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Feeling the love for Chalkdust T-shirts

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t-shirt-more-blackIn case you weren’t aware, Chalkdust make some really incredible-looking T-shirts. You can buy one here! However, the design tends to puzzle people. I haven’t yet met anyone that is good enough at curve sketching in their head to work out what the equation on it means.

 

The $\sqrt{-1}$ does give you a hint. Unless you’re an electrical ilovenyengineer, you could read it as $\mathrm{i}$ and you might guess where the rest of it is going.

But understanding what the equation means is a nice curve sketching problem.

\[ x^2+(y-\sqrt[3]{x^2})^2=1 \]

Using Wolfram Alpha is cheating! Curve sketching is a skill vastly underrated by almost everyone doing maths exams – but I think it’s hugely satisfying.

Since $x$ only appears as a function of $x^2$, the $y$ values will be the same for $x$ and $-x$ and thus the curve will be symmetric in the $y$ axis.  There’s a $y^2$ term as well, so we should expect two $y$ values for each $x$ value.

It also looks a lot like the equation for a circle – but what is that $\sqrt[3]{x^2}$ doing? If we rewrite the equation as a function of $y$ we get more of an idea:

Putting your heart back together

\[ y = \sqrt[3]{x^2} \pm \sqrt{1-x^2} .\]

The second term is just the equation of the circle, with the positive giving the upper half and the negative the lower half. So the first term is modifying both the top and bottom half of the circle. Since $\sqrt[3]{x^2}$ is increasing as you move away from the origin, it “pulls up” the curve by more as you move away from the centre.

Therefore, the top half on the circle now has two maxima, and the slope of the bottom half is much steeper. What have we drawn? A lovely heart!

Feeling the love for Chalkdust

However, our circle modifying expression $\sqrt[3]{x^2}$ isn’t unique. We need it to be even with a minimum at $(0,0)$, and to get the cusp in the middle, we need it to be not smooth at the origin. So quite a few functions of $\mid x \mid$ could work.

Paul Ma has looked at lots of different versions of equations for hearts. (However, I can’t be held responsible for anything else you might learn to plot from this article.)

henrici-crop

Henrici’s model of a cubic surface

Computers are now enormously helpful for understanding these kinds of problems – in three dimensions as well. But in the late 19th century, this was much more difficult – and serious geometers spent a lot of time thinking about strange looking surfaces. Without computers, they built intricate paper models, like this one by Olaus Henrici, which shows the surface given by \[ xyz = k^3(x+y+z-1)^3 .\]

Now that you also understand the true meaning of the Chalkdust T-shirt, you can join mathematical fashion pioneers Ian Stewart, David Colquhoun and Bearnoulli’s Principaw in getting your own!

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The wonders of mathematical crochet

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Several months ago, I learned to crochet, and I’ve been hooked ever since. And as a mathematician, one of the most fantastic things about it is that it allows you to make mathematical objects that are enormously difficult to make, or even visualise in any other way.

For those of you unfamiliar with crochet, all you need is a ball of wool and a crochet hook. You then use this hook to pull loops of wool through other loops of wool, and from this you very quickly make a stretchy, slightly holey fabric. You can either use rows of stitches to make a rectangle, or you can begin in the middle of your work and make rounds of stitches that meet at either end.

Single crochet stitch

How to make a single crochet stitch

If you’re crocheting with rounds of stitches, the most obvious shape that you can make is a circle. But how do we construct a perfect circle?

Each single crochet stitch can be thought of as a square, albeit one that can be easily squashed or stretched. (I’m using US terminology here.) For the first round of stitches, we want to make a circle with radius $1$. Therefore, the circumference will be $2\pi$. In this case, we’ll take $\pi \approx 3$ for simplicity, and so in the first round we’ll have $6$ stitches.

When we add the second round, the radius of the circle is now $2$, and so we need $4\pi$ stitches in this round, which, using our crude approximation of $\pi$, is $12$ stitches. For the next round, the circle has radius $3$ and so the round has $18$ stitches. So in each round we add $2\pi$, or $6$, more stitches—the number of stitches in each round increases linearly.

circle

A crochet circle, with linearly increasing stitches in each round.

If you try this and make a mistake in the number of stitches in a round, then you might notice that the circle stops being flat and starts to curve. This fact is actually what makes crochet such a powerful tool for creating mathematical objects—by changing the number of stitches then we change the curvature of the surface. The circle is flat, and therefore has zero curvature.

So what happens if we increase the number of stitches in a slower fashion? Each new round won’t stretch around the previous round, and hence we will create a surface with positive curvature. The classic example of a surface with positive curvature is a sphere—at every point on the sphere the surface curves away in all directions. To crochet a sphere, the number of stitches in each round increases like $\sin \theta$, where $\theta$ is the angle around the sphere. You can generate patterns for mathematically ideal crochet spheres here.

https://mspremiseconclusion.wordpress.com/2010/03/14/the-ideal-crochet-sphere/

A perfect crochet sphere – you can find the pattern here

On the other hand, what if we increase the number of stitches in each round faster than linearly? For example, what if we double the number of stitches in each round so that the length of the rounds increases exponentially? Each round is much larger than the one before it, and so the surface has to curve up and down. By continuing this pattern, you create a hyperbolic pseudosphere, which a surface of constant negative curvature. This means that if you look at any point on the surface, it will look like a Pringle—curving upwards in one direction and downwards in the other.

