Fractals/Iterations in the complex plane/Parameter plane
- 1 Parts of parameter plane
- 2 How to choose a point from parameter plane ?
- 3 How to move on parameter plane ?
- 4 Plane types
- 5 point c description
- 6 Point Types
- 7 Algorithms
- 8 Models
- 9 See also
- 10 Rerferences
Parts of parameter plane
- with respect to the Mandelbrot set
- with respect to the wakes
- inside a p/q wake
- outside any wake (???)
Parts of Mandelbrot set according to M Romera et al.:
- main cardioid
- q/p family (= q/p limb)
- period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
Not that her q/p not p/q notation is used
How to choose a point from parameter plane ?
- clicking on parameter points and see what you have ( random choose)
- computing a point with known properties.
How to move on parameter plane ?
- type of the move
- discrete = using sequence of points
- type of the curve
- along radial curves :
- external ray
- parabolic point = root point = landing point of above external ray
- internal ray which also end at the parabolic point
- along circular curves :
- boundaries of hyperbolic components
- internal circular curves
- along radial curves :
- poincare_half-plane_metric_for_zoom_animation by Claude Heiland-Allen
- youtube: Julia sets as C pans over the Mandelbrot set by captzimmo
- youtube : Julia sets about the main cardioid x 1.1 with Mandelbrot set by Thomas Fallon
- youtube: Julia Sets Relative to the Mandelbrot Set by Gary Welz
- you tube : Julia Sets of the Quadratic by Gary Welz
- youtube : Julia set morph around the cardioid / central bulb by blimeyspod
- youtube : Julia set morph / fractal animation - Beyond the Cardioid Perimiter by blimeyspod
- youtube: Julia set morph / fractal animation - Beyond a 2nd Order Bulb by blimeyspod
- Fractals: A tour of Julia Sets by corsec
- shadertoy : Julia - Distance by iq
- Evolving Julia Marco_Gilardi
// glsl code by iq from https://www.shadertoy.com/view/Mss3R8 float ltime = 0.5-0.5*cos(time*0.12); vec2 c = vec2( -0.745, 0.186 ) - 0.045*zoom*(1.0-ltime);
// glsl code by xylifyx from https://www.shadertoy.com/view/XssXDr vec2 c = vec2( 0.37+cos(iTime*1.23462673423)*0.04, sin(iTime*1.43472384234)*0.10+0.50);
// by Marco Gilardi // https://www.shadertoy.com/view/MllGzB vec2 c = vec2(-0.754, 0.05*(abs(cos(0.1*iTime))+0.8));
The phase space of a quadratic map is called its parameter plane. Here:
- is constant
- is variable
There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.
The parameter plane consists of :
- The Mandelbrot set
- The bifurcation locus = boundary of Mandelbrot set
- Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set 
- plain ( c-plane )
- inverted c-plane = 1/c plane
- lambda plane
- exponential plane ( map) 
- unrolled plain (flatten' the cardiod = unroll )  = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." ( Figure 4.22 on pages 204-205 of The Science Of Fractal Images)
- transformations 
- log : "To illustrate the complexity of the boundary of the Mandelbrot set, Figure 8 renders the image of dM under the transformation log(z - c) for a certain c e dM ? Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to
zooming in towards the point c. (Namely, c = -0.39054087... - 0.58678790i... the point on the boundary of the main cardioid corresponding to the golden mean Siegel disk.).Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. "
- "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting side-scrolling plane ^^)"
- " There are no program that can render this fractal on log(Z) plane. But you can make it in Ultra Fractal or in similar software with programmable distributive. Formula is:C = exp(D), for D - is your zoomable coordinates" SeryZone X
- Mandelbrot set projected on a shrinking Riemann-sphere by Arneauxtje
- Mandelbrot's Elephant Valley (Short Version) Timothy Chase
- Mandelbrot Buds and Branches Timothy Chase Timothy Chase
- Mandelbrot Zoom on a Sphere video by craftvid : "This a 300-trillion time zoom-in on the Mandelbrot set. The images are set on a Spherical "mobius" projection, meant to be wrapped onto a spherical surface. The image zooms in on the front center of the Sphere, while fading away on the back of the sphere."
point c description
- c value
- Cartesion description
- real part
- imaginary part
- polar description:
- (external or internal ) angle
- ( external or internal) radius
- Cartesion description
point =pixel of parameter plane = c parameter
Criteria for classification of parameter plane points :
- arithmetic properties of internal angle (rotational number) or external angle
- in case of exterior point:
- type of angle : rational, irrational, ....
- preperiod and period of angle under doubling map
- in case of boundary point :
- preperiod and period of external angle under doubling map
- preperiod and period of internal angle under doubling map
- in case of exterior point:
- set properties ( relation with the Mandelbrot set and wakes)
- inside wake, subwake
- outside all the wakes, belonging to a parameter ray landing at a Siegel or Cremer parameter,
- geometric properities
- number of of external rays that land on the boundary point : tips ( 1 ray), biaccesible, triaccesible, ....
- position of critical point with relation to the Julia set
There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.
- exterior of Mandelbrot set
- Mandelbrot set
- boundary of Mandelbrot set
- interior of Mandelbrot set
- other internal points ( points of internal rays )
partial classification of boundary points
- Boundaries of primitive and satellite hyperbolic components:
- Boundary of M without boundaries of hyperbolic components:
- non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of non-hyperbolic components would be infinitely renormalizable as well.
Here "other" has not a complete description. The polynomial may have a locally connected Julia set or not, the critcal point may be rcurrent or not, the number of branches at branch points may be bounded or not ...
- points ( coordinate)
- compute c from multiplier , period and center (inverse multiplier map)
- compute multiplier from c ( multiplier map)
- Escape time
- Discrete Velocity of non-attracting Basins and Petals by Chris King
- atom domains
- average distance between random points
- combinatorial : tuning
- codegolf SE question: mandelbrot-image-in-every-language
- codegolf SE question: generate-a-mandelbrot-fractal ( ASCI)
- rosetta code : Mandelbrot_set
- stackoverflow question: code-golf-the-mandelbrot-set
- stackoverflow questions tagged mandelbrot
- Types of Julia sets
- Exact Coordinates by Robert P. Munafo, 2003 Sep 22.
- Rational Coordinates by Robert P. Munafo, 2012 Apr 22.
- SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
- interesting c points by Owen Maresh
- Visual Guide To Patterns In The Mandelbrot Set by Miqel
- fractalforums : deep-zooming-to-interesting-areas
- Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
- Alternate Parameter Planes by David E. Joyce
- exponentialmap by Robert Munafo
- mu-ency : exponential map by R Munafo
- Exponential mapping and OpenMP by Claude Heiland-Allen
- Linas Vepstas : Self Similar?
- the flattened cardioid of a Mandelbrot by Tom Rathborne
- Stretching cusps by Claude Heiland-Allen
- Twisted Mandelbrot Sets by Eric C. Hill
- FRONTIERS IN COMPLEX DYNAMICS by CURTIS T. MCMULLEN
- youtube video : Mandelbrot deep zoom to 2^142 or 5.5*10^42. Log(z) by SeryZone X
- stackexchange : classification-of-points-in-the-mandelbrot-set
- fractalforums : tricky-mandelbrot-problem