Lakes of Wada

In mathematics, the lakes of Wada are three disjoint connected open sets of the plane with the counterintuitive property that they all have the same boundary.

Sets with a similar property are said to have the Wada property; examples include Wada basins in dynamical systems.

The construction was first published by the Japanese mathematician Kunizô Yoneyama in 1917, who credited the discovery to his teacher Takeo Wada.

Construction of the lakes of Wada

First five stages 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 a_n of all remaining dry land, where a₁, a₂, a₃,... is some sequence of postive real numbers tending to 0. 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):

1. Dig a blue lake of width 1/3 passing within √2/3 of all dry land.
2. Dig a red lake of width 1/3² passing within √2/3² of all dry land.
3. Dig a green lake of width 1/3³ passing within √2/3³ of all dry land.
4. 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.)
5. 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.

Wada basins

Wada basins of attraction for x³ − 1 = 0; all three disconnected open basins have the same boundary

Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems. A basin having the property that every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin, or said to have the Wada property. Unlike the Lakes of Wada, Wada basins are often disconnected.

An example of Wada basins is given by the Newton-Raphson method applied to a cubic polynomial with distinct roots, such as x³ − 1; see the picture.

A physical system that demonstrates Wada basins is the pattern of reflections between three spheres in contact.

See also

• Counterexamples in topology
• List of examples in general topology

External links

• An introduction to Wada basins and the Wada property
• Reflective Spheres of Infinity: Wada Basin Fractals
• Wada basins: Rendering chaotic scattering

References

• Gelbaum, Olmsted, Counterexamples in analysis, ISBN 0-486-42875-3, example 10.13.
• Kunizô Yoneyama, Theory of Continuous Set of Points, The Tôhoku Mathematical Journal 12: 43–158, 1917.
• J. Kennedy and J.A. Yorke, Basins of Wada, Physica D 51 (1991), 213-225
• Romulus Breban and H E. Nusse, "On the creation of Wada basins in interval maps through fixed point tangent bifurcation" (2005). Physica D-Nonlinear Phenomena. 207 (1-2), pp. 52-63. 10.1016/j.physd.2005.05.012.
• J. G. Hocking, G. S. Young Topology ISBN 0-486-65676-4 has a crude picture of the lakes of Wada on page 144.