Prior to 1997, it was extremely difficult to build a robust model of a hyperbolic pseudosphere, but then mathematician and crocheter Taina Daimina realised that you could easily construct one using crochet. The idea was hugely popular and led to a book, as well as a huge project creating coral reefs using hyperbolic crochet.

Photo by https://www.flickr.com/photos/pandaeskimo/

Crocheted hyperbolic pseudosphere by Panda Eskimo

The following graph shows how the number of stitches in each round should increase when making a circle, sphere or hyperbolic surface.

Crochet-graph

The existence of hyperbolic crochet is perhaps the best known example of mathematical crochet, but it only touches the surface of what crochet can represent. For example, an easy crochet project for beginners is a rectangular scarf. If you make a twist in the scarf and sew the ends together, you’ve made yourself a Möbius strip, which is a surface with only one side. (Try making one with some paper and drawing a line along the centre.) Apparently Alan Turing liked to knit Möbius strip scarves.

Mobius strips zip together to make Klein bottle

Two crochet mobius strips with twists going in opposite directions. Zip them together to get a Klein bottle!

A particularly interesting fact about Möbius strips is that if you glue the edges of two strips together, then you create a Klein bottle. However, you can’t demonstrate this physically with paper, because it’s not stretchy or flexible enough. However this is actually a very easy project using crochet—I made one recently in an afternoon or two. An alternative way to think of a Klein bottle is to glue the ends of a tube together in opposite directions. I used this approach to crochet a Klein bottle hat, using the crochet lathe to generate a pattern. This clever tool gives you a pattern for any surface of revolution.

klein-bottles

Two crocheted klein bottles. The right hand one is made by zipping together two Mobius strips

Another classic crochet project is granny squares. With lots of these squares, you can then construct polyominoes, and make a mathematical/tetris blanket like this one.

Polyominoes blanket by www.twitter.com/tkleinwalsh

Polyominoes blanket by Tina Klein-Walsh

A particularly impressive mathematical crochet feat is this model of the Lorenz manifold, created by researchers at the University of Auckland. The complexity and curvature of this surface means it is very difficult to create physically, but once again the flexibility of crochet came to the rescue.

Crochet model of Lorenz Manifold

Crochet model of Lorenz Manifold

And one final example of the possibilities of combining mathematics and crochet—hexaflexagons. Woolly Thoughts, who create beautiful mathematical knitting, have written a pattern for a crocheted hexaflexagon cushion.

hexaflexagon

I hope I’ve convinced you to pick up a hook and get crocheting! While some of these patterns are very complex, a hyperbolic pseudosphere is very simple, and was actually the first thing I crocheted. (Many thanks to Matthew Scroggs for teaching me.) What other mathematical objects can you create with crochet?

Cover image by Cheryl on Flickr.

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In conversation with Artur Avila

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Sitting in a pub in Leicester Square, talking to one of the most brilliant mathematicians of our generation, is not the way one would normally expect to spend a sultry evening in early June. But it turns out that Artur Avila, winner of the 2014 Fields Medal, takes very spontaneous holidays, and is a big fan of the pub.

In the UK, Artur certainly does not conform to the stereotype of a mathematician. He is good-looking, stylishly dressed in a white T-shirt and designer jeans and asking about the best London nightclubs. However, Avila was born and bred in Rio de Janeiro, famous for its spectacular parties and beautiful beaches. He still spends half his time there, based at the National Institute of Pure and Applied Mathematics (IMPA), and spends the other half at the French National Centre for Scientific Research (CNRS) in Paris, where the nightlife is terrible, according to Avila.

It’s perhaps unusual for an exceptional mathematician not to spend all their time in the USA or Europe, where there are higher numbers of world class research institutions, but Avila believes ‘it’s significant that I studied at IMPA because it shows that Brazil has institutions that can prepare someone to do maths at a high level, and it’s not necessarily true that you always have to go to the United States or to Europe to advance.’

And, of course, Rio has many appealing features: ‘I have several times brought collaborators to the beach with me and we would just sit and share ideas with each other, with the sound of the sea in the background.’

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In conversation with Hannah Fry

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Hannah Fry is a lecturer at the Centre for Advanced Spatial Analysis (CASA) at UCL. In addition to her research on the mathematics of social systems, Hannah also does a lot of public engagement – showing the general public some of the fascinating ways that Maths can be used in the real world. She’s given TED talks, spoken on TV and radio, made YouTube videos, and performed in science stand-up and stage shows. Most recently, she has written a book called The Mathematics of Love, and presented the BBC documentary Climate Change by Numbers.

Maths and the Real World

Would you like to tell our readers a bit about your mathematical background?

I did my undergraduate degree in Maths here at UCL and I much preferred the applied side. I then did my PhD in fluid dynamics with Prof. Frank Smith, doing lots of lovely asymptotic analysis. My postdoc was a bit different. I think fluids is a really great place to train, but it’s hard to find a really good postdoc in fluids, and you can’t pick the subject that you want to work on. So this postdoc came up using mathematics to look at social systems – things like trade, migration and security. And I just thought it was an interesting topic and came over here to CASA, and I’ve been here ever since!

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