Bubbles

Hillforts

plan of hill fort Imagine you're designing a hill fort around 500 BC, like the one on the right (which is a typical layout of a hill fort, taken from Mathematics Galore by Chris Budd and Chris Sangwin). What basic shape would you choose for your castle's outer perimeter? If you've seen round many hill forts, or seen pictures of them, you will know that many of them had a perimeter which was roughly circular in shape. Why do you suppose that was?

Like many design problems, it is a case of making the most efficient design for the least outlay. The whole point of a hill fort was that it would provide a safe place for as many people as possible when the enemy attacked, and that it would be easy to defend. Making your fort easy to defend meant minimising the number of people needed - a fort needing a lot of defenders might quickly find itself overrun by the enemy as soon as a few defenders were killed. You would also be more likely to get more commissions if this was done with the minimum outlay on materials for the outer defences.

The isoperimetric theorem

So this is really a problem about finding a shape which will enclose the maximum area possible, for the minimum length of perimeter. It needs to be a closed shape, since you wouldn't want holes in the outer fence to let attackers through. As you probably know, the shape which satisfied these conditions is a circle, and in other parts of this talk, you will see how we can use bubbles to predict that and also to predict answers to a range of similar problems.

The problem we've just answered is:

Q: What closed curve of a given perimeter encloses the greatest area?
A: A circle

This is such an important result that it has a special name. It is called the isoperimetric theorem, from the Greek words for 'same' and 'perimeter'. You can demonstrate this practically, and this is your first task.

Go to task 1

Task 1 should have given you a practical demonstration that the isoperimetric theorem is true - but, of course, this is not the same as a proof. We'll look at proving this theorem later in this session.

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