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 - a2; so zn+1 = zn2 + 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 -> 180o, 1/3 -> 120o and 1/4 -> 90o)
- 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 )
Sources ( delphi programs):
- 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
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Author: Adam Majewski adammaj1-at-o2-dot-pl