Arcadia: All the World's a Fractal
By Michael Gitter Arcadia, a stage play by Tom Stoppard, was first produced in London in the spring of 1993. It was an immediate success, subsequently staged in New York City at the Lincoln Center for the Performing Arts, then all across the United States, and around the world. Although written in this decade, Arcadia has already been hailed as a literary and dramatic masterpiece, and is even being taught at the high school and college levels (most notably by Professor Harold Bloom at Yale).
The play's remarkable reception can be easily explained. From a theatrical perspective, Stoppard makes brilliant use of stage space and props, seamlessly shifting between different periods of time without changing the space at all. This aspect of the play, along with its urbane humor, is apparent from reading the play, but the staging of the play brings it fully to life. Where the audience continually expects to see set-changes, it sees time-changes-- a true innovation. The play is also fascinating to a lover of history, as the characters of the present desperately search into the past for solution to something for which they only have intriguing clues. Furthermore, by the very fact that the play takes place in two time periods (the early 1800's and present day), the play becomes an insight into the differences and similarities between the temperaments of two distinct periods. For a philosopher, the play can be read as a lyrical treatise on love and learning, on what it means to come into one's own intellectually and emotionally. And finally, Arcadia is a beautiful look at and even example of an absorbing mathematical concept: fractal geometry.
The play involves two stories occurring in the same house (Sidley Park) in Derbyshire, England, 180 years apart, interwoven throughout. In 1809, Septimus Hodge, a witty and adulterous young man, tutors the young Thomasina, a curious and brilliant girl of thirteen whose eagerness to learn leads to important discoveries. Other significant characters are: Ezra Chater, an unfortunately atrocious poet who challenges Septimus to a duel for committing adultery with his wife; Lady Croom, the mistress of the house, with whom Septimus is in love; Richard Noakes, a landscape architect who is designing renovations to the park surrounding the house; and the actual poet Lord Byron, who does not appear in the play, but who nevertheless plays a key role. The present day story follows Hannah Jarvis, a wily, well-known writer who is writing a history of the hermitage on the property of Sidley Park; Bernard Nightingale, a would-be scholar of Byron who comes to the house on a seemingly wild goose chase of facts about Byron's life; Chloe and Valentine Coverly, sister and brother, children of the house, the former enchanted by Bernard, the latter by Hannah. Finally, there is the oddly timeless character of Gus, who never speaks in the play, but seems a reflection of Augustus, a character from the earlier time zone of Arcadia. The stories are put together marvelously, overlapping, crossing over, and finally, occurring simultaneously.
So what does this have to do with fractals? The answer is twofold. First, throughout this complex play, there are numerous references to and discussions of fractals, algorithms, and chaos theory. Secondly, the play itself reads as its own kind of fractal, manipulating through repetition the dimensions of time and space. But first things first. Let us trace the important references to this kind of mathematics in the play. The first occurs early in the first scene of the play, as Thomasina expresses confusion at the fact that it is impossible to "unstir" something once one has begun stirring:
THOMASINA: When you stir your rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backward, the jam will not come together again. Indeed, the pudding does not notice and continues to turn pink just as before. Do you think this is odd?
SEPTIMUS: No.
T: Well, I do. You cannot stir things apart.
S: No more you can, time must needs run backward, and since it will not, we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it forever.
Thomasina is continually interested in ideas of time and space, and once she gets started, there is not stopping her. And indeed, a moment later she comes to a realization that in effect is the idea of an equation for a fractal of events over time:
T: If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future; and although nobody can be so clever as to do it, the formula must exist just as if one could.
She is ahead of her time, both in terms of her age (13) and the age she live in (19th Century England). Her thoughts on math and nature are beautifully juxtaposed with the present-day character of Valentine, who is a post-grad at Oxford studying population growth among grouse, using algorithms to determine patterns.
Later in the play, Thomasina makes another startling comment about the mathematics of nature, seeing equations and graphs as applicable to forms in the physical world, much the same way that fractals occur in nature (eg, the seed patterns on a sunflower):
THOMASINA: Each week I plot your [Septimus's] equations dot for dot, x's against y's 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.
T: Then why do your equations only describe the shapes of manufacture?
S: I do not know.
T: Armed thus, God could only make a cabinet.
In the next scene, this line of thinking is reflected in the present time story. For the first half of the scene (a good four pages of text), Hannah and Valentine discuss what the latter is studying at Oxford as well as what they discover Thomasina was studying nearly two centuries earlier. Valentine explains the idea of an iterated algorithm, the formal name for what Thomasina had begun to uncover:
VALENTINE: It's how you look at population changes in biology. Goldfish in a pond, say. This year there are x goldfish. Next year there'll be y goldfish. Some get born, some get eaten by herons, whatever. Nature manipulates the x and turns it into y. Then y goldfish is your starting population for the following year. Just like Thomasina. Your value for y becomes your next value for x. The question is: what is being done to x? What is the manipulation? Whatever it is, it can be written down as mathematics. It's called an algorithm.
The play continues to explore these ideas, following Thomasina's education, as well as the mathematical thinking of other characters in the play. But the mathematics of fractals works in a different way in the play as well. For the whole play can be seen as a sort of fractal-- a series of repeated images, personalities, props, and sets which are almost as symmetrical as a Sierpinski gasket. Stoppard explains in the stage directions that the set should not change from scene to scene, regardless of the change in time. The same tortoise is on the stage for the entire time, as are Thomasina's math books, a coffee mug, etc. "Grouse" and "caro," as well as many other particular words are repeated in both of the interwoven plots. The scenes from the present heavily echo those of the past. In fact the whole play has the effect of a repeated series of ideas and actions-- an iterated function, if you will. Finally, near the end of the play, the characters in the present story practically reenact the first scene of the play, which occurs in the past story. The effect is striking: essentially, Stoppard creates his own fractal by repeating patterns, and also recalls Valentine's definition of an algorithm. Indeed, Stoppard repeatedly puts in narrative "x value" in the equation that is the play itself, gets a "y value," and plugs it back in as a new "x." Just as an image can become clear as a fractal is repeated, the full picture of the play comes to light by the end, as time and space finally come together. [ Home ][ Literature & Dimensionality ][ Our Favorite Links and Images ]