Construction of The Lakes of Wada
The Lakes of Wada are formed by starting with an open unit square of dry land (homeomorphic to the plane), and then digging 3 lakes according to the following rule:
- On day n = 1, 2, 3,... extend lake n mod 3 (=0, 1, 2) so that it passes within a distance 1/n of all remaining dry land. This should be done so that the remaining dry land has connected interior, and each lake is open.
After an infinite number of days, the three lakes are still disjoint connected open sets, and the remaining dry land is the boundary of each of the 3 lakes.
For example, the first five days might be (see the image on the right):
- Dig a blue lake of width 1/3 passing within √2/3 of all dry land.
- Dig a red lake of width 1/32 passing within √2/32 of all dry land.
- Dig a green lake of width 1/33 passing within √2/33 of all dry land.
- Extend the blue lake by a channel of width 1/34 passing within √2/34 of all dry land. (Note the small channel connecting the thin blue lake to the thick one, near the middle of the image.)
- Extend the red lake by a channel of width 1/35 passing within √2/35 of all dry land. (Note the tiny channel connecting the thin red lake to the thick one, near the top left of the image.)
A variation of this construction can produce a countable infinite number of connected lakes with the same boundary: instead of extending the lakes in the order 1, 2, 0, 1, 2, 0, 1, 2, 0, ...., extend them in the order 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, ...and so on.
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