z_3&= -12.33+0.3414 \, \mathrm{i},\\
&&&&z_4&= 150.06 – 7.2189 \, \mathrm{i},\\
&&&&z_5&= 22465 – 2165.4 \, \mathrm{i}.
\end{align*}

The characteristic segments of the Mandelbrot set

If the modulus of $z$ exceeds 100, then it has been proven that the orbit escapes to infinity. This occurs on the fourth iteration, so the colour chosen to represent the value of 4 would be plotted at the point $(-1.85,1.2)$ in the complex plane. The resulting image is shown in map 8, and also in the picture to the left.

The largest segment of the set is called the cardioid due to its heart-like shape. Attached to this are adornments called bulbs, upon closer inspection of which it is possible to see many smaller, somewhat similar, embellishments. The bulbs are not completely identical, although most exhibit a similar shape, and the main differences can be seen in their filaments. The filaments are the thin strings of bounded points that sprout like sticks from the tops of the bulbs. These sticks are extremely narrow and they appear to be coloured red, which would indicate they are not part of the set. However, if we were to zoom in closer on these regions, we would actually see black lines!

The self-similarity of the Mandelbrot set

The Mandelbrot set is self-similar, consisting of miniature Mandelbrot sets within the boundary of the largest set. By enhancing the filaments, smaller copies of the overall set appear in ‘Russian-doll’ like fashion, as seen in maps 26–30 above. Closer inspection of map 27 shows many more self-similar sets within the filaments around the perimeter of the Mandelbrot set. Magnifying the small copies of these Mandelbrot sets would yield infinite layers of self-similar sets.

Other fascinating and intricate shapes occur, for example the “seahorse valley” that is visible in maps 31–34 above. By enhancing the plot within this region we see two rows of seahorse shaped embellishments, each with “eyes” and “tails”. Further magnification of the “eyes” reveals spiral constellations of more “seahorses”.

Connection between Julia sets and the Mandelbrot set

The orbit of the critical point $z_0=0$ can be used to test the connectivity of the Julia set, and the Mandelbrot set shows the boundedness of these critical orbits. Hence, the Mandelbrot set itself indicates the connectivity of the Julia sets of all the different complex quadratics. The Mandelbrot set can be described as $M = \{ c \in \mathbb{C} \, | \, J_c \, \mbox{is connected}\} $, where $J_c$ is the Julia set of the function $z\hspace{0.3mm}^2+c$. The Julia set is a connected structure if $c$ is within the Mandelbrot set, and will be broken into an infinite number of pieces if $c$ lies outside the Mandelbrot set.

The cardioid-shaped main body contains all values of $c$ for which the Julia set is roughly a deformed circle (figure below: maps 35, 37, 38 and 40). The values of $c$ which lie in a bulb of the Mandelbrot set produce a Julia set consisting of multiple deformed circles surrounding the points of a periodic attractor. The number of subsections sprouting from a point on the Julia set is equal to the period of the bulb in the Mandelbrot set (below; maps 36, 39, 41–44).

A specific Julia set can be defined by a point in the Mandelbrot set

The nature of the convergence of points within the Mandelbrot set depends on the segment in which the point resides. Seeds within the cardioid converge to an attractive point, whereas orbits starting in the bulb lead to an attracting cycle.

Three particularly interesting cases of Julia sets are shown below. The first is when $c=0$, where the filled-in Julia set comprises of all the values within the unit circle (circle of radius 1, centred on the origin) and each of these points converges to $0$ when iterated. The Julia set is the boundary of the circle, the points of which, when iterated, remain on the boundary.

Three remarkable examples of the Julia set with $c=0$, $c=\mathrm{i}$ and $c=-2$

The second interesting case is when $c=\mathrm{i}$. Here, the Julia set is a dendrite, meaning there are no interior points. Instead, the set is just a branch of points. For this complex constant the dendrite is a single line in an almost lightning-bolt shape. The final case is $c=-2$, where the Julia set is a dendrite that lies directly on the horizontal axis between  $-2<x<2$.

Explore the sets yourself

I hope to have displayed the beauty behind these pictures by emphasising the extraordinary quantity of information contained in such a simple procedure, as well as through highlighting the complexity of each image, in the variety of fractals and colours visible, which further enhances the beauty.

If this article has sparked an interest in fractals, then why not try exploring these sets for yourself? You could do this by magnifying different sections of the Mandelbrot set to explore the countless shapes and patterns that exist within. You could also go deeper into exploring individual orbits.

All of these pictures are generated using simple quadratic formula. However, the Julia and Mandelbrot sets can be produced for a wide variety of functions in a similar manner to obtain countless pretty pictures.

