# Fractals/Iterations in the complex plane/subwake

How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is accessible from main cardioid ( M0) by a finite number of boundary crossing ?

# Key words

* Tuning
* external ray
** external angle in the binary number form

# Algorithm = Douady tuning

"The r/s internal ray in $M_{p/q}$ is the landing point of external rays $\theta _{\pm }(p/q,r/s)$ obtained from $\theta _{\pm }(r/s)$ by replacing:
* the digit 0 by repeating block ( of length q) from $\theta _{-}(p/q)$
* the digit 1 by repeating block ( of length q) from $\theta _{+}(p/q)$ "
"By repeating the same process ( which is known as 'tuning') we can compute the arguments of external rays landing on the boundary of any component which is accessible from $M_{0}$ by a finite number of boundary crossing." ( Shaun Bullett )
Douady tuning

# Examples by number of internal angles
## 1 angle

* 1/2 family

## 2 angles

Angled internal address:
describes a way:
* from center of period 1 hyberbolic component c=0
* along internal ray for angle p/q toward root point ( p/q-bond )
* along internal ray for angle r/s toward center ( nucleus) of s*q component of hyperbolic component
where:
* 1, q,s are periods of hyperbolic components
* p/q, r/s are internal angles

### (1/2, 1/3)

Angled internal address:
First compute external angles for p/q = 1/2 and r/s=1/3 wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Using this c program one gets:
Result:
$\theta _{-}(p/q,r/s)=\theta _{-}(1/2,1/3)=0.(\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/2,1/3)=0.(\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$
Check it with Mandel by Wolf Jung

### (1/3,1/25)

Angled internal address:
First compute external angles for p/q and r/s wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Using this c program one gets:
so the result is:
One can not check it using :
* program Mandel by Wolf Jung because period 75 exceeds the capacity of Mandel (verison 5.14)
* knowledgedoor calculator : "There are more than 100000 fractional digits in the new number. We are sorry, but we had to abort the calculation to control the loading on our server"
* Binary to Decimal converter from xploringbinary because : ***ERROR: invalid characters in input
Floating point decimal value can be computed ( xploringbinary calculator):
note that
so the difference is on the 23 decimal position
Using mandelbrot-symbolics:
Or using web interface with input : 1_1/3->3_1/25->75

### (1/3,1/2)

Angled internal address:
First compute external angles for p/q and r/s wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Result is :
One can check it using program Mandel by Wolf Jung :

### (1/3,1/3)

First compute external angles for p/q and r/s wakes ( here p/q=r/s) :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Result is :
One can check it using program Mandel by Wolf Jung :

### (1/3,1/4)

First compute external angles for p/q and r/s wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Result is :
One can check it using program Mandel by Wolf Jung :

### (1/3,3/4)

First compute external angles for p/q and r/s wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Result is :
One can check it in program Mandel by Wolf Jung :

### (1/3,1/89)

Angled internal address:

### (1/4, 1/5)

Input is :
First compute external angles for p/q and r/s wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Result is :
One can check it using program Mandel by Wolf Jung :

### (4/5, 1/17)

Angled internal address is :
* $1\quad {\xrightarrow {4/5}}\ 5\quad {\xrightarrow {1/17}}\ 5*17=85$
* 1-(4/5)-> 5 -(1/17)-> 85
Input is :
First compute external angles for p/q and r/s wakes :
then in $\theta (r/s)$ replace :
* digit 0 by block of length q from $\theta _{-}(p/q)$
* digit 1 by block of length q from $\theta _{+}(p/q)$
Result is :
One can not check it using program Mandel by Wolf Jung because period is too big. It gives answers only for period up to 64
One can check it online using web interface by Claude
* .(1110111101111011110111101111011110111101111011110111101111011110111101111011110111110)
* .(1110111101111011110111101111011110111101111011110111101111011110111101111011111011101)

