Geometry is one of the oldest subjects branches of mathematics on Earth; did you know it can be traced back to Pythagoras in the 6th Century BC? But nowadays geometry isn't just about straight lines, circles and angles, all sorts of shapes interest mathematicians. In this lecture Marcus talks about fractals, which are patterns that are repeated at ever-smaller scales to produce irregular shapes and surfaces, and he shows us that they are infinite and appear all around us in nature: in the shape of the coastline, a fern leaf, even in the artworks of Jackson Pollock. Next Marcus explains that mathematicians believe that it is possible to have more than three - 4 dimensions, 5, or more! - and they use this concept to imagine the shape of the universe. Taking this further, physicists think the universe may be wrapped up to make something like a 4D doughnut. If so, you could get in a spaceship, head up in a straight line and reappear back where you started.

Download a mathematical shape to cut out, decorate and stick

Dodecoration (PDF Document 412KB - new window)

Download this pdf to find out how repeating a simple instruction can lead to complicated shapes.

Fascinating fractals part one Drawing fractals (PDF Document 474KB - new window)

Download this pdf to find out how you can measure the length of a fractal. You might be surprised!

Fascinating fractals part two Infinite length (PDF Document 388KB - new window)

The story of the elusive shapes (PDF Document 164KB - new window)

View an interactive fractal generating program. Move the red dots around and use the Down & Up buttons to increase the complexity of the shape. (Requires Java.)

Create a fractal (external link - new window)

A short video on this site explains the concept behind the first three dimensions...and beyond...to the tenth dimension!

The tenth dimension (external link - new window)

This news article describes how mathematicians used fractals to identify forgeries of a famous painter's art.

Fake Pollocks and fractals (external link - new window)

Marcus captures lightning and shows that the infinite length of fractals can be useful in nature.

(Press play to start the video.)

Right, now, I'd like you to give a big Royal Institution welcome to the Prince of Darkness himself, Russell Thomas from the National Physical Laboratory. Come on down here, so welcome, welcome Russell. Now Russell is actually going to create some real lightning for us inside the lecture theatre. We had this sort of theatrical stuff, but I want some real lightning. So, when Russell is ready getting all his equipment here, we're going to get some real lightning. So how on earth are you going to do it Russell? What are you going to do for us?

Well, what we've got at NPL is a linear accelerator, one of the beams that we can produce is an electron beam, and we can use that to charge up these acrylic blocks, and I've stored this in dry ice.

So this thing has been charged up with loads of electrons?

Yes, we did that yesterday, cooled it down and hopefully now we're going to try and get that charge to come out.

Right, so this is a bit like some sort of spring then.

Yes.

And we're going to kind of release it, and then we're going to get some lightning. Now is it going to be dangerous? Should I hide behind there, being a bit of a coward?

No, you're ok.

Oh right, alright, let's see if we can create some lightning then inside the lecture theatre.

It doesn't always go the first time.

Ok. Give it a second go.

Doesn't always go the second time.

Yeah, third time lucky, prime number you see. Well, lightning you also can't predict. Oh wow, now that's fantastic and it is still going actually, look at that, all of these are electrons shooting round, that's really beautiful. Now it's quite hard to see this one, you can probably see the crackling. Can we make another one? I think that was quite exciting. Have you got another one?

I have, I've not ever tried it with this size before.

Oh right, oh my god, you're going to make a really big piece of lightning. This looks really terrifying. I think I might hide behind here.

Let's see if we can get this one. Third time lucky.

Wow, and all those sparkles again, that's amazing. So actually we've managed to capture in this acrylic some lightning, absolutely extraordinary. Now, Russell, actually we've got one that you prepared earlier which we can light and we can really see the beautiful fractal shape here. Let's put this one up here, so that is actually what we've made, we've actually managed to capture,. This is lightning inside this acrylic and you can see the sort of infinite complexity in this thing. So if we zoom in on this shape, actually it looks like it is very hard to judge the distance from it. Are you zooming in very close? Or are you far away? See we've got these branches here coming out and then we've got much thinner branches, thinner, thinner, thinner. So this really looks sort of like a fractal like we created on the landscape. Let's give Russell a big round of applause for creating some fractal lightning, great. Thank you very much Russell.

So lightning is fractal, but actually nature is full of lots of fractal shapes and we've got some over here, for example, if I take this fern. Now fern, if you look at it far away but zoom in very close, actually it's quite hard to judge the sort of distance you're at, it is sort of infinite complexity here, the little fern looks like the big fern. In fact, inside our body we also have some fractals. The human lung is also fractal in shape, so if you look this is a little bit like the lightning that we saw. We've got these large branches here, but with smaller and smaller branches coming off, and it has its infinite complexity of the fractal. Now actually these natural organisms are using the infinite length that a fractal has to maximise the amount of oxygen these things can take in.