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Christmas conundrum #1

It’s finally time for the first Chalkdust Christmas conundrum. Four lucky people who submit the correct answer to the puzzle will win Chalkdust T-shirts. The deadline for entries is Friday 8th December at 6pm.

It’s nearly Christmas and you have wrapped up five presents for your five best friends: Atheeta, Bernd, Colin, Dominika and Emma. You are especially happy this year as each present is the shape of a different Platonic solid.

But in your excitement, you have just forgotten which present is for which friend. You can only remember the following facts:

  1. Atheeta’s present has more faces than Emma’s present.
  2. Atheeta’s present has fewer vertices than Emma’s present.
  3. The faces of Colin’s present are triangles.
  4. Three faces meet at every vertex of Bernd’s present.
  5. The edges of Dominika’s present have an integer length, and her present has an integer volume.

Who is the icosahedral present for?

Once you have solved the puzzle, enter your answer below for a chance to win. The deadline for entries is Friday 8th December at 6pm. The winners will be announced in next week’s conundrum post, when you will also have a chance to win a copy of The Indisputable Existence of Santa Claus by Hannah Fry and Thomas Oléron Evans.

This competition is now closed.

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Chalkdust’s Christmas conundrums

Winter is coming. The temperature is dropping quickly (although not all the way to absolute zero) and the days are getting shorter. To cheer us up at this bleak time of year, somebody decided it was a good idea to have an annual celebration. No, not Isaac Newton’s birthday. We are talking, of course, about Christmas!

Ah, Christmas. What better way to spend the holiday season than relaxing at home, curled up by the fire with a good book. Or perhaps you’re the sort of person who prefers to unwind by tackling a Yule-themed puzzle or two. Or maybe you like to wear a fashionable, mathematical T-shirt that’s available in a variety of colours and any of the sizes that aren’t medium?

What about if you could do all three… at the same time‽

Continue reading

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Review of The Maths Behind by Colin Beveridge

You know what the biggest problem with The Maths Behind is? ‘Ice Col’ Colin Beveridge has basically covered every possible topic we can ever blog about on the Chalkdust website. This book explores the maths behind 57 everyday events: that’s more than a year of blog posts!

If this doesn’t strike you as a particularly large problem, then I reckon you will enjoy this book.

The first thing that strikes me about this encyclopedia of mathematical modelling is that it is very pretty. The publishers produce a lot of cookery books, and their designers have done a really rather good job of bringing these fifty-or-so topics to life. Most topics cover two to four pages, with large colourful graphics.

Stopping distances in The Maths Behind

Oh cool, stopping distances! What an interesting thing to write about.

Continue reading

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MathsJam 2017

Last weekend, seven us went to a conference centre near Stone in Staffordshire for this year’s MathsJam Gathering. In this blog post, we have collected together our highlights, which we present in approximate chronological order.

The train journey (Scroggs)

Playing Origins of World War I on the train.

The fun started before we even arrived. We spent the majority of the train journey playing Origins of World War I, a game taken from Sid Sackson’s A Gamut of Games. I won, which is unusual, especially when Bolt is one of my opponents.

The train journey was followed by the worst part of the weekend: carrying around 200 copies of Chalkdust over Stone station’s footbridge.

Saturday afternoon (Sean)

This was my first MathsJam, and I was amazed by the huge variety of the talks. Some people spoke about puzzley, gamey maths (which I had expected). Other people ranted or joked about mathematical errata from every day life (I should have expected this) and two people read truly beautiful poetry that genuinely moved me (I really didn’t expect this!). On top of the talks, everybody was lovely and there was an impressive supply of entirely adequate food for lunch. After my initial scepticism, I am already looking forward to next year!

Saturday lunch (Adam)

Saturday afternoon (Alison)

I strongly recommend giving a talk at MathsJam. Even if you have never attended before, you get so much out of presenting something interesting (or ranty) to a group of maths-obsessed people. It’s also the best way to meet other people after your talk: I spoke about Stupid Units and for the rest of the weekend people kept coming up to me with their favourite stupid unit. I now have enough units for five more talks!

If you are thinking of talking at MathsJam, absolutely go for it! They love it when new people speak. If you think your talk contains stuff everyone knows, trust me, it won’t. If you think it might not be mathematical enough (I was worried about this), you will be sent email addresses of people you can ask for guidance. If you are terrified of public speaking, go for it anyway! All the talks are only five minutes so it can be a great way of getting more practice talking to an audience, if that’s what you want. The moral of the story is: give a talk.

Saturday evening (Bolt)

TD and Bolt playing a game from A Gamut of Games.

This was my first MathsJam too and I also had a fantastic time. The Chalkdust delegation ran a table on Saturday evening where we taught people how to play some mathematically interesting games from Sid Sackson’s magnum opus, A Gamut of Games. Some of the other tables included: learning how to make delightful polyhedra using origami, learning how to play Go, a Markov chain game for generating new English words, and mathematical Dungeons & Dragons.

Later Saturday evening (Belgin)

On Saturday evening, after the mathematical Dungeons & Dragons was over, it was time for the MathsJam Jam: now here’s one singalong I couldn’t turn down! It was fun to sing (and laugh) along to rewritten versions of many famous songs. My favourite was Wonderwall, and I’m not even a fan of Oasis. The title was changed to Traversible, the lyrics were now about the postman tour problem, and to me it sounded better than the real thing!

The MathsJam room during the evening activities.

There were plenty of other mathematical versions of popular songs, and they were all hilarious. Popular maths indeed!

