I've been playing around with generating Thue Morses sequence in rings within rings and study the natural numbers N (including 0) represented by radial binary combinations. Some interesting geometrical properties emerges, such as:

* All odd integers will be arranged in a specific geometric order

* Even integers will be arranged in a specific geometric order

* The one complement will allays be an opposite radial combination where the two radial combinations together constitute a diagonal.

* The two complement for a radial combination can be found by symmetry.

and more

The presentation got a lot of graphics so I can´t post it here but I put it up on my blog if someone cares to take a look.

Link:

Some symmetrical properties constructing sets of natural numbers using Thue Morse sequence