| Newton's method is designed to find the roots of equations.
Science is filled with examples of new problems suggested by techiques developed to
solve other problems. Arthur Cayley
recognized that
if we already know the roots of a function, Newton's method suggests another
problem: |
| which initial guesses iterate to which roots. |
| That is, |
| what are the basins of attraction of the roots? |
| We shall see this problem leads to fractals (no surprise at that), and also a
simple experiment making an optical Sierpinski gasket using Christmas tree
ornaments. |
| Introducing the problem: finding the basins of attraction of the
roots of z2 - 1. |
| How much harder can it be to find the basins of attraction of
the roots of z3 - 1? |
| Here we illustrate how the basins change as
the roots move. |
| The crinlky edges of the z3 - 1 basins illustrate
fractal basin boundaries and the corresponding
loss of predictability. |
| Here is an exmple of complicated basins of attraction in a
mechanical system. |
| Here is an exmple of complicated basins of attraction in a
magnetic system. |
| Here is an exmple of complicated basins of attraction in a
optical system. |
|