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

Section 3 of the attached link might be relevant --- it turns out that the properties of T-M are much easier to establish via inspection of an equivalent 4-symbol sequence: https://www.dropbox.com/s/1etnzbciqhbmy2j/csp08_submission_23.pdf And while I am on the subject, the main topic of that paper is a little-known analogue of T-M christened the "Pagoda" sequence, arising as follows. The ternary T-M property of square-freedom is a special case of the more restrictive property of minimal linear complexity, that is nowhere satisfying _any_ linear recurrence of order r (for more than 2r consecutive terms). This constraint turns out impossible to satisfy for a ternary sequence; but if relaxed to allow just 2r+1 consecutive terms (linear deficiency = 2), it results in an apparently unique (up to some fairly trivial transformations) object with a rather beautiful, fractal-like number wall. [Sadly the publisher required greyscale graphics, though these walls look much nicer in colour.] The paper finishes with an improbable 2-D tiling-based proof that Pagoda satisfies deficiency = 2 , by which stage it has to be admitted that the going has become a little heavy ... [Correction: the ambitious graphic on p.14 is missing its bottom two rows, especially confusing since it was (deliberately) rotated anticlockwise to emphasise the symmetry: as a result, the initial "22" below has been omitted. Along with an initial invisible white (= 0) and black (= 1) line running up the left-hand side from the bottom, not to mention the eye-watering pixel scale, it was not a success. The Pagoda sequence should actually commence: 22010211 22011201 22010211 02211201 22010211 22011201 02210211 02211201 ... ] Fred Lunnon On 1/8/17, 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