**EggMath:**

**The Shape of an Egg**

What shape is an egg? How can we accurately
describe it? And draw it?

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 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:

**FOR FURTHER
READING...**

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.