# Fractals/Iterations in the complex plane/fprays

Parabolic fixed point of period p is a landing point of p dynamic external rays. These rays divide neighborhood into curvilinear sectors.

# On the main cardioid

Maxima CAS code :

## 1/p

In the elephant valley ( from parameter plane ) it is easy to find rays landing on the roots and dynamic external rays that land on the parabolic fixed point z.
* first choose internal angle (= combinatorial rotation number) : 1/p
* compute pair of parameter rays, angles of the wake : $({\frac {1}{2^{p}-1}};{\frac {2}{2^{p}-1}})$
* compute list of p external angles of dynamic rays : $({\frac {1}{2^{p}-1}}...{\frac {n}{2^{p}-1}})$
| internal angle of main cardioid | parameter rays | dynamic rays |
| 0/1 | (0/1; 1/1) | (1/1) |
| 1/2 | (1/3; 2/3) | (1/3; 2/3) |
| 1/3 | (1/7; 2/7) | (1/7; 2/7; 4/7) |
| 1/4 | (1/15; 2/15) | (1/15; 2/15; 4/15; 8/15) |
| 1/5 | (1/31; 2/31) | (1/31; 2/31; 4/31; 8/31; 16/31) |
| 1/6 | (1/63; 2/63) | (1/63; 2/63; 4/63, 8/63, 16/33; 32/63) |
| 1/7 | (1/127; 2/127) | (1/127; 2/127; 4/127, 8/127, 16/127; 32/127; 64/127) |
| 1/p | $({\frac {1}{2^{p}-1}};{\frac {2}{2^{p}-1}})$ | $({\frac {1}{2^{p}-1}}...{\frac {n}{2^{p}-1}})$ |
Note that :
* internal ray 0/1 = 1/1
* internal angle 1/p means that ray goes from period 1 component ( parent period = 1) to period p component ( child period = p)
* as child period grows computations are harder
* exponential growth of $2^{p}$. One can easly create a numeric value that is too large to be represented within the available storage space ( integer overflow ). For example $2^{34}$ is to big for short ( 16 bit ) and long ( 32 bit) integer.
* from period 1 to period ...
* 1/1 = cauliflower
* 1/2 = San Marco fractal
* 1/3 = Douady fat rabbit
* 1/4
* 1/5
* 1/7
* 1/10
* 1/15
* 1/20
* 1/30

## n/p

It is not so simple, as in 1/p case, to compute orbit portrait of parabolic fixed point.
Algorithm :
* choose child period p
* compute internal angle ( rational number) = n/p ( where n<p and n/p is ... ( to do ))
* compute angle of the wake
* switch to dynamic plane : use one angle from pair of parameter rays ( rays with the same angles land on the parabolic fixed point) to compute orbit portrait of parabolic fixed point
Methods for computing anglew of the wake:
* Combinatorial algorithm = Devaney's method
* step method:
* compute denominator of external angle = $2^{p}-1$
* find parameter rays that land on the root point which is on the boundary of main cardioid :
** compute all pairs for periods 1-p
** remove pairs which land not on the main cardioid ( inside < 1/3; 2/3 > wake )
** compute pairs of external angles which are not inside pairs of lower periods (see image on the right )
** choose n-th pair of angles which land on the root point

### Child period 5

From all 15 period five components only 4 components are directly connected to the main cardioid
| internal angle of main cardioid | parameter c = root point | parameter rays | dynamic rays |
| 1/5 | | (1/31; 2/31) | (1/31; 2/31; 4/31; 8/31; 16/31) |
| 2/5 | -0.504+0.568 i | (9/31,10/31) | (5/31 , 10/31 , 20/31 , 9/31 , 18/31) |
| 3/5 | | (21/31,22/31) | (11/31 , 22/31 , 13/31 , 26/31 , 21/31) |
| 4/5 | | (29/31,30/31) | (15/31 , 30/31 , 29/31 , 27/31 , 23/31) |

### Child period 7

From all 63 period seven components only 6 components are directly connected to the main cardioid
| internal angle of main cardioid | parameter rays | dynamic rays |
| 1/7 | (1/127; 2/127) | (1/127; 2/127; 4/127; 8/127; 16/127, 32/127, 64/127) |
| 2/7 | (17/127; 18/127) | (17/127; 34/127; 68/127; 9/127; 18/127, 36/127, 72/127) |
| 3/7 | (42/127; 43/127) | (42/127, 84/127, 82/127, 37/127, 74/127, 21/127, 42/127) |
| 4/7 | (84/127; 85/127) | (84/127, 41/127, 82/127, 37/127, 74/127, 21/127, 42/127) |
| 5/7 | (109/127; 110/127) | (109/127, 91/127, 55/127, 110/127, 93/127, 59/127, 118/127) |
| 6/7 | (125/127; 126/127) | (125/127, 123/127, 119/127, 111/127, 95/127, 63/127, 126/127) |

