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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