The Mandelbrot Set M is the set of complex numbers c, for which the sequence z[0] = 0, z[n] = z[n-1]^2 + c does not diverge. The set has a notoriously complicated geometric structure. Douady and Hubbard constructed an explicit conformal map between the complement of M and the complement of a unit disk; the map can be carried into the boundary of M and assigns each point near the boundary a unique "external angle". Part of M consists of a sequence of disks of decreasing size, arrayed along the negative real axis and converging on a point near -1.4. The exact end of this sequence is in the boundary of M, and points near it have external angles near 2pi * tau and 2pi * (1 - tau), where tau is the Prouhet-Thue-Morse number 0.0110100110010110100... (base 2). On Sun, Jan 8, 2017 at 3:51 PM, James Propp <jamespropp@gmail.com> wrote:
Do any of you have any favorite private facts about the Thue-Morse sequence, or any favorite links to existing content on this subject?
(I already know about the Numberphile video "The Fairest Sharing Sequence Ever".)
I'd especially like a link to an image or animation graphically depicting the self-similarity of the Thue-Morse sequence. (I have my own ideas for a GIF that would depict this, but I prefer not to create things that already exist.)
I'd also be interested in knowing whether any well-known (or not so well-known) poems use an abbabaab rhyme scheme, or whether there is any interesting music based on the Thue-Morse sequence.
(Yes, this is all for my next Mathematical Enchantments piece.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun