# Fractals/mandel

Mandel: software for real and complex dynamics by Wolf Jung. Here one can find unofficial docs about it.

# install
## from sources

* download
* extract
* qmake
* make
* ./mandel

# Features

* multiplatform ( the platform-independent )
* written in C++ with source code on GNU GPL licence
* libraries :
** GUI-toolkit Qt
* many algorithms
* many maps
* numerical precision
** floating point limited to double
** integer :

## angle

Program mandel uses own notation for binary fractions:

## maps or functions

Families of functions (= maps ):
* Mandelbrot set z^2+c ( key Ctrl+0 ). It is default map. Here map degree q = 2
* Multibrot sets z^q+c ( key Ctrl+1 ). Use key q to change map degree $3\leq q\leq 64$
* Branner-Fagella cz(1+z/q)^q ( key Ctrl+2 )
* cubic polynomials
* quartic polynomials
* rational mappings
* Newton mappings
* transcendental
* Non-analytic mappings
These famnilies are defined by classes

### complex quadratic polynomial

The monic and centered form of complex quadratic polynomial :
" The Mandelbrot set is based on the one-parameter family of quadratic polynomials fc(z) = z2 + c. " ( from help )
It can be used for :
* forward iteration $f^{1}(z)$, key f = mode 0
* 2 inverse ( or backward ) iterations ( multivalued with argument adjusted ) :
** 1-st inverse function = key a = $-f^{-1}(z)$
** 2-nd inverse function = key b = $+f^{-1}(z)$
* Paritition of dynamic plane related with kneading sequence
* 1/4
* 1/6 preperiod = 1 and period = 2
* 9/56 preperiod = 3 and period = 3
* 129/16256
See also example c code
$f_{c}^{-1}(z)={\sqrt {\frac {-c}{|c|^{2}}}}*{\sqrt {|c|^{2}-{\bar {c}}*z}}$
It is called by :

## internal address

Internal addres
* the numbers (period) must be increasing by definition

## Algorithms

* drawing
** plane ( drawmode or algoActs)
*** LSM Escape time ( algorithm 2)
*** DEM (algorithm 5)
** curves
** points
*** orbits ( iteration)
**** critical orbit
**** forward and inverse orbit
* computing :
** angles of the wake
Example programs which use code from Mandel
* Mandelbrot set in C++
* Mandelbrot set in C++ and SDL
* ArgPhi
* symbolic dynamics

### Level sets of escape time

Is it possibly, for algorithm 2 = ( level sets of ) escape time , to set parameters ( ? escape radius) such that boundaries of level sets go thru critical point ( and it's preimages ) ?
You can change the escape radius only by recompiling. But you can choose the parameter such that it is on an escape line and on the desired parameter ray, then the critical value will be on an escape line as well.

### potential

Complex potential of Julia set
The green procedure is named after the British mathematician George Green
Simplified function
The equipotential line through the current point, or for a given potential, is drawn with the key g.
Equipotential lines can be defined everywhere, but they will be a figure-eight, when they go through the critcal point z = 0 or through one of its preimages.
External rays through 0 or its preimages branch as well.
Potential :
    $log|\Phi (c)|>0$
See also:
* Potential of Mandelbrot set

### periodic and preperiodic points
#### Feigenbaum tuning

Steps:
* Start with a center of period n
* find an approximation for the center of period 2n and use Newton method to find it precisely The approximation can be done with Feigenbaum rescaling, but this did work only in the 1/2-limb, so I used a composition of the multiplier maps of periods 1 and 2 in the other sublimbs.
Keys: Ctrl+Alt+C
C++ code
See also:
* video: Feigenbaum_scaling
* Mandel demo ( renormalization) page 10
C code

#### find

To find periodic or preperiodic point z use:
* Menu/Points/Find point
* key x
then enter:
* period
* preperiod,period
It is numerical procedure so:
* it finds one and the nearest solution.
* it finds nothing ( on parameter plane it goes to origin c=0) if current point is:
** inside set ( Mandel of filled-in Julia)
** not near good solution
For example on the parameter plane there are 3 period 3 centers. If:
* current point is c=0 then result is c=0 ( nothing found)
* current point is outside Mandelbrot set :
** c = 0.350000000000000 +0.725000000000000 i then result is c = -0.122561166876654 +0.744861766619744 i period = 3 ( good result)
** c = 0.580000000000000 +0.240000000000000 i period = 0 then result is c = 0.000000000000000 +0.000000000000000 i period = 1 ( bad result !!!)
** c = -1.590000000000000 +0.170000000000000 i period = 0 then result is c = -1.754877666246693 +0.000000000000000 i period = 3 ( good result)
Use mode Del-Newton to show basins of attraction of Newton method, adjust period with q
It calls function find with
* input:
** sg ( "positive is parameter plane, negative is dynamic plane.")
** preper = preperiod. ( "the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane." Wolf Jung )
** per = period
* output:
** x = creal(z)
** y = cimag(z)
** returning value ( type int)
Returning value:
* 0 ( good result)
* 2 when (px*px + py*py > 1e100)
* 1 when (px*px + py*py > 1e100) = escaping in preperiodic loop
* -1 when (w < 1e-100)
Example of use:
Output:

