M-set and Farey sequence

• 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 - a²; so  z_n+1 = z_n² + 1/4 - a²
  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 =

1. Ford circles
   
   
   
2. "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
   
   
3. 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

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

http://republika.pl/fraktal/

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