mathematics

Lakes of Wada

Lakes of Wada is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lakes of Wada rather than just read about it. In short: In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point.

Key takeaways

Lakes of Wada belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lakes of Wada to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lakes of Wada from memory before moving on to harder problems.

Reference excerpt

In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point. More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. This property is rare in real-world systems. The lakes of Wada were introduced by Kunizō Yoneyama (1917, page 60), who credited the discovery to Takeo Wada. His construction is similar to the construction by Brouwer (1910) of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum.

Construction of the lakes of Wada

The Lakes of Wada are formed by starting with a closed unit square of dry land, 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 is open and connected and passes within a distance 1/n of all remaining dry land. This should be done so that the remaining dry land remains homeomorphic to a closed unit square. 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. (The small channel connects 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. (The tiny channel connects the thin red lake to the thick one, near the top left of the image.) A variation of this construction can produce a countably 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 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 fractal describing the basins of attraction of the Newton–Raphson method for finding the roots of a cubic polynomial with distinct roots, such as z3 − 1; see the picture.

Wada basins in chaos theory In chaos theory, Wada basins arise very frequently. Usually, the Wada property can be seen in the basin of attraction of dissipative dynamical systems. But the exit basins of Hamiltonian systems can also show the Wada property. In the context of the chaotic scattering of systems with multiple exits, basins of exits show the Wada property. M. A. F. Sanjuán et al. has shown that in the Hénon–Heiles system the exit basins have this Wada property.

See also List of topologies – List of concrete topologies and topological spaces

References Brouwer, L. E. J. (1910), "Zur Analysis Situs", Mathematische Annalen, 68 (3): 422–434, doi:10.1007/BF01475781 Yoneyama, Kunizô (1917), "Theory of Continuous Set of Points", Tôhoku Mathematical Journal, 12: 43–158

Further reading Breban, Romulus; Nusse, H E. (2005), "On the creation of Wada basins in interval maps through fixed point tangent bifurcation", Physica D, 207 (1–2): 52–63, Bibcode:2005PhyD..207...52B, doi:10.1016/j.physd.2005.05.012 Coudene, Yves (2006), "Pictures of hyperbolic dynamical systems" (PDF), Notices of the American Mathematical Society, 53 (1): 8–13, ISSN 0002-9920, MR 2189945 Gelbaum, Bernard R.; Olmsted, John M. H. (2003), Counterexamples in analysis, Mineola, N.Y.: Dover Publications, ISBN 0-486-42875-3 example 10.13 Hocking, J. G.; Young, G. S. (1988), Topology, New York: Dover Publications, p. 144, ISBN 0-486-65676-4 Kennedy, J; Yorke, J.A. (1991), "Basins of Wada", Physica D, 51 (1–3): 213–225, Bibcode:1991PhyD...51..213K, doi:10.1016/0167-2789(91)90234-Z Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation", Nature, 399 (6734): 315, Bibcode:1999Natur.399..315S, doi:10.1038/20573

External links An experimental realization of Wada basins (with photographs), andamooka.org An introduction to Wada basins and the Wada property www-chaos.umd.edu Reflective Spheres of Infinity: Wada Basin Fractals, miqel.com Wada basins: Rendering chaotic scattering Deprecated link archived 2012-06-30 at archive.today, astronomy.swin.edu.au

Illustrations

Lakes of Wada: A snapshot of the process converging to the lakes of Wada — A snapshot of the process converging to the lakes of Wada

Lakes of Wada: Animation of digging lakes up to day 5 — Animation of digging lakes up to day 5

Worked examples

Example 1 — a first encounter with Lakes of Wada

Start with the simplest possible case. Write down what Lakes of Wada claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lakes of Wada before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lakes of Wada ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lakes of Wada

In research: Lakes of Wada appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lakes of Wada in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lakes of Wada is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fractals, Topology, so understanding it makes those chapters shorter.
In everyday life: Look for Lakes of Wada outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lakes of Wada in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lakes of Wada means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lakes of Wada out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lakes of Wada in simple terms?

Why does Lakes of Wada matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lakes of Wada?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lakes of Wada.