What Shape is an Egg?

A module of Egg Math for Chickscope: in which we discuss surfaces of revolution and methods for drawing ovals.

The first thing to notice is that an egg is a very symmetric object. If we look at the egg from lots of different directions, then the view from many of the directions is the same. Suppose, for example, that you glue the the egg to the top of a table, with the middle of the fat end stuck to the table and the the egg pointing straight up, as in the picture. If you now walk around the table, you notice that the egg looks exactly the same no matter from which direction you look at it. (Move your mouse on top of the picture, to see an animation of this motion.)

You get the same effect if you stick a long needle right through the egg, from the middle of the thick end through to the middle of the thin end, as in the picture at the top of the page. If you hold the needle straight up in front of your eyes and slowly rotate it, the view you see does not change at all as you rotate the egg around. The needle is called an axis of rotational symmetry for the egg.

Here are some questions to think about:

• Does the egg have any other axis of symmetry?
• Can you think of other shapes which have axes of symmetry?

The symmetry of the egg is very helpful in describing its shape. Here's how: we can imagine slicing the egg in half from top to bottom by taking a sharp knife and running it along the axis of symmetry. If we separate the two halves, each now has a flat surface. The surface we see is called the cross-section of the egg. We can reconstruct the shape of the whole three-dimensional egg simply by rotating the two-dimensional cross-section around the axis of symmetry. In mathematical language, the surface of the egg (the egg shell) is thus a surface of revolution.

All we have to do now is to describe the shape of this cross-section! The shape is called an oval (the word `oval' comes from the Latin word for egg, and means `having the shape of an egg'). You can see that it looks like an ellipse which has been slightly squished at one end. Nevertheless, an ellipse is pretty close to the right shape. So for our first attempt at an accurate description of the shape of an egg, let's explore ellipses.

There is a well-known method for drawing ellipses which uses pins and string. It can be used to derive equations for an ellipse.

As pretty as the ellipses are, you can easily see that they are not really egg shaped! Both ends of an ellipse look the same, but the cross section of an egg is sharper at one end than at the other. So how can we do better? Here are two types of curves which look more like the cross-section of a real egg.

• Cartesian ovals
• ovals of Cassini

Actually, there are many other possibilities. If you want to find out more about curves, you might want to take a look at `A Book of Curves' by Edward H. Lockwood (Cambridge University Press, 1961), or, if you can't drag yourself away from cyberspace, there is a nice site about plane curves in Scotland.