Parts of parameter plane

• with respect to the Mandelbrot set
  • exterior
  • boundary
  • interior
• with respect to the wakes
  • inside a p/q wake
  • outside any wake (???)

Parts of Mandelbrot set according to M Romera et al.:^[1]

• • main cardioid
  • q/p family (= q/p limb)
    • period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
    • shrub

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.
  • For (parabolic point ) choose hyperbolic component ( period , number) and internal angle (= rotation number) then compute c parameter.
  • Misiurewicz points ^[2]
  • see also known regins in ^[3]
  • morphing
  • intresting areas^[4]
• zoom
  • autozoom

How to move on parameter plane ?

• 
  Values of C for each frame evaluates by equation: C=r*cos(a)+i*r*sin(a), where: a=(0..2*Pi), r=0.7885. Thus, parameter С outlines circle with a radius r=0.7885 and a center at origin of the complex plane.

 // 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));

Plane types

The phase space of a quadratic map is called its parameter plane. Here:

• ${\displaystyle z0=z_{cr}\,}$ is constant
• ${\displaystyle c\,}$ 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 ^[5]

There are many different types of the parameter plane^{[6] [7]}

• plain ( c-plane )
• inverted c-plane = 1/c plane
  • Mandelbrot set inversion revisited by Arneauxtje
• lambda plane
• exponential plane ( map) ^[8][9]
• unrolled plain (flatten' the cardiod = unroll ) ^[10][11] = "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)^[12]
• transformations ^[13]
• 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. "^[14]

• • "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 ^^)"^[15]
  • " 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

• 
  c-plane
• 
  inverted c plane = 1/c plane
• 
  ${\displaystyle {\frac {c^{2}}{c^{4}-0.25}}}$ plane
• 
  Unrolled main cardioid of Mandelbrot set for periods 7-13

• 
  lambda plane

point c description

• c value
  • Cartesion description
    • real part
    • imaginary part
  • polar description:
    • (external or internal ) angle
    • ( external or internal) radius

Point Types

point =pixel of parameter plane = c parameter

Algorithms

• points ( coordinate)
  • compute c from multiplier , period and center (inverse multiplier map)
  • compute multiplier from c ( multiplier map)
• dynamics
  • Escape time
  • DEM/M
  • Discrete Velocity of non-attracting Basins and Petals by Chris King
  • atom domains
  • average distance between random points^[17]
• zoom
  • Perturbation_theory
• combinatorial : tuning
  • wake
  • principle Misiurewicz points for the wake k/r of main cardioid
  • subwake, tuning and internal address
  • roots, islands and Douady tuning
  • Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection

Examples:

• 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
• benchmark

Models

• 
  Topological model of Mandelbrot set( reflects the structure of the object ). Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)
• 
  Shrub model of Mandelbrot set
• 
  Topological model of Mandelbrot set using Lavaurs algorithm up to period 12

See also

• Types of Julia sets
• Exact Coordinates by  Robert P. Munafo, 2003 Sep 22.
• Rational Coordinates   by Robert P. Munafo, 2012 Apr 22.

Rerferences

1. ↑ SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
2. ↑ interesting c points by Owen Maresh
3. ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
4. ↑ fractalforums : deep-zooming-to-interesting-areas
5. ↑ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
6. ↑ Alternate Parameter Planes by David E. Joyce
7. ↑ exponentialmap by Robert Munafo
8. ↑ mu-ency : exponential map by R Munafo
9. ↑ Exponential mapping and OpenMP by Claude Heiland-Allen
10. ↑ Linas Vepstas : Self Similar?
11. ↑ the flattened cardioid of a Mandelbrot by Tom Rathborne
12. ↑ Stretching cusps by Claude Heiland-Allen
13. ↑ Twisted Mandelbrot Sets by Eric C. Hill
14. ↑ FRONTIERS IN COMPLEX DYNAMICS by CURTIS T. MCMULLEN
15. ↑ youtube video : Mandelbrot deep zoom to 2^142 or 5.5*10^42. Log(z) by SeryZone X
16. ↑ stackexchange : classification-of-points-in-the-mandelbrot-set
17. ↑ fractalforums : tricky-mandelbrot-problem