Re: M-set: from internal angles to external

The Linus.org isn't coming up on this end of the Internet.
I'm also having trouble with Australia today...

The natural cyclotomic rotations are:
r(n,m)=Exp(I*m*2*Pi/n)
They are solutions to the
x^n -1=0
cyclotomic polynomials.
Linus Vepstas seems to be using some 2 based scale
that I don't actually understand: you should ask him.

The usual way to get the "cycles" in a Mandelbrot set is to do
a constant iterative procedure: ( from page 129 of Complex Dynamics by Carleson and Gamelin, Spring, 1991)
P[1,c]=c
P[2,c]=c+c^2
p[3,c]=c+c^2+2*c^3+c^4
p[4,c]=c+c^2+2*c^3+5*c^4+6*c^5+6*c^6+4*c^7+c^8
croot[m_,n_]=c/. Table[Solve[P[n,c]==0,c],{n,1,4}][[m,n]]
is solved for the roots and the Julia of the roots gives
Douady type angular cycles: one of the roots give Douady's rabbit.
As you can see these polynomial are not only different than the
cyclotomic polynomials, but have very different solutions.
If Linus Vespas has a better way, I'd like to see it.
That gives angles on the circle of 2*Pi radians of :
rational[m,n]=Arg[croot[m,n]]/(2*Pi)

Maybe you mean some other angles?
adam majewski wrote:

Roger Bagula napisał(a):

Several people have done work on this type of problem if I understand your question right. In "The Beauty of Fractals" Peitgen et al,Springer-verlag 1986 A. Duuady : Julia Sets and the Mandelbrot set pape 161 decusses such angles.
http://math.bu.edu/people/bob/
Bob Devaney has some papers related at:
http://math.bu.edu/people/bob/papers.html
Geometry of the Antennas in the Mandelbrot Set
With Monica Moreno Rocha

This paper proves that one can determine (using harmonic measure) the p/q bulb in the Mandelbrot set by looking at the geometry of its antenna. Fractals. 10 (2002), 39-46.

    * Postscript Version
    * pdf Version: http://math.bu.edu/people/bob/papers/monica.pdf
adam majewski wrote:

Hi
How compute externla angles of 2 landing rays on the root point of secondary components on the main carioid?

Adam Majewski

republika.pl/fraktal


Thx for that informations.
In Devaney method is conversion from binary number ( intinerary) to real number ( external angle).
Do you know different methods?
Linas Vepstas shows method that doesn't use conversion:
http://linas.org/art-gallery/escape/phase/arcs.html

rotation number=p/q
for p:=1 external_angle_1=1/(2^q - 1)
externa;_angle_2:=2/( 2^q -1)
is it possible to compute external angles for any rotation number using this method?

Adam Majewski
republika.pl/fraktal


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Relevant Pages

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