Two Mathematical Ideas in Arcadia

Gove Effinger, Department of Mathematics

    At the opening of Scene Four of Arcadia, Hannah Jarvis reads from a notebook written more than 180 years ago by the play's budding young genius:

    "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 32 + 42 = 52 (i.e., 9 + 16 = 25). There are lots of other sets of three whole numbers which form right triangles; for example try 5, 12 and 13 (since 25 + 144 = 169), or try 8, 15, and 17, and so on. Such sets of three numbers are called "Pythagorean triples", and it turns out that you can keep finding more and more new triples as you go up; there are infinitely many of them.

    There is probably no mathematical result more often referred to in our culture than this one. For example, in the wonderful movie The Wizard of Oz, when the scarecrow finally gets his ̉brainÓ from the wizard, he immediately states the Pythagorean Theorem. (Incorrectly, however! Listen carefully the next time you watch the 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 Arithmetica by the Greek mathematician Diophantus (200 A.D.), wrote one of the most famous (and infamous) margin notes in history upon which Thomasina Coverly modeled her own marginal note. What Fermat wrote in the margin, translated from Latin to English, was:

    "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, 33+ 43 = 91, which is not a perfect cube; and Fermat is saying you can never find an example which "works" as long as the power is above 2.

    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 theory of iterated algorithms and fractals. Thomasina Coverly's interest in Fermat's Last Theorem is, well, marginal, but she is keenly interested in the question of how numbers can be used to produce the true shapes of nature. Consider the following dialogue between Thomasina and her tutor Septimus Hodge from Scene Three of Arcadia:

    The idea which Thomasina discovers, as we find out later from the present day discussions of Hannah and Valentine, is that of iterated algorithms, that is: the idea of starting with a number (or point), processing it somehow to obtain a new number or point (which you record), and then feeding that new number or point back into the process. You do this "feedback mechanism" again and again, and after a long time you see the pattern which emerges.

    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? Now, repeat this process starting at your new dot! You obtain a new new dot. Now repeat again, and again, and again . . .

    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 fractal. Fractals are the visual objects which result from iterated algorithms, and they have the characteristic that they are "self-similar" in the sense that if you change their scale and look at them, they appear exactly as they did before. Each of the three smaller triangles in the Sierpinski triangle looks just like the big one, and each of their three triangles looks like them, and so on down the line.

    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

For Further Study: