Scene Four Examples

Hannah, in the present, is reading from Thomasina's primer.
Hannah   "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. This margin being too mean for my purpose, the reader must look elsewhere for the New Geometry of Irregular Forms discovered by Thomasina Coverly."
(pg 43)
Recall (pg 6) Thomasina's observation that the similar note Fermat wrote in the margin of his Arithmetica, that he had a proof of Fermat's last theorem too large for the margin, was a joke.
 
Describing part of Thomasina's primer, Valentine says
Valentine   Each graph is a small section of the previous one, blown up. Like you'd blow up a detail of a photograph, and then a detail of the detail, and so on, forever.
(pg 43)
This is the notion of self-similarity or scale invariance, a central concept of fractal geometry.
 
Next, Valentine describes Thomasina's method of iterating a function.
Valentine   ... 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. (pg 44)
A particularly easy way to visualize this is graphical iteration.
Valentine was trying to use a similar representation to study the grouse population of the Coverly estate, using the game books as samples of the population.
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.
(pg 45)
 
Also in that scene we find
Valentine   When your Thomasina was doing maths it had been the same maths for a couple of thousand years. Classical. And for a century after Thomasina. Then maths left the real world behind, just like modern art, really. Nature was classical, maths was suddenly Picassos. But now nature is having the last laugh. The freaky stuff is turning out to be the mathematics of the natural world.
(pg 45)
Among many examples, here is one inside each of us.
Hannah   What did you mean you were doing the same thing she was doing? What are you doing?
Valentine   Actually I'm doing it from the other end. She started with an equation and turned it into a graph. I've got a graph - real data - and I'm trying to find the equation which would give you the graph if you used it the way she's used hers. Iterated it.
(pg 45)
Valentine is referring to the return map.
 
Still later in that scene
Valentine   ... The unpredictable and the predetermined unfold together to make everything the way it is. It's how nature creates itself, on every scale, the snowflake and the snowstorm. It makes me so happy. To be at the beginning again, knowing almost nothing. People were talking about the end of physics. Relativity and quantum looked as if they were going to clean out the whole problem between them. A theory of everything. But they only explained the very big and the very small. The universe, the elementary particles. The ordinary-sized stuff which is our lives, the things people write poetry about - clouds - daffodils - waterfalls - and what happens to a cup of coffee when the cream goes in - these things are full of mystery, as mysterious to us as the heavens were to the Greeks. We're better at predicting events at the edge of the galaxy or inside the nucleus of an atom than whether it'll rain on auntie's garden party three Sundays from now. Because the problem turns out to be different. We can't even predict the next drip from a dripping tap when it gets irregular. Each drip sets up the conditions for the next, the smallest variation blows prediction apart, and the weather is unpredictable. When you push the numbers through a computer you can see it on the screen. The future is disorder. A door like this has cracked open five or six times since we got up on our hind legs. It's the best possible time to be alive, when almost everything you thought you knew is wrong.
(pg 47-48)
Valentine if discussing sensitivity on initial conditions, one of the main features of chaos.

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