### algorithm 9 : zeros of qn(c)

* Parameter plane colored according to sign of 5-th iteration. It uses algorithm 9, order n=5.
* Dynamic plane colored according to sign of 8-th iteration. It uses algorithm 9, order q=8
* dynamical plane q = 5
Description:
* "It is meant to show the location of precritical points, i.e., the preimages of 0. So it shows in four different colors, in which quadrant f^{n-1}(z) is. This means that the real axis is pulled back n times, so it gives an approximation to the checkerboard when the parameter is real." ( Wolf Jung )
* It "allows locating zeroes of the iterated polynomial at a certain period where 4 colours meet."
It can be used for:
* on the dynamic plane
** making parabolic checkerboard ( chessboard) when c is a parbolic parameter
** showing center of components of Filled Julia sets, when c is a center of hyperbolic component and z=0 is a one of attracting periodic orbit
This is radial 4th-decomposition of plane ( compare it with n-th decomposition of LSM/M )
4 colors are used because there are 4 quadrants :
* re(z_n) > 0 and im(z_n) > 0 ( 1-st quadrant )
* re(z_n) < 0 and im(z_n) > 0 ( 2-nd quadrant )
* re(z_n) < 0 and im(z_n) < 0 ( 3-rd quadrant )
* re(z_n) > 0 and im(z_n) < 0 (4-th quadrant ).
"... when the four colors are meeting somewhere, you have a zero of q_n(c), i.e., a center of period dividing n. In addition, the light or dark color shows if c is in M." ( Wolf Jung ) See function quadrantqn from class mndlbrot in mndlbrot.cpp file
The differences from standard loop are :
* there is no bailout test (for example for max value of abs(zn) ). It means that every point has the same number of iterations. For large n overflow is possible. Also one can't use test Iteration==IterationMax
* Iteration limit is relatively small ( start for example IterationMax = 8 )
Compare :
* code in Sage
Here is fragment of c code for computing 8-bit color for point c = cx + cy * i :

### Newton for qn(c)

It can be used with:
* Menu/Draw/Algorithm/Newton for Qn(c)
* key n
Key q lets you change n ( period)
Let:
* q_n(c) be the n-th iterate of z^2 + c with z = 0
* N_n(c) the corresponding Newton method.
Then:
* The zeros of q_n are the centers of period dividing n and at the same time the superattracting fixed points of N_n . There is another fixed point at infinity , which is repelling.
* The critical points of q_n are poles of N_n . In many cases these are located in hyperbolic components of period less than n .
* Each immediate attracting basin of N_n is connected to infinity with at least one channel. Therefore I do not see a relation to the atom domains.
* It seems that each hyperbolic component of period n is contained completely in one attracting basin. I do not have a proof for this.

### Misiurewicz point

"... the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane." ( Wolf Jung )
Finding principal Misiurewicz points of the wake k/r of main cardioid
* go to the parameter plane
* mark the wake by drawing 2 external rays landing on the root point :
** use Menu/Rays/Angles_of_the_wake
** input fraction k/r in lowest turns ( k/r = rotation number = int_angle in turns )
* click outside of the Mandelbrot set but inside a wake
* find point
** Menu/Points/Find point or key x
** enter preperiod,period
* if the wake is to small ( = you don't see a free space between rays ) then zoom in
Compare with:
* How to compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm ?

