Hello, Everyone! Last time, I gave you this problem: you have four towns, in a one-mile square– what is the minimum amount of road you need to connect those four towns together? Well, let’s look at a few of the possibilities. So we could connect the towns together like this, so every town is connected to every other town. Now, each side of the square has length one, and the diagonals– well, with a quick bit of Pythag you can work out that the diagonals are root two–the square root of two. So all together, this has length 4 plus 2 root 2– that’s about 6.82. We could connect the towns together in a big circle, perhaps. I’ve already told you the diagonal has a length of root two. So the circle has a length of pi root 2. That’s about 4.44. We could connect the towns together in a big square. That has length 4. Even better than that, we could connect them in that sort of U shape, which has length 3, but it’s not a very efficient way, especially to get from A to B. It’s not a very efficient way to get around. Perhaps a slightly better answer would be to connect them into a H, Which would still have length 3, but perhaps slightly more efficient. But this isn’t the minimum answer, either. You might have thought the smallest answer would have been a cross. So those two diagonals–well, the diagonals have length root two, so this has a length of 2 root 2, which is about 2.82. That’s certainly the smallest answer so far, but it isn’t the minimum solution. Let’s find out what is. Working out the minimum length of a problem like this is called ‘calculus of variations,’ and can be quite difficult. But this is where nature is going to help us. I’ve had a prop made for me. It’s actually two layers of Perspex® with four pegs in it. And these four pegs are going to represent our four towns. And I’m going to dip this into this bowl of soapy water. And what it will do, is it will create a film of soap and this soap film will naturally try to minimize its free energy. So, in other words, it’s going to minimize its surface area, like a bubble! So, if I do this, this is going to give our minimum solution. Let’s do it! If I dip this in and I’ll lift it out, and if I hold this up to the camera, you can see that this is our minimum solution. And it might not be the solution you expect. This actually has length one plus root three. And that’s about 2.73. If I tell you that those roads intersect at an angle of 120 degrees, then some of you might like to try and confirm that answer. So let’s do this again! Let’s do this for three towns this time. I’ve made another prop here, for three towns, and if I dip this into my soapy water, let’s see what answer we get this time. If I hold this up to the camera, you see that I create a extra point there in the middle of the triangle. And these extra points that you create are called ‘Steiner points.’ And if you want to solve problems like this, they always look the same. They’re always three roads connected together at angles of 120 degrees. So this was our solution for three towns. Our solution for four towns looked like this, so this time, we have two Steiner points. And we can keep going. For five towns, the answer is this. And this is the answer for six Now, for eight towns, well–here’s a solution, this is the internal solution for eight towns, but it isn’t the smallest. In fact, if you connect the perimeter together, the answer is slightly smaller. So the internal solution is just a local minimum. You can think of these minimums as like valleys, and you can sit quite happily in one of those valleys, but it might not be the lowest answer. But to get to the lowest answer, you have to put in some energy first. And we can do this with our soap film Let’s try it out! If I take our four cities again– if I put this into the soapy water and I pull this out in a slightly different way, I’ve got now a different solution. It’s just a U shape like we’ve seen before. And this is one of our local minimums. And if you want to get the true answer you might have to do this a number of times and get a few of these local minimums and then you need to check which one is the true minimum. But perhaps I can do something about this, because if I take this local minimum solution and blow on it, I now get our minimum solution. This has real world application. So if you want to find the minimum length of motorways or building gas pipes, or anything like that– even if you had an obstacle that you have to build around, all you need to do is just cut out a section of your Perspex® plate there, and the soap film will go around it too. It’s very clever! And if you have been, thanks for watching.