Fractals/Iterations in the complex plane/Parameter plane

Structure of parameter plane

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

• $z0=z_{cr}\,$ is constant
• $c\,$ 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

Structure of parameter plane from programmers point of view:

• exterior of M-set
• Each component is surrounded by an atom domain ( disc with radius 4 times bigger, for cardioids radius has about the square root of the size).
• Each component has a nucleus at its center, which has a periodic orbit containing 0.

curves

The Mandelbrot set contains smooth curves:

• the intersection with the real axis M ∩ R = [−2, 1/4] = real slice of Mandelbrot set
• the main cardioid of M, which is the set of parameters c for which fc has an attracting or indifferent fixed point (of course, this is smooth except at the cusp c = 1/4).
• boundary of period 2 hyperbolic componenet, which is a circle

real slice

" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band $B_{n}$ and the Misiurewicz point $m_{n}$ that separates the chaotic bands $B_{n=1}$ and $B_{n}$ "

$B\prec MF\prec A$

where:

• B is a chaotic region from -2 to MF
• $c=MF=m_{\infty }=b_{\infty }=-1.4011551890......$ = Feigenbaum point = the Myrberg-Feigenbaum
• A is a periodic region : from MF to 1/4
• $\prec$ is the precedes symbol. The binary relationship: "x precedes y" is written: $x\prec y$. It is used to distinguish other orders from total orders.

chaotic region

$B_{0}\prec B_{1}\prec B_{2}\prec ...\prec B_{\infty }$

where:

• $B_{m}$ is chaotic band $B_{m}=(2n+1)\cdot 2^{m}$

 $3*2^{m}\prec 5*2^{m}\prec 7*2^{m}\prec ...\prec \infty$
 

See also

• separators between bands
• It is a part of Sharkovsky ordering

periodic region

period doubling cascade

• 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."

$2^{\infty }\prec ...\prec 2^{2}\prec 2^{1}\prec 2^{0}$

named parts of Mandelbrot set

Valley:

• double spiral
• spindle
• seahorse valley/coast ( or Seahorse Valley East )
  • main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part.
  • disk 3 seahorse valley = Gap between period 1 and period 3
• Elephant Valley = Gap near cusp of main cardioid. Here the antennae resemble the trunks of elephants
  • elephant coast = boundary of main cardioid near cusp
• Scepter Valley = gap between period 2 and period 4 component, Also known as Seahorse Valley West or Sceptre Valley. "Scepter Valley" where double spirals have scepters coming (echoing the spindle) coming off of all their tips.
  • double spiral valley =
  • double spiral coast = boundary of period 2 component near main cardioid
• triple spiral coast ( valley) - boundary of period 3 componnet near period 1 cardioid

Parts

• Peacock eye = shrub ( decoration) from Seahorse valley = Principal Misiurewicz point with all branches
• seahorse
• elephant

• 

  Gap between the head and the body = seahorse valley

• 

  Double-spirals on the left, seahorses on the right

• 

  Fragment of Mandelbrot set called the elephant valley

Mini Mandelbrot sets

Names:

• midget
• mini mandelbrot set
• baby Mandelbrots
• island = island mu-molecule = island mu-unit
• primitive component

Parameter rays of mini Mandelbrot sets

" a saddle-node (parabolic) periodic point in a complex dynamical system often admits homoclinic points, and in the case that these homoclinic points are nondegenerate, this is accompanied by the existence of infinitely many baby Mandelbrot sets converging to the saddle-node parameter value in the corresponding parameter plane." Devaney

See also

• How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accesible from main cardioid ( M0) by a finite number of boundary crossing ?
• distorted islands

Julia island = Embedded Julia Set = virtual Julia Sets

• "A structure comprised of filaments, resembling a Julia set in appearance, which has a higher delta Hausdorff dimension than filaments in the immediately surrounding region. They are sometimes also called Julia islands or virtual Julia Sets." Robert Munafo
• "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot."

• 
• 

hyperbolic componenets

• Components of the parameter plane

order of hyperbolic componenets

• Myrberg 1963
• Sharkovsky 1964
• Metropolis 1973
• internal adress by Lau and Schleicher 1994
• rotation number by Devaney 1997
• R2 naming system by R Munafo 1999
• orbit portrait by Milnor 2000
• shrubs by M Romero et al 2004

shape

• ON THE SHAPES OF ELEMENTARY DOMAINS OR WHY MANDELBROT SET IS MADE FROM ALMOST IDEAL CIRCLES by V. Dolotin A.morozov
• Algebraic Geometry of Discrete Dynamics. The case of one variable by V.Dolotin, A.Morozov

Parts of parameter plane

• with respect to the Mandelbrot set
  • exterior
  • boundary with different point types: Cremer, Siegel, parabolic, Misiurewicz, Feigenbaoum, ...
  • interior
• 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)
    • periodic part: period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
    • the Myrberg-Feigenbaum point
    • chaotic part: shrub

Not that her q/p not p/q notation is used

Components:

• wake / limb
• migdet / island / mu-molecule
  • distorted = warped
• hyperbolic component of Mandelbrot set

Names source

• mu-ency
• Mandlebrot

How to choose a point from parameter plane ?