### Child period 11

| internal angle of main cardioid | parameter c = root point | parameter rays | parabolic fixed point z | dynamic rays |
| 1/11 | | | | |
| 5/11 | -0.6900598700150440+0.2760264827846140i | (681/2047, 682/2047) | -0.4797464868072486+0.1408662784207147i | (341, 597, 661, 677, 681, 682, 1194, 1322, 1354, 1362, 1364)/2047 |
* parameter plane t=5/11
* Dynamic plane t=5/11
Maxima CAS code using doubling map:
Other version :

### child period 21

See also wake 10/21
result without sort(Orbit)

### Child period 27

internal angle 13/27
* c = -0.739880396515927 +0.115700424748225 i
* fixed point alpha z = -0.496619178870972 +0.058046457062615 i
* period 27
The 13/27-wake of the main cardioid is bounded by the parameter rays with the angles
* 44739241/134217727 or p010101010101010101010101001
* 44739242/134217727 or p010101010101010101010101010

### Child period 34

See image by Arnaud Cheritat
Angles of the wake external rays ( on parameter plane ) in different formats :
$4985538889/17179869183={\frac {4985538889_{10}}{17179869183_{10}}}=p0100101001001010010100100101001001=.(0100101001001010010100100101001001)=0.(0100101001001010010100100101001001)=0.{\overline {0100101001001010010100100101001001}}_{2}={\frac {100101001001010010100100101001001_{2}}{1111111111111111111111111111111111_{2}}}$
$4985538890/17179869183=p0100101001001010010100100101001010=.(0100101001001010010100100101001010)=0.(0100101001001010010100100101001010)$
There are 8 589 869 055 components of period 34.
External angle landing on the root points :
$e={\frac {n}{d}}$
where denominator d is :
$d=2^{34}-1=17\ 179\ 869\ 183$
| internal angle of main cardioid | parameter c = root point | external angles of the wake (decimal fractions) | external angles of the wake ( binary fractions) | parabolic fixed point z | dynamic rays ( orbit portrait , only numerators) |
| 1/34 | | (1/d; 2/d) | | | |
| 13/34 | -0.392571548476155+0.585781365897037i | (4985538889/d ; 4985538890/d) | (p0100101001001010010100100101001001; p0100101001001010010100100101001010) | -0.3695044586103295 ; 0.3368478218232787 | [4985538889,9971077778,2762286373,5524572746,11049145492,4918421801,9836843602, 2493818021,4987636042,9975272084,2770674985,5541349970,11082699940,4985530697, 9971061394,2762253605,5524507210,11049014420,4918159657,9836319314,2492769445, 4985538890,9971077780,2762286377,5524572754,11049145508,4918421833,9836843666, 2493818149,4987636298,9975272596,2770676009,5541352018,11082704036] |
Widest sector ( which incudes critical component ) is :
Last componet = component to the left of component including critical point zcr = 0.0. This component is almost not changing when iPeriodChild is increasing , see this video

### 89
#### 34/89

See image by Arnaud Cheritat
Let's find some info using program Mandel by Wolf Jung :
denominator or external angle ( computed with this program ) :
$d=(2^{89}-1)_{10}=618970019642690137449562111_{10}=11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111_{2}$
Location of root point using book program by Claude Heiland-Allen:
External angles of rays that land on the root point one can compute with book program by Claude Heiland-Allen :
Converting to other forms using gmp :
Note that difference in floating form of external angles :
is on 27-th decimal digit after decimal sign
and on 88-th binary digit after decimal sign.
Numerators of the orbit portrait ( external angles of rays landing on the fixed point alfa ) :

# Code

* program Mandel by Wolf Jung: MainMenu\Rays\Orbit Portrait, input first angle of dynamic ray ( or wake angle)

# References

* ↑ muency : elephant valley
* ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
* ↑ integer number in wikipedia
* ↑ Integer overflow in wikipedia
* ↑ planetmath : San Marco fractal
* ↑ wikipedia : Douady rabbit
* ↑ wikipedia : orbit portrait
* ↑ Parameter rays of root points of period p components by A Majewski
* ↑ Parameter rays of root points of period p components by A Majewski
* ↑ Arnaud Cheritat - gallery
* ↑ knowledgedoor calculators: convert_a_ratio_of_integers
* ↑ Arnaud Cheritat - gallery