## Algorithms for the Lakes of Wada

The Lakes of Wada partitions the unit square in to three regions, all of whom share a common boundary. The Wikipedia entry (http://en.wikipedia.org/wiki/Lakes_of_Wada) gives a construction approach, and a picture of a first few steps of the construction.

Is there a good algorithm available somewhere to explicitly list the partition up to some specific iteration? Or a closed form expression for the boundary in the limit of the process?

I'm hoping for something I can implement myself, or a piece of software that already does it, and in the end get a picture to arbitrary high levels of detail and arbitrary high iterations of the construction.

-

Wada lakes can be obtained following a recipe given by Plykin. An algorithm is explained in an online article in the Notices of the AMS. You only need to iterate a single explicit function to get the picture. So, this can be done using any fractal generator (chaospro,fractint...). Note that zooming on the boundary of the lake is not so interesting. You only get straight interlaced color bands after a few zooms.

The result can be seen online (scroll to the section about Wada Lakes). There is also a movie.

Actually there are four regions on the pictures/movie. You can merge two of them if three is enough for your need.

Cheers

-
 But if I merge two regions, then those two will no longer be homeomorphic to disks, right? – Mikael Vejdemo-Johansson May 16 2010 at 21:09 Yes, sorry. Merging is not the correct word. You can use the method to produce any finite number of lakes, just by adding/removing attracting periodic points. You can get three lakes by removing the perturbation at the origin, for example. – coudy May 17 2010 at 11:13 I also wonder whether you are making the distinction that Wikipedia is making between Lakes of Wada and *Basins of Wada*; what I really want is a partition of the unit square, preferably in roughly equal size pieces, with the wada property. – Mikael Vejdemo-Johansson May 17 2010 at 17:07 I believe a countably infinite number of lakes can actually be produced. – Jon Paprocki Jun 12 at 17:18