• take point ( and check it's properities)
  • taking parameter choosing by other people (visual choose)
  • clicking on parameter points and see what you have ( random choose)
• computing a point from it's known properties
  • For (parabolic point ) choose hyperbolic component ( period, number) and internal angle (= rotation number) then compute c parameter.
  • Misiurewicz points
  • see also known regins in
  • morphing
  • interesting areas
• zoom
  • autozoom
  • math.stackexchange question: value-to-use-as-center-of-mandelbrot-set-zoom

Is parameter tweaking an acquired art or just random chance?

 "From my own experience with monocritical polynomials and Lyapunov diagrams, all my images I found purely by chance.
 For z^2+c e.g. as long as you're in the same hyperbolic component, the shape changes only in the sense, that a fat spiral might become thinner, but the number of arms stays constant. 
 If you move the c value out of that    component into another - and if this 2nd component is not directly attached to the first, then, I'm not aware of a direct way of telling what it would look like there.

Usually I perform a parameter walk just computing black and white escape images with periodicity. Then for  the interesting shapes/periods I apply a color walk with some gradients that looked fine in previous images. 
But that's  more or less guessing.

If you want to turn more into the constructing-an-image from a vision you have, you might try two articles: genetic algorithm and Leja points"  marcm200

Mandelbrot set : z^6+ A*z+c How do you find such intereseting examples ?

" I'm running from time to time an A,c-parameter space walk (brute force) in a rather wide grid (-2..+2 in ~0,01 or larger steps) for the family z^n+A*z+c, adding a small random dyadic fraction to the 4d coordinates to get variation. 
Following numerically the orbits of the critical points with a rather high max it of 25000 it's possible to get the number of attracting cycles and their length to some accuaracy level in a decent time. If those A,c-parameter pairs pass some filters (mostly sum of length of cycles and diversity) I scan through small overview pictures manually.
Then I use interesting A,c pairs (shape-wise or from the filter values) and some small deviations from it to compute level 10-12 TSA images, as sets wiith similar shapes can show a different dynamical behaviour w.r.t. the level at which interior cells emerge.
I'll take the fastest one and see how many cycles can be detected up to level 18-19." marcm200

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.

• 
• 

  zoom and translation towards Feigenbaum point

• Implosion : from circle thru cauliflower to imploded cauliflower
• 

  c=0

• 

  c=1/4

• 

  c= 1/4 + 0.05

• 

  c = 1/4 + 0.029

• 

  c = 1/4 + 0.035

How to choose step of a move ?

• fixed size step / adaptive size step
• minimal size of the move

"For c values that could not be represented accurately by C++ double data type I calculated the images using interval arithmetics with tiny intervals (border values being fractions of 2^25, interval width 2^-25) encompassing the published value (as needed for real or imaginary part or both) (values were computed using wolfram-alpha to multiply large integers). Given is the left (smaller) border value, the right is obtained by adding one." marcm200

What is the reason for : 2^-25  ?

"When I started with this article back in March this year, my initial formulas were z^2+c and z^2+c*z.

In expanded form with real coordinates: z=(x+i*y) and c=(d+e*i):

( x²-y² + d, 2xy + e )  or ( d*x + x² - e*y - y² , e*x + d*y + 2x*y )

To accurately represent a sum, the widest two terms can be apart is 53 bits (mantissa precision for C++ double) and all other must lie in this range.

The smallest non-zero value of x,y is "axis range / pixel count", i.e. 4 (escape radius of 2, hence axis -2..+2) divided by 2^refinement level.

So for x^2 this goes to 2^-26 as the lowest possible value for x. And since d,e are multiplied with x in the 2nd formula, the same goes for d and e. As I do not like to work "on the edge" I used a buffer of 1-2 bits and came to  the lowest value of 2^-25 for d,e (and refinement level limit of 27 which is currently outside a reasonable range).

For the 1st formula z^2+c, the seed value could go as little as 2^-48 (stated in the article) as it is only added.