Later Saturday evening (TD)

While everyone else was having a little sing-song, I productively spent the evening learning about medieval French poetry with Adam Atkinson. One of the highlights was the following masterpiece:

Un petit d’un petit
S’étonne aux Halles
Un petit d’un petit
Ah! degrés te fallent
Indolent qui ne sort cesse
Indolent qui ne se mène
Qu’importe un petit
Tout gai de Reguennes.

(Hint: Read it aloud in your worst French accent. Spoiler at the bottom of the post.)

Even later Saturday evening (Scroggs)

After the MathsJam Jam, it was already quite late, but there was still plenty of time to try out a variety of the games that people had brought along. Highlights included Hare and Tortoise, Pinguin Party, and Dobble. I finally went to bed at around half past midnight, as Bolt was just starting a game of Settlers of Catan. He assures me that he won.

Sunday breakfast (Belgin)

Weetabix review: Tasteless! You have to add some honey/nuts/fruit/other stuff to make your breakfast taste of anything. And it soaks up milk like a sponge, so you’re left with no milk in your bowl. Maybe I’ll go for Wheat Bisks instead..

Sunday (Adam)

MathsJam weekend is a wonderful combination of everything I enjoy about pop maths. By sheer coincidence, the ethos of the weekend was distilled beautifully into session 2c on Sunday. Let me tell you about it. Six talks; five minutes each. If you exceed your time, Rick Astley plays you off over the speakers.

First, Glen Whitney brings out half a dozen huge 3D shapes that we helped make the night before. How many holes does it have? He whips out some play-doh, and shows that T-shirts are topologically the same as these giant creations.

Pedro in the middle of his trick.

Next, Sue de Pom is dressed up as Ada Lovelace and tells us some stories we don’t know about her (see issue 07). She ends with the sad tale of Ada’s death. No-one knows, but she is half a second from running over time, and Matt Parker deftly kills Rick before he can launch into this wildly inappropriate moment. It’s a win for the hidden tech.

Pedro Freitas shows us a card trick. Somehow cards can do subtraction? We debate how the trick works on the train home. Matthew and John Bibby do an amusing father-and-son two-hander, where only one of them has the slide clicker, and Geoff Morley shows us some classic portrait A4 slides musing over irrational or negative number bases. (Question: do you need a minus sign in a negative base?)

One of the many interesting object left on the table of things to play with (but give back afterwards).

Finally, Adam Atkinson, the aforementioned distributor of fine medieval French poetry, contemplates how large a statue of a hedgehog ought to be if he wants to place it on top of some obscure mountain near Catania. The audience erupts in applause.

This session was funny; it was touching; it was puzzling; it made you want to go and have a play. This spirit, one of earnestness and inclusivity, is not easy to cultivate. It’s extremely difficult to invite people from everywhere, based only on the criterion that they “like maths”, and then get them to bond over an assortment of five-minute talks. Among the delegates you’ll find amateurs who just enjoy playing with puzzles; you’ll find people working in universities; you’ll talk to people who engage with the public professionally. And yet what Katie, Matt and especially Colin have done is to make none of those differences matter. The conference is so full of things to do, ways to get involved, and people to make friends with, that you wonder why other conferences are so bad. Maybe the secret really is Rick Astley.

Sunday afternoon (Alison)

The winning cake.

Just before the final talks, it was time for the competition prizes. First, the prize for the best cake in the MathsJam Bakeoff. This was deservedly awarded to mathematician, crossnumber enthusiast and occasional poet Sam Hartburn.

After this, it was my job to award the prize for the competition competition. The competition was strong, but the winner of the competition competition was Katie Steckles, who beat all the competitions in competition with her competition. Competition competition competition competition competition competition. Competition.

Maybe the highlight of the prize giving was when Matt Parker won the prize for the most entries in a competition. His prize was a £5 voucher to spend on Maths Gear, an excellent shop for mathematical toys and games that just happens to be run by Matt Parker himself.

Another train (Scroggs)

After the end of the talks, it was time to get the train home. See you all next year!

 

 

Medieval French poetry (spoiler)

When read in a French accent, the poem sounds exactly like Humpty Dumpty. It is taken from Mots d’Heures, a collection of English nursery rhymes that have been “translated” homophonically into French.

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Beghilos

Every schoolchild (and former schoolchild) has played around with typing numbers on a calculator, turning it so the display screen is upside down and giggling at the (usually rude) word spelled out. These words are called beghilos, named for the only letters you can make out of numbers on a standard calculator: b from 8, e from 3, g from 6, h from 4, i or l from 1, o from 0, and s from 5. Some variations use a 2 as Z and o as a capital D.

Using the traditional beghilos, there are over 200 possible words, although some of them are a bit rubbish like GI ( a martial arts uniform) and GHOLE (an anarchic term for ghoul, but sounds like it should be something else). Continue reading

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Dispersion on the dark side of the moon

For the last fifty years, one of the most reliable ways of making sure that your artwork is seen by thousands of people around the world has been to stick it on the front of a (successful) album. Images used by artists like The Beatles, The Velvet Underground and The Clash are so famous not only because they adorn the fronts of CDs and LPs, but also because of the huge amount of branded merchandise that they have spawned. T-shirts, tote bags, coasters and Sex Pistols Virgin Money credit cards have ensured that album covers provide some of the most enduring modern works of art.

Given this, it might be easy to dismiss album art as a purely commercial exercise. But many artists also look to link their cover art with their music by visually representing some of the central themes. Perhaps it is not surprising, then, that some of the best album covers can be looked at through a mathematical lens. After all, mathematicians have always had ways to visualise abstract ideas, and there are easy connections to be made between physics and the psychedelic and cosmic imagery that was popular during the ‘golden era’ of the concept album. Let’s take a look at one of the most famous examples… Continue reading

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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.

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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”.