Chaos, fractals, and Arcadia

Chaos, Fractals, and Arcadia

Robert L. Devaney
Department of Mathematics
Boston University
Boston, MA 02215


Tom Stoppard's wonderful play, Arcadia, offers teachers of both mathematics and the humanities the opportunity to join forces in a unique and rewarding way. The play features not one but two mathematicians, and the mathematical ideas they are involved with form one of the main subthemes of the play. In particular, such contemporary topics as chaos and fractals form an integral part of the plot, and such stalwarts as Fermat's Last Theorem and the Second Law of Thermodynamics also play important roles.

The play is set in two time periods, the early nineteenth century and the present, in the same room in an English Estate, Sidley Park. As the play opens, we meet Thomasina, a young thirteen year old girl who struggles with her algebra and geometry under the watchful eye of her tutor, Septimus Hodge. But Thomasina is not your typical mathematics student; as becomes clear as the play unfolds, she is a prodigy who not only questions the very foundations of her mathematical subjects, but also sets about to change the direction of countless centuries of mathematical thought. In the process, she invents "Thomasina's geometry of irregular forms" (aka fractal geometry), discovers the second law of thermodynamics, and lays the foundation for what is now called chaos theory.

In the modern period, we meet Valentine, a contemporary mathematician who is attempting to understand the rise and fall of grouse populations using iteration. As luck would have it, Valentine is heir to Sidley Park and part of his inheritance is a complete set of game books that go back to Thomasina's time. These books detail the precise number of grouse shot at the estate each year. Gradually, he becomes aware of some of the old mysteries surrounding Sidley Park, including Thomasina's discoveries, and this sets the stage for a unique series of scenes that hop back and forth between the early nineteenth century and the present.

Mathematics is not the only theme of this play, of course, but the ideas of regular versus irregular geometry or chaos versus order seem to pervade all of the other events occurring at Sidley Park. We are thrust into a debate about emerging British landscape styles featuring the orderly classical style versus the irregular, ``picturesque'' style. Valentine's fiancee Hannah Jarvis methodically proceeds to uncover Sidley Park's secrets, in stark contrast to her nemesis, Bernard Nightingale, who jumps from one theory to another with reckless abandon. Indeed, the entire play pits the rationalism of Newton against the romanticism of Byron.

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