For long double and float128 one could go lower in both formulas, but I haven't explored that."marcm200

Compare

• Numerical calculations and rigorous mathematics
• Reliable Computing, Interval Computations

Types

• type of the move
  • continuous
  • discrete = using sequence of points
• type of the curve
  • along radial curves :
    • escape route
    • external ray
    • parabolic point = root point = landing point of above external ray
    • internal ray which also ends/lands at the parabolic point
  • along circular curves :
    • equipotentials
    • boundaries of hyperbolic components
    • internal circular curves
  • loops

The dynamics of the polynomials of moving along curves is :

• for external ray: ”stretch” the dynamic on the basin of infinity. The argument of φPa,b (2a) stay unchanged ( fixed)
• for equipotential: twist the dynamics in the annulus between the Green level curves of the escaping critical point and of the critical value. The modulus of φPa,b (2a is fixed

Examples

• 

  point c moves along boundary of main cardioid toward c=0.75 ( root point of period 2 component of Mandelbrot set) using a sequence

External ray

• From Cantor to Semi-hyperbolic Parameters along External Rays
• along parameter external ray for angle 9/31 by David Madore : Its external argument is constantly 9/31 and it approaches the bud's root (~ −0.481763 + +0.531657i) exponentially slowly.
• external angle 1/3 by David Madore

Boundary of the component

• boundary of the largest bulb of the Mandelbrot set by David Madore

Others

• morphing
• 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));

Escape route

• 

  escape route 1/2

Escape route 0

Escape route for internal angle 0/1

Steps

• nucelus of period 1 component ( c = 0) Fixed point alfa is supperattracting fixed point. Julia set is connected.
• along internal ray 0. Imaginary part of parameter c is zero. 0 < cx < 0.25. Fixed point alfa is attracting fixed point. Julia set is connected.
• parabolic point c = 1/4. Fixed point alfa is parabolic fixed point. Julia set is connected.
• along external ray 0. Imaginary part of parameter c is zero. 0.25 < cx. Fixed point alfa is repelling fixed point. Julia set is disconnected

Here parabolic implosion/ explosion ( from connected to disconnected ) occurs. In parabolic point child periodic points coincides with parent period points

evolution of dynamics along escape route 0 ( parabolic implosion)

parameter c	location of c	Julia set	interior	type of critical orbit dynamics	critical point	fixed points	stability of alfa
c = 0	center, interior	connected	exist	superattracting	atracted to alfa fixed point	fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling	r = 0
0<c<1/4	internal ray 0, interior	connected	exist	attracting	atracted to alfa fixed point	alfa is attracting, beta is repelling	0 < r < 1.0
c = 1/4	cusp, boundary	connected	exist	parabolic	atracted to alfa fixed point	alfa fixed point equal to beta fixed point, both are parabolic	r = 1
c>1/4	external ray 0, exterior	disconnected	disappears	repelling	repelling to infinity	both finite fixed points are repelling	r > 1

Stability r is absolute value of multiplier at fixed point alfa:

$r=|m(z_{\alpha })|$

c = 0.0000000000000000+0.0000000000000000*I 	 m(c) = 0.0000000000000000+0.0000000000000000*I 	 r(m) = 0.0000000000000000 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.0250000000000000+0.0000000000000000*I 	 m(c) = 0.0513167019494862+0.0000000000000000*I 	 r(m) = 0.0513167019494862 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.0500000000000000+0.0000000000000000*I 	 m(c) = 0.1055728090000841+0.0000000000000000*I 	 r(m) = 0.1055728090000841 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.0750000000000000+0.0000000000000000*I 	 m(c) = 0.1633399734659244+0.0000000000000000*I 	 r(m) = 0.1633399734659244 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1000000000000000+0.0000000000000000*I 	 m(c) = 0.2254033307585166+0.0000000000000000*I 	 r(m) = 0.2254033307585166 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1250000000000000+0.0000000000000000*I 	 m(c) = 0.2928932188134524+0.0000000000000000*I 	 r(m) = 0.2928932188134524 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1500000000000000+0.0000000000000000*I 	 m(c) = 0.3675444679663241+0.0000000000000000*I 	 r(m) = 0.3675444679663241 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1750000000000000+0.0000000000000000*I 	 m(c) = 0.4522774424948338+0.0000000000000000*I 	 r(m) = 0.4522774424948338 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2000000000000000+0.0000000000000000*I 	 m(c) = 0.5527864045000419+0.0000000000000000*I 	 r(m) = 0.5527864045000419 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2250000000000000+0.0000000000000000*I 	 m(c) = 0.6837722339831620+0.0000000000000000*I 	 r(m) = 0.6837722339831620 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2500000000000000+0.0000000000000000*I 	 m(c) = 0.9999999894632878+0.0000000000000000*I 	 r(m) = 0.9999999894632878 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2750000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.3162277660168377*I 	 r(m) = 1.0488088481701514 	 t(m) = 0.0487455572605341 	period = 1
 c = 0.3000000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.4472135954999579*I 	 r(m) = 1.0954451150103321 	 t(m) = 0.0669301182003075 	period = 1
 c = 0.3250000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.5477225575051662*I 	 r(m) = 1.1401754250991381 	 t(m) = 0.0797514300099943 	period = 1
 c = 0.3500000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.6324555320336760*I 	 r(m) = 1.1832159566199232 	 t(m) = 0.0897542589928440 	period = 1