## 3 angles
### (1/2, 1/3, 1/4)

We go thru the list of angles from right to left
First compute (1/3,1/4) wake which will be used as a a new r/s wake :
After that compute 1/2 wake ( most left), which will be used as a p/q wake :
then in $\theta (r/s)$ replace :
* digit 0 by repeating block (of length q, color blue) from $\theta _{-}(p/q)$
* digit 1 by repeating block (of length q, color red) from $\theta _{+}(p/q)$
Result is :
in plain text format ( for copy ) :
One can check this wake in program Mandel by Wolf Jung using ray to point command ( Ctrl+e) :

### (1/3, 1/4, 1/5)

Input is a list :
We go thru the list of angles from right to left and divide list into 2 sublists :
First compute (1/4,1/5) wake which will be used as a a new r/s wake :
After that compute 1/3 wake ( most left), which will be used as a p/q wake :
then in $\theta (r/s)$ replace :
* digit 0 by repeating block (of length q, color blue) from $\theta _{-}(p/q)$
* digit 1 by repeating block (of length q, color red) from $\theta _{+}(p/q)$
Result is :
One can check it using program Mandel by Wolf Jung :

## 4 angles list
### (1/2, 1/3, 1/4, 1/5)

Input is a list :
so the internal adders should be :
One can not check it using program Mandel because it is limited to period 64.
We go thru the list of input angles from right to left and divide list into 2 sublists :
First compute (1/3, 1/4, 1/5) wake which will be used as a a new r/s wake :
After that compute 1/2 wake ( most left), which will be used as a p/q wake :
then in $\theta (r/s)$ replace :
* digit 0 by repeating block (of length q, color blue) from $\theta _{-}(p/q)$
* digit 1 by repeating block (of length q, color red) from $\theta _{+}(p/q)$
Result is ( to check !!!!) :
$\theta _{-}(1/2,1/3,1/4,1/5)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$
$\theta _{+}(1/2,1/3,1/4,1/5)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$
One can check it visually using book program by Claude Heiland-Allen

## multi angle
### period 776

Try mandelbrot-web

### period 20

Info from Mandelbrot-web by Claude
* External Angle ( wiggled external ray )
** binary = .(00000000001111111111)
** decimal in Rational form = 1/1025
** Period = 20
* Kneading 1111111111000000000★
* Internal Address : 1→11→12→13→14→15→16→17→18→19→20
* Landings .00000000001111111111 1/1025 .00000000010000000000 1024/1048575
Angled internal addres:
$1\xrightarrow {1/11} 11\xrightarrow {1/2} 12\xrightarrow {1/2} 13\xrightarrow {1/2} 14\xrightarrow {1/2} 15\xrightarrow {1/2} 16\xrightarrow {1/2} 17\xrightarrow {1/2} 18\xrightarrow {1/2} 19\xrightarrow {1/2} 20$
Info from program Mandel by Wolf Jung
* The angle 1023/1048575 or p00000000001111111111
* has preperiod = 0 and period = 20.
* The conjugate angle is 1024/1048575 or p00000000010000000000 .
* The kneading sequence is AAAAAAAAAABBBBBBBBB* and
* the internal address is 1-11-12-13-14-15-16-17-18-19-20 .
* The corresponding parameter rays land at the root of a primitive component of period 20 with center c = 0.329617350093832 +0.042415693708911 i

# Code

Output:

# See also

* islands
* Mandelbrot symbolics in the browser by Claude Heiland-Allen
* automatically_finding_external_angles by Claude Heiland-Allen
* book program by Claude Heiland-Allen
* Trees of visible components in the Mandelbrot set by Virpi K a u k o - the tree structures of the sublimbs of the Mandelbrot set M, using internal addresses of hyperbolic components

# References

* ↑ Shaun Bullett : Lectures on one-dimensional complex dynamics' (7th-10th November 2005), lecture 4, page 43, section 4.5
* ↑ Ordered orbits of the shift, square roots, and the devil's staircase by Shaun Bulletta and Pierrette Sentenac
* ↑ knowledgedoor calculator
* ↑ exploring binary : binary-converter
* ↑ mandelbrot web