Chaos, fractals, and Arcadia

Chaos, Fractals, and Arcadia

Thomasina's Geometry of Irregular Forms

Thomasina does not like Euclidean geometry. Early in the play she chides her tutor, Septimus, "Each week I plot your 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 bell, then there must be an equation for a bluebell, and if a bluebell, why not a rose?" So she decides to abandon classical Euclidean geometry in order to discover the equation of a leaf.

Years later, Hannah discovers Thomasina's workbooks in which she has written, "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 number alone." Thomasina has discovered the mathematical procedure that is now called an iterated function system. Hannah asks Valentine how she does this. Val explains that she uses "an iterated algorithm."

"What's that?" Hannah inquires.

With the precision that only a mathematician can muster, Val responds "It's an algorithm that's been....iterated."

Then the fun begins. Val goes on to explain that an algorithm is a recipe, that if you knew the recipe to produce a leaf, you could then easily iterate the algorithm to draw a picture of the leaf. "The math isn't difficult. It's what you did at school. You have an x and y equation. Any value for x gives you a value for y. So you put a dot where it's right for both x and y. Then you take the next value for x which gives you another value for y......what she's doing is, every time she works out a value for y, she's using that as her next value for x. And so on. Like a feedback....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 know where to expect the next dot. But gradually you'd start to see this shape, because every dot will be inside the shape of this leaf."

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