These images are already becoming dated, having been taken for granted for so many years since they were first produced on the big bulky computers of the 1980s. The Julia set of the quadratic function, and the corresponding Mandelbrot set, could be inspiration for pretty pictures which are yet to be fully explored, or even discovered. Largely, the discoveries discussed here have been recorded in recent years. Furthermore, there could still be vast amounts of information within these sets that are yet to be discovered. Could you be the one to make a discovery?

References

  1. Franke, H. (1986). Refractions of Science into Art. In: H. Peitgen and P. Richter, ed., The Beauty of Fractals, 1st ed. Berlin: Springer-Verlag, pp.181-187.
  2. Mandelbrot, B. (2004). Fractals and chaos. 1st ed. New York: Springer.
  3. Peitgen, H., Jurgens, H. and Saupe, D. (1992). Fractals for the Classroom: Part 2: Complex Systems and Mandelbrot Set. 1st ed. New York: Springer-Verlag, pp.353-473.
  4. Hall, N. (1992). The New Scientist guide to chaos. 1st ed. London: Professional Books.
  5. Douady, A. (1986). Julia Sets and the Mandelbrot Set. In: H. Peitgen and P. Richter, ed., The Beauty of Fractals, 1st ed. Berlin: Springer-Verlag, pp.161-173.
  6. Fraser, J. (2009). An Introduction to Julia Sets. 1st ed. [ebook] Available at: http://www.gvp.cz/~vinkle/mafynet/GeoGebra/matematika/fraktaly/linearni_system/julia.pdf [Accessed 4 Apr. 2017].
  7. Peitgen, H., Jurgens, H. and Saupe, D. (1992). Fractals for the Classroom: Strategic Activities Volume two. 1st ed. New York: Springer-Verlag.
  8. Devaney, R. (2006). Unveiling the Mandelbrot set | plus.maths.org. [online] Plus.maths.org. Available at: https://plus.maths.org/content/unveiling-mandelbrot-set [Accessed 4 Apr. 2017].
  9. Moler, C. (2011). Experiments with MATLAB. 1st ed. [ebook] MathWorks, p.Chapter 13 Mandelbrot Set. Available at: https://uk.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/exm/chapters/mandelbrot.pdf [Accessed 20 Apr. 2017].
  10. Peitgen, H. and Richter, P. (1986). The Beauty of Fractals. 1st ed. Berlin: Springer-Verlag.
  11. Mandelbrot, B. (1986). Fractals and the Rebirth of Iteration Theory. In: H. Petigen and P. Richter, ed., The Beauty of Fractals, 1st ed. Berlin: Spring-Verlag, pp.151-160.
  12. Uk.mathworks.com. (2017). Illustrating Three Approaches to GPU Computing: The Mandelbrot Set – MATLAB & Simulink Example – MathWorks United Kingdom. [online] Available at: https://uk.mathworks.com/help/distcomp/examples/illustrating-three-approaches-to-gpu-computing-the-mandelbrot-set.html?searchHighlight=mandelbrot%20set&s_tid=doc_srchtitle [Accessed 20 Apr. 2017].

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Four things you didn’t notice in Issue 05

With just a couple of days to go until we launch issue 06, we thought it’d be fun to share a few bits and pieces that we hid around issue 05. If this gets you excited for issue 06, why not come to the launch party on Thursday?!

Scorpions

Since we published the horoscope in issue 03, scorpions have been running around all over Chalkdust HQ. Seven of them managed to sneak into issue 05.
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The kind of problems black mathematicians wish didn’t need solving

John Derbyshire, columnist for the National Review, wrote an essay implying that blacks are intellectually inferior to whites: only one out of six blacks is smarter than the average white. Derbyshire pulled these figures from a region near his large intestine.

One of Derbyshire’s claims, however, is true: there are no black winners of the Fields medal, the ‘Nobel prize of mathematics’. According to Derbyshire, this is “civilisationally consequential”.

Derbyshire implies that the absence of a black winner means that blacks are incapable of genius. His ilk are only able to sustain such lies because 150 years of racial terrorism have ensured that few dare to challenge them, and, when we do, the consequences are dire. His ilk can get away with thinking that Euclid and Eratosthenes were not Africans working in Africa (even “sub-Saharan Africa”, if they want to make that idiotic distinction), but Greeks with blond hair and alabaster complexion (much like Jesus).

In reality, black mathematicians face career-retarding racism which white Fields medallists never encounter. It’s hard to focus on abstract algebra after Belgian King Leopold has hacked off your hands. Three stories will suffice to make this point. Continue reading