Mandelbrot set topology
How to read informations from images ?
How to describe hyperbolic component of Mandelbrot set ? ( escpecially island )
- period of main pseudocardioid
- center of main pseudocardioid
- window ( radius and center) of parameter plane
- angles of external rays that land on cusp of the pseudocardiod
- size of main pseudocardioid
- orientation
- distortion
For each island:
- find period
- find center of component ( nucleus) = pseudocardioid_nucleus
- find shape ( psudocircle or pseudocardioid)
- find cusp of pseudocardioid = pseudocardioid_cusp
- find root point between 2 main componnents = pseudocardioid_root_half
- find center of second component ( nucleus) = second_bulb_nucleus
- compute distortion of the island
Here are few examples of islands.
- first image ( LastIteration = 108): shows interior points
- second image (Period = 107): Only main (pseudo)cardioid of period p and main component of period 2p is drawn
- for more description see the output the new txt file m.txt and output the old txt file m.txt
#define kMax 21 // number of examples, see plane_examples
//plane : plane_center_x, plane_center_y, plane_radius, period
double plane_examples[kMax][4] = {
{-0.4, +0.0, 0.8, 1}, // k = 0
{+0.29254, -0.01497, 0.00015, 16},
{-1.763, +0.0, 0.016, 3},
{-0.15842, +1.03335, 0.008, 4},
{+0.358431, +0.643507, 0.005, 5},
{+0.442990, +0.374, 0.003, 6},
{+0.432259, +0.2275, 0.002, 7},
{+0.404879, +0.146216, 0.0015, 8},
{+0.378631, +0.098841, 0.001, 9},
{+0.356854, +0.069659, 0.0007, 10},
{+0.339454, +0.050823, 0.0005, 11}, // k = 10
{+0.325631, +0.038164, 0.0005, 12},
{+0.260270, +0.00167, 0.00002, 32 },
{+0.2524945, +0.0001973, 0.0000025, 64},
{+0.25061329, +0.00002399, 0.0000003, 128},
{+0.250151979, +0.000002959, 0.000000036, 256},
{+0.250037823, +0.0000003673, 0.000000004, 512},
{+0.2500094342, +0.000000045694786520646, 0.0000000005, 1024}, //
{+0.25000235583, +5.701985912706845832e-09, 0.00000000007, 2048}, //
{+0.2500005886144, +0.0000000007122, 0.0000000000083, 4096}, //
{+2.500001471109009610e-01, +8.897814201389663379e-11, 1.1e-12, 8192} // long time
};peroid = 1 = whole Mandelbrot set ( using LastIterarion method with interior detection)
Here unknown pixels are marked by red big pixels. They are boundary points.
With important points: pseudocardioid_nucleus, pseudocardioid_cusp, pseudocardioid_root_half, second_bulb_nucleus
Here distortion angle = 0.0000000000000070 ( it is = 0, the number 7 on the penultimate position is numerical error)
Here are 2 main components : period 16 pseudocardioid and period 2*16 pseudocircle. Note that the image differs from that computed with LastIteration method ( above). The components are smaller and do not touch in root points
Here is the comparison using ImageMagic ( see bash file)
Here is a comparison between boundaries of LastIterarion method and Period method. The difference is big. It was caused by to low value of iMax = 2000; ( see function GivePeriodByIteration line 676 )
After increasing iMax (10 000 and 50 000) ( and orbit[OrbitLength]; // length(orbit) = iMax + 1 ) new image looks better
With
- important points: pseudocardioid_nucleus, pseudocardioid_cusp, pseudocardioid_root_half, second_bulb_nucleus
- 2 axis, one thru pseudocardioid_nucleus and pseudocardioid_cusp, second thru pseudocardioid_nucleus and pseudocardioid_root_half
- distorsion angle is denoted with arc
Here distortion angle = 10.2217850853740675 degrees = 0.0283938474593724 turns = 585.6651445133592233 radians