### critical orbit

If you want to draw critial orbit for more iterations then Mandel has as a standard
* on parameter plane choose internal angle ( key c or Manu/Poins/bifurcate)
* go to dynamic plane
* redraw image to remove critical orbit
* press and keep on pressing keys Ctrl-F. Each key hit shall give 4000 iterations;
Functions :
* in qmnshell.cpp in the function QmnShell::dFinished see : dplane->drawOrbit(f, x, y, 0, 4000);
* in QmnShell::map(QAction *act) = Ctrl-F

### Tiling of parabolic Julia set interior

It works for interior of parabolic Julia sets, only for cases when :
* rotation number is 0 or 1/2 ( key C or MainMenu/Point/Bifurcation ), then c is the root of a satellite component.
* Period of parent component is 1
* map is complex quadratic polynomial ( MainMenu/ Map or key= Ctrl+0 )
It makes parabolic checkerboard

### equipotential

* key G
* Main Menu/ Rays/Equipotewntial
"... to draw equipotential lines. ... Boundary scanning is slow and the Newton method has three problems: finding a starting point, drawing several lines around a disconnected Julia set, and choosing the number of points depending on the chosen subset of the plane. Therefore I am using boundary tracing with starting points on several lines through the image. See QmnPlane::green. Still, sometimes not the whole line is drawn, or not all components are found."
" Hit the key g to draw the equipotential line through the current point z (outside of Kc) or c (outside of M), or to specify the potential level."

### Argument of Phi or External angle

It can be applied with:
* Menu/Draw/Algorithm/Argument of Phi
* key 6
Here is the turn function for class mndlbrot

### External rays

Methods :
* Backwards iteration
* Newton method
* Argument tracing

#### Algorithms

* ray to point
* external ray

##### Lamination

Hit o to draw an orbit portrait, which is shown as a lamination when you give the command from the parameter plane, and Ctrl+o for the precritical lamination.
If you are:
* on the parameter plane then choosing lamination and angle 1/7 draws lamination of the unit disk in the right window
* on the dynamic plane and choose lamination and angle 1/7 draws lamination of the dynamic plane in the right window
Lamination of:
* the unit circle
* the dynamic plane
* parameter plane ( QML)

##### Orbit portrait

Steps
* Main menu/Rays/Orbit portrait
* qmnshell.cpp
* portraitAct

##### Spider algorithm

"The Thurston Algorithm is iterating the critical orbit backwards to compute the parameter c corresponding to a given external angle. To choose the correct preimages, a connecting path between point configurations is used by Chéritat for slow mating. The Spider Algorithm employs rays to infinity. Use s or Ctrl+s to illustrate these algorithms. With d you may move the critical value along some path to define a Dehn twist or a recapture. Special examples are available with Ctrl+d. Hit Return repeatedly to observe the iteration. Use Home or enter 0 to quit." ( from help )

##### Wake

In program mandel:
* from Main Menu / Rays / Angles of the wake
* keys: Ctrl+k
Original code from mandel 5.19
Examples :
* angles of the wake
* subwake

#### Methods
##### Argument tracing
# Colors

* purple = RGB( 255,0,255)= HTML(ff00ff) = points which do not escape and do not fall into fixed attractor ( udecidable). For tiling of parabolic Julia set it can show repelling petals in case of low max number of iteration ( for example 100 )
In most drawmodes, Mandel uses a palette of colors 0 to 15.
In mode 3, a value from 0 to 255 is interpreted as a hue. See line 160 in qmnplane.cpp:
To get grayscale, you may replace it with:
4 colours:
* Cyan / Aqua = (0,255,255), green = ( 0,255,0), red = ( 255,0,0), blue = (0,0,255)

### Change colors

This function can be started
* by main menu File/Change colors
* key F5
User may put 2 numbers separated by a comma
* first number is the color which should change
* second number is the color which should be

# Code
## classes

Classes types :
* base
* derived from base classes
Base classes :
* mndynamics = abstract base class (escape to infinity)
* mndScale : scaling of the parameter plane at Misiurewicz points
* mndAngle : external angles
* mndCombi : kneading sequences or internal addresses
* mndPath : spider with legs or with path

