Proceed to Safety


Robert P. Munafo, 2023 Jun 17.

[AMC] Art Matrix Corp, PO Box 880 Ithaca NY 14851-0880.

This small company sells beautiful color pictures, postcards, slides, etc. of Mandelbrot Set images. They have lots of their own pictures, and they will do custom pictures if you give them the parameters. The prices are quite reasonable, and the quality is better than most people could do themselves, even with a good camera and a super high-res RGB monitor. They can also put you in touch with the authors of MANDELZOOM-type programs for other brands of personal computers.

[AMY] Amygdala, PO Box 219 San Cristobal, NM 87564-0219.

This is a newsletter about the Mandelbrot Set and other fractals. Write to them for more info. Issue #22, A Mandelbrot Set Lecture Tour, by Philip & Philip, and Issue #19, The Taming of the Shrew, also by Philip & Philip, covers many of the same things described here.

[PM1] P.J. Myrberg, "Sur l'itération des polynomes quadratiques", Journal des Mathématiques Pures et Appliquées 41 (1962).

The first publication of the value commonly called the Feigenbaum point, -1.401155.

[PM2] P.J. Myrberg, "Iteration der reellen Polynome zweiten Grades III", 1963, Annales Academiae Scientiarum Fennicae Band 336 (1963).

Based on the iteration Zn+1=Zn2-C, he gives numerical values for the nuclei of two islands on the real axis: period-3 R2F(1/2B1)S and period-4 R2F(1/2B1)FS[2]S; and he period-4 mu-atom R2.1/2.1/2a. The values given are 1.7548776662, 1.9407998065, and 1.3107026413 respectively. The period-8 nucleus of R2.1/2.1/2.1/2a is also given as 1.3815474844.

[SL1] N. J. A. Sloane. A Handbook of Integer Sequences. New York: Academic Press, 1973.

This book lists nearly every integer sequence which has ever been mentioned in scientific literature, although by now it is a bit old. Several important sequences are related to the Mandelbrot Set, including the powers of 2, Euler's totient function, and others.

[MB1] Benoit B. Mandelbrot. The Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1983. ISBN 0-7167-1186-9.

This book discusses dozens of fractals that Mandelbrot has worked on over the years. The book, which is an updated and expanded version of the earlier Fractals: Form, Chance, and Dimension, contains lots of useful information but does not serve well as a reference book due to its rather terse and complex style. Mandelbrot's primary goal in the book seems to be to simulate the common forms of nature with mathamatics, and at this he succeeds in most cases.

Most of my preferred terminology for features of the Mandelbrot set (such as filament, mu-atom, continent, island, lemniscate and so on) comes from this book. See page 183, in the sections entitled "self-squared Julia curves in the plane (Mandelbrot 1980n)" and "mu-atoms and mu-molecules"; plus the captions for plates 188 and 189 (on pp. 186-189).

[SO1] Peter R. Sørensen. "Fractals — Exploring the rough edges between dimensions". Byte, Sep. 1984 pp. 157-172.

Article discusses a few of Mandelbrot's fractals and gives a program for drawing Julia Set plots. Photo 1 (page 158) shows a zoom sequence of 4 images of the Mandelbrot set using the Lambda map coordinate transform.

[DE1] A. K. Dewdney. "Computer Recreations". Scientific American August 1985.

This article was the first description of the Mandelbrot Set and how to compute images of it in a well-known magazine; thus it is responsible for most of the interest in the Mandelbrot Set that came soon after. In other issues of Scientific American, this column discussed other simple, interesting computer programs, including several other types of computer art. The column no longer runs.

[DO1] A. Douady. "Algorithms for Computing Angles in the Mandelbrot Set". Chaotic Dynamics and Fractals pp. 155-168 (1986). Rather technical description of the relationships between external angles in different Julia Sets and in corresponding parts of the Mandelbrot Set.

[PE1] H. O. Peitgen and P. H. Richter. The Beauty of Fractals. New York: Springer-Verlag Inc. ISBN 0-387-15851-0.

This book has a dual purpose. To mathematicians and scientists it describes the chaotic behavior of complex systems, to which the Mandelbrot Set and Julia Sets are closely related. To the rest of us, it presents dozens of beautiful, high-resolution pictures and describes in a fairly simple manner how to program the computer to duplicate them. It describes the structure of the Mandelbrot Set in a more complete manner than Mandelbrot's book.

Bodil Branner. "The Mandelbrot set." In Proceedings of Symposia in Applied Mathematics (Chaos and Fractals: The Mathematics Behind the Computer Graphics) 39 75-105 (1989)

John Milnor. "Hyperbolic components in spaces of polynomial maps." SUNY Stony Brook (1992). at

(also in Conformal dynamics and hyperbolic geometry 573 (2012): 183-232.)

[CG] L. Carleson and T.W. Gamelin, Complex Dynamics, Springer, Berlin Germany (1993)

Robert L. Devaney, "The Mandelbrot Set and The Farey Tree." (1997). at

Curtis T. McMullen. "The Mandelbrot set is universal". (preprint) Mathematics Department, Harvard University (1997).

Also in Tan L, ed. The Mandelbrot set, Theme and Variations. London Math Soc Lecture Note Ser. 274. Cambridge University Press (2000) 1–17.

Robert L. Devaney, "The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence." The American Mathematical Monthly, 106:4 pp. 289-302 (1999). DOI: 10.1080/00029890.1999.12005046

Dierk Schleicher. "On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets", (1999). at

[DO2] Adrien Douady, Xavier Buff, R. Devaney, and Pierrette Sentenac. "Baby Mandelbrot sets are born in cauliflowers." London Mathematical Society Lecture Note Series (2000): 19-36.

Shows the nested bifurcation rings in the embedded Julia sets in the cardioid cusp of any island, and the first to use the term "cauliflower" for them.

[PJS] Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, Inc. (2004) ISBN 0-387-97903-4.

This book is an excellent reference on current knowledge throughout the entire field of chaos and fractals at the time. There is also quite a bit of discussion on related topics such as pi (π). Several BASIC programs, extensive footnotes and bibliography.

[BEK] Xavier Buff, Adam L. Epstein, and Sarah Koch. "Böttcher coordinates." Indiana University Mathematics Journal (2012): 1765-1799.

[RPO] Miguel Romera, Gerardo Pastor, A. B. Orue, A. Martin, M. F. Danca, and Fausto Montoya. "A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set." Chaos-Fractals Theories and Applications, Volume 2013, article ID 105283 (2013).

[DS] Schinella d’Souza. "Counting hyperbolic components in the main molecule". at

revisions: 19960305 oldest on record; 20101021 add notes on [MB1]; 20230418 clean up formatting; 20230419 add Douady2000; 20230616 add Branner1988 and McMullen1997; 20230617 Add Schleicher1999

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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