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.



Algorithms for the Lakes of Wada
Wada lakes can be obtained following a recipe given by Plykin.
An algorithm is explained in an online [article](http://www.ams.org/notices/200601/fea-coudene.pdf) 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](http://www.math.univ-brest.fr/perso/yves.coudene/dyn3.html) (scroll to the section about Wada Lakes). There is also a [movie](http://www.math.univ-brest.fr/perso/yves.coudene/lakes.avi).


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


Cheers


