Parameter rays of mini mandelbrot sets








midget = mini mandelbrot set = primitive component
boundary of the central bulb of the Mandelbrot set is a cardioid given by the equation
c = {1 - (e^{it}-1)^2}/4
The Mandelbrot set contains an infinite number of slightly distorted copies of itself
and the central bulb of any of these smaller copies is an approximate cardioid.


Nm(p) = Number of mini Mandelbrot sets (for given period ) :
= Number of all cardioids (for given period )
= Number of all components (for given period ) - Number of components (for given period ) that are not cardioids

Nm(p) = Na(p) - Nd(p)
Nm(1)=1 there is only one period 1 cardioid = main cardioid
Nm(2)=0 there is no period 2 cardioid
Nm(3)=1 there is only one period 3 cardioid,
Nm(4)=3 there are three period 4 cardioids,,
Nm(5)=11,
Nm(6)=20,
Nm(7)=57,
Nm(8)=108,
Nm(9)=240,
Nm(10)=472,
Nm(11)=1013,


Period 3


on main antenna ( real slice of mandelbrot set ):
(3/7,4/7) primitive period 3,
cusp c= -1.75 = -7/4
center c=-1.754877666246693

Period 4


Mini Mandelbrot set with:
- period of cardioid = 4 ( primitive period),
- center c = - 0.15652 + 1.03225*I
- pair of angles of external rays landin on the root point ( cusp) : ( 3/15 ; 4/15 )



number of iteration increased to 1000

(Similar midget ( miror symmetry around real axis ) with primitive period 4 one can find for angles (11/15,12/15) and center c=-0.15652 - 1.03225 I )


Here are also drawn rays of angles 167/819 and 164/819 which land on the root point of period 12 component [M1]

pair of rays (7/15, 8/15) landing on the root point (cusp): c = -1.940799806529485 +0.000000000000000*i ( on main antenna ) with center c=-1.940799806529488


Period 5


on main antenna :

Mini Mandelbrot set with:
- period of cardioid = 51 ( primitive period),
- center c = -0.154089799234924 + 1.03061700413726*i
- pair of angles of external rays landin on the root point ( cusp) : ( 3/15 ; 4/15 )
e1 = 450359962736947/2251799813685247 = 450359962736947/(2^51 - 1)= 0.1999999999999545252649113535679094088732170888131495770499274359
e2 = 450359962736948/2251799813685247 = 450359962736948/(2^51 - 1)= 0.1999999999999549693541212036307227935521895781802086632115492383
Next cardioid :
e1 = 321685687669320/2251799813685247

Midgets on main antenna ( real slice of mandelbrot set ):


images made with Mandel - program for DOS by Wolf Jung, version 3.5
Probably mathemathically most advanced program for drawing mandel / Fatou / Julia sets

1< enter > ; q = 2 ; e = 0/1 < enter > ; F9 ; < end >
changed to grayscale, resized and converted to jpg using IrfanView




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

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