- Point of exterior of M-set has one external angle.
- M-set

- Point of boundary of M-set has external angle ( one or more) and internal angle ( only one ).

Boundary point which has 2 external angles ( 2 external rays land on this point) is called biaccessible. It is a root of component. - Point of M-set of interior of M-set has an internal angle ( only one).

- Point of boundary of M-set has external angle ( one or more) and internal angle ( only one ).

So we have

- complex angle of complex point := Argument ( c)
- last point phase of orbit = Argument ( Zn) where Zn := F
_{J}( Z0=0,n,c ) it is used in decomposition of exterior of M-set - external angle of Mandelbrot set:= Arg
_{M}(c) = Arg ( Phi_{M}( c) ) where Phi_{M}it is a Boettcher map - internal angle of hyperbolic component of Mandelbrot set := Argument ( multiplier (c) )

Angle is a real number.

angle = Argument ( complex_number)

Argument ( c) : C --> R

because :

c := Re(c) + Im (c)* i =

Log(c) := Ln ( r ) + i * ( angle + 2 * Pi* k )

Argument(c) := Im ( Log ( c) )

Angles can be measured in degrees, radians and turns.

360 [degrees] = 2 * Pi [radians] = 1 [ turn]

angles are measured in counterclockwiserotation

and modulo 1

it means 0,2 = 1,2 mod 1 = 2,2 mod 1

Forms of angles:

- binary form ( sequence of binary digits)
- decimal form :

- vulgar fraction = rational numbers
- irreducible (or in lowest terms) fraction
- external angles can be in normalized form : numerator / (2
^{p}-1)

where p is the period of hyperbolic component

form n/d where d=(2^{p}-1) is normalized form

form n/(2^{p}-1) is explicit normalized form

"if the denominator is 2^{p}-1, the external angle leads to the root of a mu-atom of period p.

All other denominators correspond to external angles that lead to branch points, tips, etc. The external angles with powers of 2 in the denominators correspond to the bifurcation in rotational symmetry of the filaments surrounding all islands. This also shows up in the symmetry of embedded Julia sets and their paramecia." from External Angle Robert P. Munafo

- vulgar fraction = rational numbers

It is a curve, sometimes a line.

External ray of Mandelbrot set is called

External ray of Julia set is called

R(c,angle) = { c | Arg

Rays are labeled by the coresponding angles.

Autor: adammaj1-at-o2-dot--pl

Adam Majewski

Feel free to e-mail me. (:-))

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