### mndynamics

mndynamics is an abstract base class (escape to infinity)
Classes derived from mndynamics
* with critical relations and escape to infinity :
** mndlbrot : z^2 + c , Mandelbrot set
** mndmulti : z^q + c , Multibrot set
** mndbfpoly : cz(1 + z/q)^q (Branner-Fagella)
** mndcubic : various cubic polynomial families
** mndquartic : various quartic polynomial families
** mndmating : various quadratic rational families (mating), some of these check only for periodicity, there is no escape.
** mndmenage : c(z + 0.5/z^2) , rotationally symmetric (Henriksen)
** mndsingpert: z^2 + c/z^2 , singular perturbation of z^2
** mndexpo : various exponential families
** mndtrigo : various trigonometric families
** mndsurge : piecewise polynomial, modeling quasi-conformal surgery
** mndtricorn : (z*)^q + c , anti-analytic, Tricorn or Multicorn
** mndrealcubic: cubic poly., real-analytic para., 2 indep. cr.pt.
** mndifs : iterated function system for Liouville roots
* mndsiegel = abstract derived class (persistent Siegel, two orbits, one may escape)
** mndcubicsiegel : cubic polynomials (Zakeri-Buff-Henriksen & 2-per), derived from mndsiegel
** mndquartsiegel : quartic polynomials (additional critical relation)
** mndexposiegel : exponential
** mndtrigosiegel : trigonometric
** mndmatesiegel : quadratic rational
** mndnewtonsiegel: quartic Newton mappings with Siegel cycle
** mndnewtonpara : cubic Newton maps with 2 roots and 1 parabolic basin
** mndnewtonqara : cubic Newton maps with 1 root and 2 parabolic basins
** mndnewtonrara : cubic Newton maps with 0 roots and 3 parabolic basins
* mndnewton = Abstract derived class (cubic or quartic Newton mappings, with critical relations)
** mndcubicnewton : Newton for cubic polynomials, derived from mndnewton
** mndquarticnewton : Newton for quartic polynomials, derived from mndnewton
* mndherman = cubic rational with Herman ring. Derived from mndynamics (special case with escape to 0 or infinity)
* mndhenon : Henon mappingDerived from mndynamics (special case, escape to -infinity, no cr.pt.)
* mndlambda : Henon mapping. Derived from mndynamics (special case, escape to squares, no cr.pt.)
Most classes describe a one-parameter family of complex functions :
$f_{c}(z)$
where :

### mndAngle

The class mndAngle represents an angle (in turns) by a fraction of unsigned long long integers , which are <= 2^64 - 1 . They are normalized to the denominator 2^K * (2^P - 1) when possible, i.e., K + P <= 64 . Otherwise P = 0 . Many applications only need the static functions.
* normalize returns the period.
* twice doubles the angle modulo 1.
* conjugate computes the conjugate angle, returns period of the char.pt.
* wake computes the angles of a limb.
* radians gives the angle in radians.
* lambda computes e^{core entropy h} for dyadic angles.
* realSpider checks if the root is real and computes the center c .
* truncatedTuning truncates (u1, u2)*w to the next n-periodic angle.

### mndCombi

The class mndCombi represents a *-periodic kneading sequence with period P <= 64 and the corresponding internal address. They are stored each in the bits of an unsigned long long integer . QString conversion is done by QmnCombiDialog . In the address, the first bit (2^63) is period 64 and the last bit (2^0) is period 1. Example:
since the upper kneading sequence is AABBAAA
The kneading sequence is always the upper one: setKneading ignores the last entry, and the last entry from getKneading shall be printed as a '*' . The set-functions compute the internal address from the kneading sequence or vice versa.
* fromAngle computes the combinatorics of a periodic angle.
* renorm gives the lowest period of simple renormalization.
* count returns the number of realizations, assuming admissibility.
* failsBS returns the first failure of the Bruin-Schleicher admissibility condition, see http://arXiv.org/abs/0801.4662 . 0 means that there is no evil orbit and the combinatorial data corresponds to a planar Hubbard tree with an orientation-preserving mapping, thus to a quadratic polynomial realizing these combinatorics. The corresponding angles are computed in toAngles by backwards iteration for each period, always choosing the 1/q-limb when q > 2 . The simple algorithm fails for the first time at 1-2-6-7-13-14 , so the wake is subdivided by calling backAngles recursively. See the explanation in mndcombi.cpp .

## Global variables

* drawmode uint. Defined in mndynamo.h
* maxiter uint. Defined in mndynamo.h
* a
* b
* A double. Defined in mndynamo.h
* B double. Defined in mndynamo
* sign int. Defined in mndynamo.h . "positive is parameter plane, negative is dynamic plane."
* subtype int. Defined in mndynamo.h { subtype = subtype0; A = 0; B = 0; bailout = 16.0; temp = new double[5]; }
* signtype = +- subtype : its modulus remembers the chosen subtype, and its sign denotes the current plane; pass its value for sign.
* Period uint. Defined in mndynamo.h
* *temp double. Defined in mndynamo.h
* bailout double. Defined in mndynamo.h
* k preperiod ( of critical value not critical point). It means that c= -2 has k=1 and p=1 so it is M_1,1 in this notation not M_2,1 This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point,
and the same angles are found in the parameter plane.
* p period
* plane parameters
** mdouble hmid = horizontal mid = plane center x
** vmid = vertical mid = plane center y
** pw = plane width
** ph = plane height
** uint drawmode;
** int imid ( horizontal pixel coordinate of center
** int kmid ( vertical coorinate of the center
** imax = horizontal maximal pixel coordinate
** kmax = vertical maximal pixel coordinate
** mode
** alpha;

