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 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

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 :

- (13/31,18/31)
- (14/31,17/31)
- (15/31,16/31)

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 ):

- (3/7,4/7) primitive period 3,

cusp c= -1.75 = -7/4

center c=-1.754877666246693 - primitive period 4 :

pair of rays landing on the root point (cusp): (7/15, 8/15)

with center c=-1.940799806529488 - primitive period = 5
- (13/31,18/31)
- (14/31,17/31)
- (15/31,16/31)

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

- enumeration of features by Robert P. Munafo
- Midget Mandelbrot Set Images © by Jay Hill, 1997
- "A count of maximal small copies in Multibrot sets." R Perez ,A. Bridy,
- largest islands by Robert P. Munafo
- Fractint PAR file centered on midgets by Jay Hill

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

About