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 backwards, 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?
Thomasina speaks this quote to Septimus in the first scene of the play. This quote reveals Thomasina's scientific curiosity, even while eating her rice pudding. Thomasina attempts to find scientific explanations for the world around her and, quite successfully, sees the basis of chaos theory in the movement of the jam in her bowl. The trails of jam move toward a larger disorder that cannot be stirred back together by going the other direction. This move toward greater and greater disorder is characteristic of chaos theory as explained by Valentine later on. Chaos theory, Valentine explains to Hannah, helps scientists get closer to the everyday happenings of things around them from "what happens in a cup of coffee." Like the repeated algorithm in iteration, the jam continues to be more disorder because of where it came from; each equation uses the answer of the last equation for its unknown values. In other words, the iteration or stirring continues and the equation continually changes.
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 cleaver to do it, the formula must exist just as if one could.
Thomasina makes this claim to Septimus in Scene One. In their discussion of free will, Thomasina tells Septimus that God must be a Newtonian. By her theory—that all things in the universe can be put into proper equations—this seems somewhat correct. As Thomasina suggests, if there are formulas for all life, than life is predictable. In this quote, Thomasina is again hinting at chaos theory. In chaos theory, equations and forms of algebra can explain life. Of course, the problem of chaos theory and determinism itself is free will, as Septimus suggests and as Thomasina later figures out. Like Chaos theory, all actions cannot be specifically predicted, but general trends can be pointed out. For instance, Thomasina knows that heat eventually destroys itself and burns out; however the cause or instance of heat or when something may spark or have more friction is unknown. Thus, the mysteries of life can be plotted somehow, as Thomasina suggests, but they cannot be predicted, as she will later conclude.
If you knew the algorithm and fed it back say ten thousand times, each time there'd be a dot somewhere on the screen. You'd never where to expect the next dot. But gradually you'd start to see this shape .
Here, Valentine explains the chaos theory to Hannah Jarvis. Like the story of Hannah and the story of Thomasina, the stories are chaotic—they feed into themselves with unpredictable results. Chaos theory is one of the central metaphors in Arcadia and influences the structure of the work itself. What is important about Valentine's description of chaos is that it links mathematics directly into the world of art, the same place Stoppard desires to take it. Valentine describes the new math as he does a Picasso, with the language of modern art. Like Thomasina's algorithm, the structure of Arcadia is equally plotted and planned. Stoppard leaves behind old modes of theatre—the three act well made play—and presents a chaotic, if not unpredictable algorithm of stories. Stoppard creates his own set of points and equations that form a picture, not unlike Thomasina's. Arcadia is a test of the algorithm forming a fractal form of a play that attempts to imitate nature through the realities of mathematics.
SEPTIMUS It will go to infinity or zero, or nonsense
THOMASINA No, if you set apart the minus roots they square back to sense
Thomasina explains her theory to Septimus in Scene Seven. It is by this quote that the audience is sure Thomasina knows exactly what she is doing, she understands that sense will eventually arise from the disorder she has created with her equations. The "rabbit equation" Thomasina doesn't have time to extend is an iterated algorithm, a process of equations that describe a rabbit. Thomasina's equation cannot be completed without the aid of a computer to create the final fractal image, but it is clear she can somehow know the end result and picture her algorithm will create. Thomasina understands the eventual predictability of her algorithm and the eventual way her equation will "square back to sense."
When we have found all the meanings and lost all the mysteries, we will be alone, on an empty shore.
In the final, climactic moment of the plot, Stoppard beautifully combines the realization of death with an understanding of Thomasina's heat diagram. Valentine and Septimus both finally see the reality of the picture that Thomasina has created, and the men understand that Thomasina has drawn a picture of a reality where all humanity is doomed and destined for a fiery end. There is a sudden urgency for time and moment as the play draws to its end, and even Hannah Jarvis must submit and beg for a dance. The characters realize the unfulfilling, if not damning, end of academia and brace close to the mysteries of relationships and others around them that may allow them stolen time and leave room for wonder. The eagerness of Thomasina—the creator of the frightening picture of mortality—to dance reveals that there are other types of knowledge to be had in the world and new mysteries to be solved. With dance, with love, with carnal knowledge, one might avoid the empty shore. Thomasina suggests that the cold, emptiness can be overcome with heat energy and with waltzing and dancing that will allow new knowledge and fulfillment.