M-set and Farey sequence

From Farey sequence to M-set:

- draw a
**line**from 0 to 1 ( points are numbered from 0 to 1 ) - draw a Ford Circles on a line
- make a circle from a line ( glue point 0 and 1, so 0 = 1 )
- draw circles on the
**circle**( = buds on the main cardioid ) - make a
**cardioid**from circle

( see "square" parametrisation c =f(a)= 1/4 - a^{2}; so z_{n+1}= z_{n}^{2}+ 1/4 - a^{2}

the main cardioid of the M-set turns into a circle with radius r = 1/2. primary bulb attaches to the main circle ( cardioid) at an internal angle = 2* Pi* m / n [ degrees ]

where m/n is rotation number (e.g. 1/2 -> 180^{o}, 1/3 -> 120^{o}and 1/4 -> 90^{o}) - ad period doubling and other sequences of circles
- result is
**"theoretical M-set"**

Anne M Burns gives algorithm for**Fake Mandelbrot Set**= M-set without hairs, filaments and primitive hyberbolic componennts. ( see her paper and www page below). - Topological model of Mandelbrot set( reflects the structure of the object )

- A Delphi unit implementing Bresenham's Circle/Ellipse/Line algorithm by Finn Tolderlund
- Drawing Ford Circles in Delphi
- Drawing Circles on circle in Delphi =

- Ford circles

- "Mandelbrot" circles ( = circles on circle)

Compare this image with that on Robert L. Devaney's page about Rotation Numbers and Internal angles of the Mandelbrot bulbs . Do you see the similarity?

radius of secondary components ~= (S/2)* sin(Pi * p / q) / (q*q)

Where S is a diameter (= 2 * radius) of main circle

- Theoretical Mandelbrot set ( = circles on cardioid) - (in future )

radius ~= S * sin(Pi * p / q) / (q*q) for children of cardioid

where p/q is the internal angle of the child

and S is the size of the parent, measured (roughly) from its cusp to the bond point with its 1/2 child.

It is topological model of Mandelbrot set

- Mandelbrot Bud Maths by Linas Vepstas
- Square parametrisation by Robert L. Devaney
- "The Mandelbrot Set and the Farey Tree " by Robert L. Devaney
- Mu atome sizes by Robert P. Munafo
- other ways to draw mandelbrot? in sci.fractals
- drawing ellipse by John Kennedy
- The 1/n2 rule by Michael Frame, Benoit Mandelbrot, and Nial Neger
- The Structures of the Mandelbrot Sets by Thayer Watkins
- Burns A M: Plotting the escape- An animation of parabolic bifurrcation in the mandelbrot set. Mathematics Magazine: Volume 75, Number 2, Pages: 104-116 page 104

Bifurcation sets and critical curves A preprint version of a “Mathematical graphics” column from Mathematica in Education and Research . Mark McClure
- The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653
- Bifurcation sets and critical curves by Mark McClure
- Construction of IFS Fractals of Specific Similarity Dimension By Roger Bagula

Feel free to e-mail me!

Author: Adam Majewski adammaj1-at-o2-dot-pl

http://republika.pl/fraktal/