[seqfan] Mandelbrot Iteration from "Road to Reality"

Brad Klee bradklee at gmail.com
Sat Dec 8 18:09:09 CET 2018

Hi Seqfans,

While there is much to be said regarding jagged boundary curves and the failure
of fragile materials such as glass, the results of such experiments do not seem
to fit with the general OEIS directive. However there are many entries having to
do with //fractals//.

If you're feeling adventurous today, perhaps check out a book from Roger
Penrose. Chapter four of "The Road to Reality" discusses "Magical Complex
Numbers", and section 4.5 in particular, "How to construct the Mandlebrot set".
The iteration, z -> z^2 + C, is simple ( simple enough for a Casseur? ),
and it leads right into the entries A143848 and A143849.

If we change the constant from C=i to C=1, and use the initial condition
z_0=1+i, then we may extend the records by a couple more sequences:

Complex Sequence: z -> z^2 + 1, z_0 = 1 + i .
Re: 1, 1, -2, -11, -134, -105947, 2325485914 . . .
Im: 1, 2, 4, -16, 352, -94336, 19989232384 . . .

One valid complaint about the style of "Road to Reality" would be that
Penrose doesn't include very much in the way of example calculation.
This becomes most cumbersome to the reader at chapter 8 and
thereafter. It could be quite a fun and useful project to create a map
between this book and the OEIS, also a difficult challenge. A search
returned only four citations: A014985, A049006, A010123, A124507.

Best wishes for a peaceful and productive holiday season -- Brad

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