Mandelbrot-set-topology

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

To do

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

Example islands

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 
	 
};

Period 1

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)

Period 16 island

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

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)

Period 4

Here distortion angle = 3.1678661702170463 degrees = 0.0087996282506029 turns =

Period 5

Period 6

Period 7

Period 8

Period 9

Period 10

Period 11

Period 12

Period 32

Period 64

Period 128

Period 256

Period 512

Period 1024

Period 2048

Period 4096

Period 8192

Results

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

Files

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

Algorithms

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)

Key words

distortion
computational topology
digital topology
2D
raster graphic
computer graphic
binary image
Connected-component labeling

Git

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

Repo

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/

Subdirectory

mkdir png
git add *.png
git mv  *.png ./png
git commit -m "move"
git push -u origin main

then link the images:

![](./png/n.png "description")

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/

Github

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README.md

README.md

Mandelbrot-set-topology

To do

Example islands

Period 1

Period 16 island

Period 3

Period 4

Period 5

Period 6

Period 7

Period 8

Period 9

Period 10

Period 11

Period 12

Period 32

Period 64

Period 128

Period 256

Period 512

Period 1024

Period 2048

Period 4096

Period 8192

Results

Files

Algorithms

Key words

See also

Git

Repo

Subdirectory

Github

Files

README.md

Latest commit

History

README.md

File metadata and controls

Mandelbrot-set-topology

To do

Example islands

Period 1

Period 16 island

Period 3

Period 4

Period 5

Period 6

Period 7

Period 8

Period 9

Period 10

Period 11

Period 12

Period 32

Period 64

Period 128

Period 256

Period 512

Period 1024

Period 2048

Period 4096

Period 8192

Results

Files

Algorithms

Key words

See also

Git

Repo

Subdirectory

Github