Period 3 island looks like whole Mandelbrot set
With important points, axes and distortion angle
Here distortion angle = 0.0000000000000066 degrees = 0.0000000000000000 turns = 0.0000000000003803 radians ( it is = 0, the numbers on the penultimate and last positions are numerical errors)
Here distortion angle = 3.1678661702170463 degrees = 0.0087996282506029 turns =
Period 1 nucleus = +0.0000000000000000 +0.0000000000000000 distortion angle = 0.0000000000000070 degrees = 0.0000000000000000 turns = 0.0000000000004020 radians island sizes: HalfRoot-Cusp = 1.0000000000000000
Period 3 nucleus = -1.7548776662466927 +0.0000000000000000 distortion angle = 0.0000000000000066 degrees = 0.0000000000000000 turns = 0.0000000000000001 radians island sizes: HalfRoot-Cusp = 0.0185291524676852
Period 4 nucleus = -0.1565201668337551 +1.0322471089228318 distortion angle = 3.1678661702170463 degrees = 0.0087996282506029 turns = 0.0552896949298424 radians island sizes: HalfRoot-Cusp = 0.0082966360777899
Period 5 nucleus = +0.3592592247580074 +0.6425137371385423 distortion angle = 4.9405365983730638 degrees = 0.0137237127732585 turns = 0.0862286304522303 radians island sizes: HalfRoot-Cusp = 0.0047414665489602
Period 6 nucleus = +0.4433256333996235 +0.3729624166628465 distortion angle = 6.1617377889422356 degrees = 0.0171159383026173 turns = 0.1075426120557851 radians island sizes: HalfRoot-Cusp = 0.0029786183826876
Period 7 nucleus = +0.4323761926419945 +0.2267599044353486 distortion angle = 7.0832512125542779 degrees = 0.0196756978126508 turns = 0.1236260553982713 radians island sizes: HalfRoot-Cusp = 0.0019783278097542
Period 8 nucleus = +0.4048996651751222 +0.1458203637665893 distortion angle = 7.8032252414405772 degrees = 0.0216756256706683 turns = 0.1361919727305068 radians island sizes: HalfRoot-Cusp = 0.0013689191712135
Period 9 nucleus = +0.3786081241329295 +0.0985580111183820 distortion angle = 8.3732692725677360 degrees = 0.0232590813126882 turns = 0.1461411179544790 radians island sizes: HalfRoot-Cusp = 0.0009793927717717
Period 10 nucleus = +0.3568172484923120 +0.0694528654668303 distortion angle = 8.8278041145438362 degrees = 0.0245216780959551 turns = 0.1540742475115654 radians island sizes: HalfRoot-Cusp = 0.0007208567191598
Period 11 nucleus = +0.3394108199955985 +0.0506682851626426 distortion angle = 9.1924920908597691 degrees = 0.0255347002523882 turns = 0.1604392534403695 radians island sizes: HalfRoot-Cusp = 0.0005437231668462
Period 12 nucleus = +0.3255895095506603 +0.0380478809347557 distortion angle = 9.4870876692376047 degrees = 0.0263530213034378 turns = 0.1655809162446018 radians island sizes: HalfRoot-Cusp = 0.0004189608583239
Period 16 nucleus = +0.2925037532341934 -0.0149250689983438 distortion angle = 10.2217850853740675 degrees = 0.0283938474593724 turns = 0.1784038051613850 radians island sizes: HalfRoot-Cusp = 0.0001743936505473
Period 32 nucleus = +0.2602618199285007 +0.0016677913209265 distortion angle = 10.9845361225518747 degrees = 0.0305126003404219 turns = 0.1917163221324199 radians island sizes: HalfRoot-Cusp = 0.0000205403237492
Period 64 nucleus = +0.2524934589775105 +0.0001971526796077 distortion angle = 11.1719775049860850 degrees = 0.0310332708471836 turns = 0.1949877914102099 radians island sizes: HalfRoot-Cusp = 0.0000024544115792
Period 128 nucleus = +0.2506132008410751 +0.0000239693264251 distortion angle = 11.2171220053156571 degrees = 0.0311586722369879 turns = 0.1957757115800861 radians island sizes: HalfRoot-Cusp = 0.0000002990710383
