| The Wada Property is most easily understood by an
image of an island (white) in a red sea. Imagine the island
contains a blue lake and a green lake. |
| First, dig a canal from the sea so no point of the
island is farther than 1/2 mile from the canal. |
| Next, dig a canal from the blue lake
so no point of the island is farther than 1/4 mile from this canal. |
| Now, dig a canal from the green lake
so no point of the island is farther than 1/8 mile from this canal. |
| Now, dig another canal from the sea so no point of the
island is farther than 1/16 mile from this canal. |
| Continue in this fashion, until infinitely many canals have been dug. |
| Between canals of any two colors, we find a canal of the third color.
Always. No matter how small the canals. |
|
| |
| To illustrate the Wada property for Newton's method, consider these three
images, using reverse graphical iteration to find the basins of attraction of
each of the three roots pictured here. |
|
| The blue is part of the reverse graphical iteration
of the left-most root (Do you see why?). |
| For example, observe that between the left-most shown
red and green branches
we find the blue. |
| Continuing the reverse iteration, the picture becomes very complicated,
but we can show that between any two colors we always find the third. |
