Category Archives: Mathematics

The Chaotic Pendulum

Double pendulums are pretty awesome. Funnily enough, they’re pretty much what they sound like – a pendulum, with another one attached to the bottom. But the cool part is come from their twin nature, but the type of motion they exhibit – chaotic motion.

People often confuse chaotic behaviour with random behaviour; random behaviour has no deterministic qualities (so you can’t tell from previous physical conditions what is going to happen), whereas chaotic behaviour is actually deterministic, meaning it is possible to predict. The difference between chaotic deterministic behaviour and regular deterministic behaviour, however, is that tiny, tiny changes in initial conditions in a chaotic systems can bring about massive changes, unlike a normal deterministic system, where tiny changes in initial conditions will bring about tiny changes overall. You may have heard of something called “the butterfly effect” – where a butterfly flapping its wings on one side of the world can cause a hurricane on the other. This is a chaotic system, as even something as small as the miniscule breeze from a butterfly flapping its wings can cause massive changes in weather, which may at first appear random but actually do have a deterministic cause.

The fact that double pendulums are chaotic is that go can set two different pendulums up to be almost exactly the same, but not quite, and after a period of time they’ll be totally different from each other. It also makes the path a double pendulum traces out amazingly beautiful and crazy, almost like those geometric roses, or the drawings you made using spirographs as a kid.

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There are chaotic systems everywhere in nature, from geology to robotics, and biology to economics. Chaotic systems can be very difficult to detect, because they can appear so similar to random systems unless you measure them closely. Because of this, there are certain traits of dynamical systems that scientists will look for to see if a system is chaotic. These are things like the fact that there isn’t any periodic (repetitive) behaviour in time, but there  is in something called phase space, which is a bit like the variation of the speed of the pendulum with its displacement. Also, the fact that tiny initial differences between two pendulums will grow exponentially with time. There are other chaotic properties, like the fact it’s nonlinear, and that it must have at least three independent, first-order autonomous differential equations which describe it, but those are harder to explain within bringing up the ridiculously long and complicated equations of motion, which are what you use to describe how the pendulum moves (funny that, what with their name).

Chaos is such a wonderful thing, because at first it appears completely crazy, and random, and completely unordered. However, if you have the ability to actually look close enough, it’s perfectly deterministic, and not at all unpredictable. A bit like so many messy things in life, if you have the patience and the time to look for the source, and untangle the enigma, you’ll be able to to decode the whole thing, and know what’s actually going to happen.


 

Okay, so this week, I’ve been very busy with my summer project – building a simulation of a double pendulum. Double pendulums are pretty damn cool, exhibiting chaotic behaviour and being hypnotising to watch. So, I thought it’d be more interesting to write a quick thing about them (and a lot easier for me, given that I’ve got my final report deadline on Monday, so am spending most of my time thinking about them), and also show you guys some of the simulations I’ve built. It might be a little bit more mathsy that usual, but I’ve tried to keep it so that the majority of it is all understandable.

 This does mean that I don’t actually have a shoot to go with this piece, and given that I’ll be working on the project till the end of Monday, I probably won’t have time to do one. Don’t despair, though, I’ll still be giving you pretty pictures! You like that one gif up there? Well, on Wednesday you’ll get to see gifs of double pendulums on all the different planets and more – it’s pretty cool. My favourite one is the moon – it’s much slower than the pendulum on Earth, and really relaxing to watch. 

Also, I must give credit to my amazing project partner, Kiyam Lin, who pretty much did all the confusing code whilst i just sat there learning all about chaos. Everything would have been far more confusing without him. 

 

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The Fractal Universe

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Think about the coastline of Britain. Yeah, I know, it’s probably not your choice of scintillating subject (unless you’re a geologist), but hopefully that’s going to change. Now, if you were to measure the coastline of Britain with a 200 km ruler, you’d get a value around 2400km. If you measure it with a 50km ruler, you add about 1000km on to that number – the coastline is now 3400km. And as the ruler gets smaller, the coastline gets bigger and bigger, all the way to infinity. To put it in mathematical terms, as the length of the ruler tends to zero, the length of the coastline tends to infinity.