## viewport

Wolf Jung uses center and width for describing rectangle viewport of complex plane:
* center of the rectangle : xmid + i ymid
* the midpoint of the right side are given by : rewidth + i imwidth
So initial values of corners are :
* parameter plane
* dynamic plane : -2 <= Re(z) <=2, -2 <= Im(z) <=2

## normalize

compile
run:
result:

# Images

There is a category on commons : Category:Fractals created with Mandel

# opening images

One can open png image in Mandel and check it.
Image must be :
* png type
* square ( width = height )
* size must be typical ( 2000 not 2001 pixels)
* world coordinate should be :
** parameter plane ( to do )
** dynamic plane : -2 <= Re(z) <=2, -2 <= Im(z) <=2
Steps :
* open Mandel
* adjust image size ( Menu/File/Resize )
* if you open image of dynamic plane change coordinate on parameter plane ( Menu/Points/Coordinates )
* open image ( Menu/File/Load.png )
Then you can :
* draw : orbit, ...
* save
You can't:
* zoom
* redraw

# Demos

See :
* Main Menu/Help/
* src code : qmndemos.cpp

## Demo 2 : periodic points and bifurcations
### page 7

Finding c parameters (on the boundary of main cardioid) with Siegel Disk :
* start with internal angle t = Golden ratio conjugate
* compute c from t
* go to the next t ( here y is increased by 0.01) with Go key
Here is the original code :

### page 10

elephants demo: there is just a rotation and rescaling on a small region instead of the non-linear transformation, like in unrolling
The method of parabolic renormalization was used to show that the boundary of the Mandelbrot set has Hausdorff dimension 2, and recently to construct a Julia set of positive area. For another application of parabolic implosion, recall that the 1/r-limb is attached to the main cardioid at a parameter c, such that the fixed point $\alpha _{c}$ is parabolic with combinatorial rotation number 1/r. These limbs converge to the root c = 0.25 of the main cardioid. Pierre Lavaurs has studied these limbs using the Fatou coordinates and the related horn map of Ecalle cylinders. It is conjectured that the limbs converge to a limiting "elephant" under a suitable rescaling. The left image shows the 1/2-limb rotated by 90$^o$. All limbs will be shown with increasing magnification and appropriate rotation, such that they point upwards from the boundary of the main cardioid. (Which is rotated such that it is horizontal at the root of the satellite component.) The right image shows the filled Julia set for the corresponding center of period r. Hit Return or push the Go-button to increase the period r of the limbs. This kind of scaling is not available in the main program. See page 9 of Chapter 5 for another scaling phenomenon at parabolic parameters.))
Here:
* rotation number 1/r is t = 1/n
* n is increasing by 1 at every go button click
* setPlane(mdouble xmid, mdouble ymid, mdouble rewidth, mdouble imwidth, bool nowayBack)

## Demo 3: external rays
### page 1

Parametr plane : Mandelbrot set with external rays for angles ${\frac {j}{16}}$ where $j\in [0,16)$

## Demo 5: renormalization
# Videos

videos by Wolf Jung produced with FFmpeg, combining images made with Mandel

## Mandelbrot

* Bifurcation of Julia sets
* Similarity and self-similarity of the Mandelbrot set
* Visualization of the Thurston algorithm

### Approaching the airplane root with Fatou-Lavaurs translation (1/σ ~ time)

The variable sigma is related to the multiplier of the approximately parabolic point, and 1/sigma is going to infinity when approaching the root.
1/sigma is also approximately the number of iterations from inside the approximate basin to the outside. In one video the time is proportional to 1/sigma, which means that the spiral is turning at constant speed. In the other video, the parameter c is moving at constant speed, which makes 1/sigma go faster and faster, such that it reaches infinity in a finite time.

