Summary

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

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You are free:

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• to remix – to adapt the work

Under the following conditions:

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https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

Software

Program :

• is made with c / gcc
• uses GMP for arbitrary precision rationals
• MPFR for arbitrary precision floating point
• OpenMP

Image

Algorithms

• exterior :
  • Smooth colouring with continuous escape time
  • grid based on integer escape time and binary decomposition
  • Atom domains
  • external rays : the Newton Method^[3]
• interior :
  • Interior coordinates^[4]
  • Atom domains^[5]
• boundary : distance estimation ( DEM/M)

All algorithms are described in the book : "How To Write A Book About The Mandelbrot Set" by Claude Heiland-Allen^[6]

External rays

Wolf Jung descibes the test for drawing external parameter rays : three external parameter rays for angles (in turns)

• 321685687669320/2251799813685247 (period 51, lands on c1 = -0.088891642419446 +0.650955631292636i )
• 321685687669322/2251799813685247 ( period 51 lands on c2 = -0.090588078906990 +0.655983860334813i )
• 1/7 ( period 3, lands on c3 = -0.125000000000000 +0.649519052838329i )

Angles differ by about ${\displaystyle 10^{-15}}$, but the landing points of the corresponding parameter rays are about 0.035 apart. ^[7] It can be computed with Maxima CAS :

(%i1) c1: -0.088891642419446  +0.650955631292636*%i;
(%o1) 0.650955631292636*%i−0.088891642419446
(%i2) c2:-0.090588078906990  +0.655983860334813*%i;
(%o2) 0.655983860334813*%i−0.09058807890699
(%i3) abs(c2-c1);
(%o3) .005306692383854863
(%i4) c3: -0.125000000000000  +0.649519052838329*%i$
(%i5) abs(c3-c1);
(%o5) .03613692356607755
(%i6) a3:1/7$
(%i7) float(abs(a3-a1));
(%o7) 4.440892098500628*10^−16

1/7

Informations from W Jung program :

The angle  1/7  or  p001 has  preperiod = 0  and  period = 3.
The conjugate angle is  2/7  or  p010 .
The kneading sequence is  AA*  and the internal address is  1-3 .
The corresponding parameter rays are landing at the root of a satellite component of period 3.
It is bifurcating from period 1.

321685687669320/2251799813685247

The angle  321685687669320/2251799813685247  or  p001001001001001001001001001001001001001001001001000 has  preperiod = 0  and  period = 51.
The conjugate angle is  321685687669319/2251799813685247  or  p001001001001001001001001001001001001001001001000111 .
The kneading sequence is  AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABB*  and the internal address is  1-49-50-51 .
The corresponding parameter rays are landing at the root of a primitive component of period 51.

Angled internal adress :

  ${\displaystyle 1\quad {\xrightarrow {16/49}}\ 49{\xrightarrow {1/2}}98{\xrightarrow {1/2}}...MF^{2}({\frac {16}{49}}*{\frac {1}{2}}){\xrightarrow {}}M(49,1){\xrightarrow {}}51}$

The 16/49-wake of the main cardioid is bounded by the parameter rays with the angles
80421421917329/562949953421311  or  p0010010010010010010010010010010010010010010010001  and
80421421917330/562949953421311  or  p0010010010010010010010010010010010010010010010010 .

Misiurewicz point

 M(49,1) = c = -0.088578934561802  +0.650191316455881 i

/mandelbrot-numerics/c/bin$ ./m-describe -0.088578934561802  +0.650191316455881
the input point was -0.088578934561802006 + +0.650191316455881019 i
the point escaped with dwell 13913.14281
and exterior distance estimate 1.65230e-23
nearby hyperbolic components to the input point:
- a period 1 cardioid
  with nucleus at +0.000000000000000000 + +0.000000000000000000 i
  the component has size 1.00000e+00 and is pointing west
  the atom domain has size 0.00000e+00
  the nucleus is 6.56197e-01 to the south of the input point
  the input point is exterior to this component at
  radius 1.00213064069366764 and angle 0.326615441968459541 (in turns)
  the multiplier is -0.463997292336526423 + +0.888241146154282180 i
  a point in the attractor is -0.231998646168263212 + +0.444120573077141090 i
  external angles of this component are:
  .(0)
  .(1)
- a period 3 circle
  with nucleus at -0.122561166876653638 + +0.744861766619744237 i
  the component has size 1.88914e-01 and is pointing north
  the atom domain has size 3.74124e-01
  the nucleus is 1.00585e-01 to the north-north-west of the input point
  the input point is exterior to this component at
  radius 1.06496264182355826 and angle 0.058440999696630087 (in turns)
  the multiplier is +0.993969869373938830 + +0.382320974125887503 i
  a point in the attractor is -0.272481525828079219 + +0.083247619635819975 i
  external angles of this component are:
  .(001)
  .(010)
nearby Misiurewicz points to the input point:
- a point with preperiod 1 and period 1
  is at -2.000000000000000000 + +0.000000000000000000 i
  the point is 2.01898e+00 to the west-south-west of the input point
- a point with preperiod 2 and period 2
  is at -0.228155493653961816 + +1.115142508039937308 i
  the point is 4.85450e-01 to the north-north-west of the input point
- a point with preperiod 1 and period 3
  is at +0.395014052076694544 + +0.555624571005995938 i
  the point is 4.92753e-01 to the east of the input point
- a point with preperiod 4 and period 3
  is at +0.164679751707792033 + +0.630301832272442519 i
  the point is 2.54038e-01 to the east of the input point
- a point with preperiod 7 and period 3
  is at +0.073739340457323027 + +0.642958116764365095 i
  the point is 1.62479e-01 to the east of the input point
- a point with preperiod 10 and period 3
  is at +0.025776654392151672 + +0.646754862998201507 i
  the point is 1.14407e-01 to the east of the input point
- a point with preperiod 13 and period 3
  is at -0.003720933850675586 + +0.648246033473079630 i
  the point is 8.48803e-02 to the east of the input point
- a point with preperiod 16 and period 3
  is at -0.023652091976878179 + +0.648928351287613503 i
  the point is 6.49391e-02 to the east of the input point
- a point with preperiod 50 and period 1
  is at -0.088233647532228787 + +0.650688541738126180 i
  the point is 6.05356e-04 to the north-east of the input point
- a point with preperiod 50 and period 3
  is at -0.088233647532228801 + +0.650688541738126069 i
  the point is 6.05356e-04 to the north-east of the input point
- a point with preperiod 50 and period 49
  is at -0.088609218415842989 + +0.650468589049570056 i
  the point is 2.78921e-04 to the north of the input point

