Jim, surely you already have this, but, using Allan's definition of tau below, tau/2, interpreted base 4, is a "squarefree" (stutter-free) string on 0123. Replacing 2 with 1, then 3 with 2 gives a squarefree string on 012. (The "same" string as in the standupmaths video?) But my favorite "fractoid", repeated here before, is that if H(t) is the Hilbert curve mapping [0,1] onto [0,1]×[0,i] (the unit square in the complex plane), then tau and 1-tau are the only solutions x to H(x) = x+i. For this, and some analytic expressions for tau, see http://www.inwap.com/pdp10/hbaker/hakmem/series.html Item 122, The parity number. --rwg Date: 2017-01-08 14:10 From: Allan Wechsler <acwacw@gmail.com> 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

standupmaths (https://www.youtube.com/watch?v=prh72BLNjIk)

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