### Bifurcation of preperiodic points and rays at c = γM(5/12)

Video
* Bifurcation of Julia sets : Bifurcation of preperiodic points and rays at $c=\gamma _{M}(5/12)$ ( c is parameterized by the angle θ of the external ray landing at the critical value)
The parameter c is moving to the left on the real axis. As it crosses the Misiurewicz parameter with angles 5/12 and 7/12 , the critical value has the same angles and the critical point has the angles 5/24 , 7/24 , 17/24 , 19/24 ; these rays are landing in two pairs in a different pattern for parameters right or left of this one.
Note that bifurcations happen all the time: parts of the Julia sets move down and up toward 0 and then to the left and right.

### Feigenbaum scaling

Video: Mandelbrot set and quadratic polynomials\ Slideshow of Feigenbaum scaling
Feigenbaum rescaling in the 1/2-limb
Description from the Madel demo 5 page 10:
The sequence of repeated period doubling leads to the infinitely renormalizable Feigenbaum parameter cF ≈ -1.401155189, the nested small Mandelbrot sets are scaled asymptotically by the first Feigenbaum constant $\delta \approx 4.669201609$, and the small Julia sets around z = 0 (not z = c) by the second Feigenbaum constant $\alpha \approx 2.502907875$ . The decorations in the parameter plane are becoming more hairy, and converging to a dense subset of the plane. This conjecture by John Milnor was proved by Mikhail Lyubich, and Curt McMullen has shown convergence of blow-ups of the Feigenbaum Julia set.
Hit the Return key or push the Go-button a few times to rescale the parameter plane by δF around c = cF and the dynamic plane by αF around z = 0. In the main program, Feigenbaum scaling is available in the parameter plane with the key Ctrl+Alt+f. You will have to increase the number of iterations with n. The corresponding tuning of center parameters is obtained with the key Ctrl+Alt+c. Renormalization with the key r works best in the primitve case.
See also:
* Feigenbaum_tuning

## Symmetric
# Wishlist

* showing window (viewport) parameters : center and radius
* Main Menu/Rays. What about changing it to Main Menu/Curves ? It is uded to draw rays and equipotential curves
* saving description of the image, maybe inside png image or in different txt file
* making one image from various algorithms ( like adding boundary made by DEM or IIM to other images), layers of the image
* scripting language form making complex images/operations
* OpenMP , GPU
* more description in the code and about algorithms
* change the silent error checking to explicit information
* options to do long operations without keeping the key preseed. For example try to draw critical orbit for bifurcation from period 1 using t = 34/89. It draws only some points.
* dictionary
** A Qt Linguist phrase book = qph file extension. These are human-readable XML files containing standard phrases and their translations. These files are created and updated by Qt Linguist and may be used by any number of projects and applications.
** Qt Linguist Manual
* not use of finite binary expansion but infinite version, because it describes better the dynamics. For example, now is: "The angle 1/4 or 01 has preperiod = 2 and period = 1.". Better version is :"The angle 1/4 or 00p1 has preperiod = 2 and period = 1."
* description how to add a new fractal algorithm or coloring method
* internal rays for parameter and dynamic plane

## Bugs

* "after loading certain png-images from another source, things like changing colors and drawing do not work any longer ..."

# Translation

Example A Qt Linguist phrase book (qph file). It is a human-readable XML file containing standard phrases and their translations. This file is created and updated by Qt Linguist and may be used by any number of projects and applications.

# Papers with images made with Mandel

* Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions by Mary E. Wilkerson
* THE DYNAMICAL ANDRE-OORT CONJECTURE: UNICRITICAL POLYNOMIALS by D. GHIOCA, H. KRIEGER, K. NGUYEN, AND H. YE

## University

* Montana State University, Department of Mathematical Sciences :class 2016-fall-m597/index.html

# References

* ↑ Mandel by W Jung
* ↑ Mandelbrot set in C++
* ↑ Mandelbrot set in C++ and SDL
* ↑ ArgPhi
* ↑ symbolic dynamics
* ↑ periodicity_scan_revisited by Claude
* ↑ Multiplatform cpp program Mandel by Wolf Jung
* ↑ Formation of Escape-Time Fractals By Christopher Olah
* ↑ rapidtables: RGB Color Codes Chart
* ↑ wikipedia : Golden_ratio_conjugate
* ↑ Parabolic Renormalization by Hiroyuki Inou and Mitsuhiro Shishikura
* ↑ The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets by Mitsuhiro Shishikura
* ↑ Quadratic Julia Sets with Positive Area by Xavier Buff, Arnaud Cheritat
* ↑ Douady, Adrien : Does a Julia set depend continuously on the polynomial?
* ↑ Mandelbrot set and quadratic polynomials - videos by Wolf Jung