Period 256 nucleus = +0.2501519680089798 +0.0000029549623259 distortion angle = 11.2283621916479994 degrees = 0.0311898949768000 turns = 0.1959718898401125 radians island sizes: HalfRoot-Cusp = 0.0000000368869447
Period 512 nucleus = +0.2500378219137853 +0.0000003668242053 distortion angle = 11.2312909069159588 degrees = 0.0311980302969888 turns = 0.1960230055643890 radians island sizes: HalfRoot-Cusp = 0.0000000045794813
Period 1024 nucleus = +0.2500094340031834 +0.0000000456947865 distortion angle = 11.2320980460567430 degrees = 0.0312002723501576 turns = 0.1960370927999167 radians island sizes: HalfRoot-Cusp = 0.0000000005704648
Period 2048 nucleus = +0.2500023558032561 +0.0000000057019859 distortion angle = 11.2323756629140412 degrees = 0.0312010435080946 turns = 0.1960419381281353 radians island sizes: HalfRoot-Cusp = 0.0000000000711847
Period 4096 nucleus = +0.2500005886128087 +0.0000000007121327 distortion angle = 11.2321190246282825 degrees = 0.0312003306239675 turns = 0.1960374589450624 radians island sizes: HalfRoot-Cusp = 0.0000000000088904
Period 8192 nucleus = +0.2500001471109010 +0.0000000000889781 distortion angle = 11.2329396859086224 degrees = 0.0312026102386351 turns = 0.1960517821864472 radians island sizes: HalfRoot-Cusp = 0.0000000000011108
- m.c - one file c program
- m.sh - bash file used to compile, run the m.c and image manipulations using ImageMagic console programs
- Makefile file for make which runs all
- output txt file m.txt - to read about the results
- old code
- images are in the png directory
- math SE question: function-to-calculate-the-period-of-a-mandelbrot-set-point
- atom domains for period (= period domains) of Mandelbrot set hyperbolic components
- period of Mandelbrot set hyperbolic componnets
- multiplier map
- interior detection
- The Lyapunov exponent
- Interior distance estimation - DEM
- rendering-mandelbrot-set-mu-units: How to extract a mu-unit of a given period using distance estimation colouring, pruning off the outer filaments?
- mu-atom - mu-atom mapping: period p hyperbolic components of the Mandelbrot set can each be mapped conformally to the unit disc, by the derivative d/dz of the periodic limit cycle where f_c^p(z_0) = z_0.
- period-doubling-in-minibrots
- shape estimation
Description
- all calculations are in a point-sampling, numerical ( double precision), non-rounding controlled manner
- execution time of DrawImage for LastIteration = 31 for Period = 525 MultiplierMap = 5 seconds for example 2 ( period 3)
- distortion
- computational topology
- digital topology
- 2D
- raster graphic
- computer graphic
- binary image
- Connected-component labeling
create a new repository on the command line
echo "# " >> README.md
git init
git add README.md
git commit -m "first commit"
git branch -M main
git remote add origin git@github.com:adammaj1/Mandelbrot-set-topology.git
git push -u origin main
Change:
- in general settings
- add Social Preview Image ( Images should be at least 640×320px (1280×640px for best display))
- in repository details ( near About) add
- description
- website
- Topics (separate with spaces)
Local repository
~/Dokumenty/Mandelbrot-set-topology/
mkdir png
git add *.png
git mv *.png ./png
git commit -m "move"
git push -u origin main
then link the images:
to overwrite
git mv -f
the other files
git mv ./src/*.c ./src/modified/bash/
git mv ./src/*.sh ./src/modified/bash/
git mv ./src/Makefile ./src/modified/bash/