resses and ingredients destined for the entrée ending up in the dessert.
So, giving a nod to popular culture, we turn to the chaotic bedlam of the
kitchen for a look at some sophisticated chaos mathematics you can do
with a rolling pin.
Imagine that one has a cylinder of dough, and we roll it out so that it is
twice as long as it was originally, and then cut it in half and place the right
half on top of the left half. We then repeat this rolling and cutting pro-
cess, which is called the baker’s transformation. Think of the pie dough
as occupying the segment of the real line between the integers 0 (where
the left end of the dough is located) and 1 (where the right end is located).
The baker’s transformation is a function B(x), which tells us where a
point that was originally located at x is located after the rolling, cutting,
and placing. B(x) is defined by
B(x) 2 x 0 x ^1 ⁄ 2
B(x) 2 x 1 1 ⁄ 2 x 1A simple description is that a point on the left half of the dough moves
to twice its distance from the left end after the baker’s transformation,
but a point on the right half doubles its distance from the left end after
the rolling, and then is moved one unit toward the left end after the cut-
ting and placing.
This doesn’t look very complicated, but surprising things happen. Two
points that are originally located very close together can end up very far
apart quite quickly. I’ve chosen two different points that are initially very
close to each other, and to honor my wife, who was born on September 1,
1971, her birthday is the first starting point, x.090171. The second
starting point is located at .090702, only^1 /1,000,000 of a unit to the right of
the first starting point.
After one iteration, the two points have drifted .000002 apart, and even
after twelve iterations they are only about .004 apart. But after the six-
teenth iteration, one of the points is in the left half of the dough, and the
other is now in the right half. The next iteration moves them widely
apart—the first point is not far away from the right end, whereas the sec-
ond point is very near the left end.
Start Iteration
1 121314151617
0.090171 0.180342 0.340416 0.680832 0.361664 0.723328 0.446656 0.893312
0.090172 0.180344 0.344512 0.689024 0.378048 0.756096 0.512192 0.024384The Disor ga nized Universe 177