File:Part of parameter plane with external 3 rays landing on the Mandelbrot set.png
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Contents
Summary
DescriptionPart of parameter plane with external 3 rays landing on the Mandelbrot set.png |
English: Part of parameter plane with external rays 1/7, 321685687669320/2251799813685247 and 321685687669322/2251799813685247 landing on the Mandelbrot set |
Date | |
Source | Made with c program[1] by Claude Heiland-Allen[2] |
Author | Adam majewski |
Licensing
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
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 :
- 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 , 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 :
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 :
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
- ↑ c program by Claude Heiland-Allen
- ↑ Claude Heiland-Allen - blog
- ↑ An algorithm to draw external rays of the Mandelbrot set by Tomoki KAWAHIRA
- ↑ Interior coordinates in the Mandelbrot set
- ↑ Modified Atom Domains
- ↑ Mandelbrot Notebook
- ↑ Wolf Jung's test for precision of drawing parameter rays
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