This sounds bizarre – am I really trying to say that if you measured the coastline of Britain with perfect accuracy that it would have infinite length? Yes. Yes I am, however counterintuitive that may sound. And funnily enough, the coastline of Britain isn’t some sort of anomaly of nature – shapes like these show up everywhere, from clouds to bark to lightening strikes. They’re called fractals, and are characterised in ‘the broken, wrinkled and uneven shapes of nature’ (in Mandelbrot’s words, who was actually the person to coin the word ‘fractal’), and on a deeper level the characteristic of self-similarity within an object, where smaller part of that object look the larger bits.

To fully understand this, it’s probably best to give an example of a classic fractal, the Koch snowflake. For the Koch snowflake you first start with a normal equilateral triangle. Then you put a triangle a third of the side in the middle of each of the edges, then a triangle a third of the size of that triangle along all the new sides. Lather, rise and repeat an infinite amount of times, and you have the Koch snowflake! Magnify any part of the Koch snowflake, and you see the pattern at the macro scale repeated at the micro.

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The steps to create the Koch Snowflake fractal, but you’d kept on going infinitely so that no matter how far you zoomed in you get the same patterns. From here.

Fractals are far more representative of the real world than the traditional ideal Euclid forms. They are not perfect in the sense of being smooth and impeccably formed – they are perfect in their complexity, in their pitted and splintered nature. From the minimalist Sierpinski’s gasket, to the (initially apparently simple) infinitely intricate Mandelbrot set, fractals are mesmerising, and oddly familiar. Once you’ve got the concept in your head, you start seeing fractals everywhere, both in the physical world, and in art. Complex subtext in literature doesn’t have a linear relationship, it has a decidedly fractal one. More than one dimension but less than two, fractals have a kind of convoluted cohesion that springs up everywhere in nature, and lends itself equally to both the scientific and artistic eye.

Fractals are everywhere around us – in the sky, in the plants in your garden, even in your food. (Believe it or not, broccoli is fractal – that’s why smaller broccoli florets look like shrunken heads of broccoli!) You’ve probably seen fractals hundreds of times, and maybe even admired their beauty, whether it be appearing in the bark on trees or in clouds in the sky, and learning and understanding the mathematics behind the art just makes it even more mesmerising. Learning the mechanics behind the artistry of the universe does nothing to detract from it’s beauty, as some would like to claim, but simply adds to it. Ignorance, in this case, is most definitely not bliss; knowledge is.


If you’ve enjoyed this blog post, I strongly recommend ‘Introducing Fractals: A Graphic Guide‘, which was one of the first books/comics I read explaining the subject, and was such a great form of media to get into fractals via – it’s definitely a subject in which the use of images helps out! (I also recommend ‘Introducing … : A Graphic Guide’ series in general – they’re very good at explaining a subject in brief without overly simplifying the concept!)

Also, if you’re loving the beauty behind fractals, there are so many mesmerising videos out there which zoom in on the border of the Mandelbrot set which I could watch for hours – here’s one I found quite quickly on Google. I must also give credit to Theo Emms (who’s does theoretical physics with me, and lives in the same halls) for the enlightenment that broccoli is fractal!

This blog post was inspired by the topics I babbled on about in the recording for the fashion video for The Elegant Universe editorial which just came out (and I’m the star of – let’s not forget to mention that!), most of which got cut out in the final video (as I babbled on for over an hour!). If you love the influence of science on art, and vice versa, you should definitely check out the editorial – it’s in V magazine, with so many amazing people who worked on it. Nick Knight shot it, Amanda Harlech styled, Sam McKnight was on the hair, Peter Philips did the make-up, Marian Newman did the nails, and last but not least, Kev Stenning did the 3D scans (yep, there are 3D scans!).