 ./mandelbrot_describe_external_angle '.0010010010010010010010010010010010010010010010001(0010010010010010010010010010010010010010010010010)'
binary: .0010010010010010010010010010010010010010010010001(0010010010010010010010010010010010010010010010010)
decimal: 45273235722436040546370715649/316912650057056787424222380032
preperiod: 49
period: 49
a@zelman:~/book/code/bin$ ./mandelbrot_describe_external_angle '.0010010010010010010010010010010010010010010010010(0010010010010010010010010010010010010010010010001)'
binary: .0010010010010010010010010010010010010010010010010(0010010010010010010010010010010010010010010010001)
decimal: 45273235722436603496324136959/316912650057056787424222380032
preperiod: 49
period: 49

321685687669322/2251799813685247

The angle  321685687669322/2251799813685247  or  p001001001001001001001001001001001001001001001001010 has  preperiod = 0  and  period = 51.
The conjugate angle is  321685687669329/2251799813685247  or  p001001001001001001001001001001001001001001001010001 .
The kneading sequence is  AABAABAABAABAABAABAABAABAABAABAABAABAABAABAABAABAA*  and the internal address is  1-3-51 .
The corresponding parameter rays are landing at the root of a satellite component of period 51.
It is bifurcating from period 3.

Angled internal adress :

  ${\displaystyle 1\quad {\xrightarrow {1/3}}\ 3\quad {\xrightarrow {1/17}}\ 51}$

Bash source code

"... be aware that the code is in alpha state and might change making the examples incompatible " Claude Heiland-Allen

#!/bin/bash

# bash script file to run program by Claude Heiland-Allen
# from https://gitorious.org/maximus/book/
# chmod +x jung_in.sh
# run ( from code directory) : :
# time ./examples/jung_in.sh

# left upper c = -0.093513494934392  +0.655423979708980 i 
# right down c = -0.084366835109596  +0.646262824385254 i   
# center c = -0.087053056317652  +0.652712041637700 i    
# zoom view

view="-0.087053056317652  0.652712041637700  0.01"

# Test for parameter rays by Wolf Jung 
# http://mndynamics.com/indexp.html
# draw 3 parameter external rays which 
# "... the angles 1/7 (of period 3) and 321685687669320/2251799813685247 (of period 51) differ by about 10-15, but the landing points of the corresponding parameter rays are about 0.035 apart. 
# On this scale, the rays cannot be drawn well by tracing the argument, unless high-precision arithmetics is used. 
# The following image shows the parameter ray for 1/7 and two rays of period 51 in the slit betwen the main cardioid and the 1/3-component. "
#
# script uses heredoc syntax 

./render $view && ./colour > out.ppm && ./annotate out.ppm j_in.png<<EOF
rgba 1 1 1 1
line ./ray_in 1/7 250 | ./rescale 53 53 $view 0
line ./ray_in 321685687669322/2251799813685247 250 | ./rescale 53 53 $view 0
line ./ray_in 321685687669320/2251799813685247 250 | ./rescale 53 53 $view 0
EOF

References

1. ↑ c program by Claude Heiland-Allen
2. ↑ Claude Heiland-Allen - blog
3. ↑ An algorithm to draw external rays of the Mandelbrot set by Tomoki KAWAHIRA
4. ↑ Interior coordinates in the Mandelbrot set
5. ↑ Modified Atom Domains
6. ↑ Mandelbrot Notebook
7. ↑ Wolf Jung's test for precision of drawing parameter rays