At the opening of Scene Four of

"I, Thomasina Coverly, have found a truly wonderful method whereby all the forms of nature must give up their numerical secrets and draw themselves through numbers alone. This margin being too mean for my purpose, the reader must look elsewhere for the New Geometry of Irregular Forms by Thomasina Coverly." |

In this quote, Stoppard ties together the two main mathematical ideas which appear in his fascinating play: Fermat's Last Theorem and the theory of iterated algorithms and fractals. This brief guide will introduce these two ideas to you.

First, Fermat's Last Theorem. This story starts with the very famous Pythagorean Theorem (credited to the Greek geometer Pythagoras, who lived around 550 B.C.), which says that a triangle is a right triangle (i.e., contains a right angle) if and only if the sum of the squares of the lengths of its two shorter sides (called the "legs") equals the square of the length of its longest side (called the "hypotenuse.")

Suppose, for example, that side A is 3 feet, side B is 4 feet, and side C is 5 feet, then the triangle they form is a right triangle since 3

There is probably no mathematical result more often referred to in our culture than this one. For example, in the wonderful movie

But what does all this have to do with Fermat? Well, in about 1637, the French mathematician Pierre de Fermat, while reading about the Pythagorean Theorem in a Latin translation of the book

"To write the cube of a number as a sum of two cubes, or the fourth power as a sum of two fourth powers, or any power above 2 as a sum of two like powers, is impossible. I have a truly wonderful proof of this fact, but the margin is too narrow to contain it."

In other words, Fermat claimed to have a proof that if the power n is greater than 2, then there are no solutions in positive whole numbers to the "Pythagorean-like" equation:

For example, 3

Making a very long mathematical story short, no proof by Fermat of this idea was ever discovered among his writings. Fermat stated many "theorems" without supplying their proofs, and after his death each of these was subsequently proved by other mathematicians one by one until finally only one remained: The Last Theorem, this one of the famous margin note. The Last Theorem was finally proved in 1994 (by the mathematician Andrew Wiles of Princeton University), 357 years after Fermat wrote his marginal note, and one year after Arcadia premiered.

Let us turn now to the

- Thomasina: . . . Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?
- Septimus: We do.
- Thomasina: Then why do your equations only describe the shapes of manufacture?
- Septimus: I do not know.
- Thomasina: Armed thus, God could only make a cabinet.
- Septimus: He has mastery of equations which lead into infinities where we cannot follow.
- Thomasina: What a faint-heart! We must work outward from the middle of the maze. We will start with something simple. (
*She picks up the apple leaf.*) I will plot this leaf and deduce its equation. You will be famous for being my tutor when Lord Byron is dead and forgotten.

The idea which Thomasina discovers, as we find out later from the present day discussions of Hannah and Valentine, is that of

Here is an example of what Thomasina proposed to do, but it is simpler and thus results in a figure which is simpler than her apple leaf. Place three dots on a page labeled A, B and C. Now starting at any point on the page (called "the seed"), do the following process: Roll a die. If it comes up 1 or 2, move halfway from the seed toward dot A and make a new dot, if it comes up 3 or 4, move halfway toward dot B, or if it comes up 5 or 6, move halfway toward dot C. With us so far?

When doing this "by hand", as Thomasina had to do with her apple leaf, it is extremely difficult to get far enough to see what pattern (if any!) emerges. But with the aid of the modern digital computer, tens of thousands of iterations like this become a simple matter. Here is what you eventually see emerge from this particular iterated algorithm (it is called the Sierpinski Triangle):

The Sierpinski Triangle is an example of what's called a

We are led to presume that Thomasina went as far as she could in producing her apple leaf. Near the beginning of Scene Four, Valentine says of Thomasina's notebook: "The left-hand pages are graphs of what the numbers are doing on the right-hand pages. But all on different scales. Each graph is a small section of the previous one, blown up. Like you'd blow up a detail of a photograph, and then a detail of a detail, and so on, forever. Or in her case, till she ran out of pages." He goes on to mention that he is using the same technique (iterated algorithms) to study the grouse population over time at Sidley Park.

The iterated algorithm needed to get a leaf is quite a bit more complicated than that for the Sierpinski Triangle, so we can't go into it here. Below, however, is a fractal fern leaf image, as an example of the fascinating world into which Thomasina was venturing.

A fern leaf after 5,000, 10,000, and 50,000 iterations

- On Fermat's Last Theorem, a wonderful source is the recent Nova presentation entitled "The Proof" (available on video tape), which focuses on Andrew Wiles lonely and dramatic odyssey to find an answer to this 350 year old question.
- On iterated algorithms and fractals, there are many books available. A good one, for example, is Michael Barnsley's
*Fractals Everywhere*(Academic Press 1988). - On mathematics in
*Arcadia*, the Web site math.bu.edu/DYSYS/arcadia, produced by the mathematician R. L. Devaney at Boston University, discusses iterated algorithms and fractals in more detail and includes some excellent animations.

Gove Effinger has been at Skidmore since 1986, teaching a variety of courses in mathematics, computer science and liberal studies. He has written a book and several articles about prime numbers and another book about computer programming. In his spare time he enjoys doing lots of sports, especially distance running.