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
�brain� 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:
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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 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:
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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.