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. |