• evolution of dynamics along internal/external ray 0
• 

  center = superattracting

• 

  attracting

• 

  parabolic

• 

  repelling

• 
• 
• 
• 

• target set

• Internal level sets

• binary decomposition

• Full tile = binary decomposition and internal level sets

Real slice of parameter plane: imag(c) = 0

parameter c value	description of c locations	fixed points	Julia set	basins	target set ( petal)
1/4 < c	point c is on the external ray 0	both fixed points are repelling	disconnected	only ona basin of attraction ( infinity)
c = 1/4	cusp of main carioid	both fixed points are parabolic ( belong to Julia set)	connected = Cauliflower		circle
0 < c < 1/4	inside main cardioid, along internal ray 0	Treść komórki	Treść komórki
c = 0	center of main cardioid	Treść komórki	Treść komórki
0 < c < 3/4	inside main cardioid, along internal ray 1/2	Treść komórki	Treść komórki
c = 3/4	root point ( parabolic)	Treść komórki	Treść komórki
3/4 < c < 1.0	inside period 2 component, along internal ray 0	Treść komórki	Treść komórki
c = 1.0	center of period 2 component	Treść komórki	Treść komórki
1/0 < c < 5/4	inside period 2 component, along internal ray 1/2	Treść komórki	Treść komórki
c = 5/4	root point ( parabolic)	Treść komórki	Treść komórki

stability index of period 1 points	period 1 points on dynamic plane	period 1 points on parameter plane
changes from attractive through indifferent to repelling	moves from interior of Kc to its boundary	moves from interior of componetnt of M-set to its boundary

Escape route 1/3

• evolution of dynamics along internal1/3
• 

  parameter plane with escape route 1/3

• animation from center to parabolic

• 

  superattracting = center for period 1

• 

  parabolic = boundary

• 

  superattracting = center for period 3

Plane types

Criteria for plane classifications

• fractal formula ( function): c-plane, lambda-plane
• plane transformations

There are many different types of the parameter plane

• by function
  • plain c-plane: $z=z^{2}+c$
  • plain lambda plane $z=\lambda z(1-z)$ where $c={\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}$
• by transformations
  • inverted c-plane = 1/c plane: $z=z^{2}+c^{-1}$
  • 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)

• 

  c-plane

• 

  inverted c plane = 1/c plane

• 

  ${\frac {c^{2}}{c^{4}-0.25}}$ plane

• 

  Unrolled main cardioid of Mandelbrot set for periods 7-13

• 

  lambda plane

• 

  1/lambda plane

Parameter space types by dimensions

• 1D ( 1 real parameter):
• 2D ( 1 complex parameter): standard Mandelbrot set, here space is a 2D plane
• 4D ( 2 complex parameters) : the family f(z) = z^n+A*z+c by marcm200
• 6D ( 3 complex parameters) : six dimensional space of the complex parameters m, b, and d used in the formula f(x)=mx(1-x)(x+b)/(x+d) by Valannorton

One can only show 2D slice in the multi dimensional space.

How to describe c point ?

Numerical description

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

Symbolic description

• set realtion: Julia set interior / boundary / exterior

Point Types

point =pixel of parameter plane = c parameter

Criteria

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
• set properties ( relation with the Mandelbrot set and wakes)
  • interior
  • boundary
  • exterior
    • inside wake, subwake
    • outside all the wakes, belonging to a parameter ray landing at a Siegel or Cremer parameter,
• geometric properties
  • number of external rays that land on the boundary point : tips ( 1 ray), biaccesible, triaccesible, ....
  • position of critical point with relation to the Julia set
• Renormalization

Classification

There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.

Simple classification

• exterior of Mandelbrot set
• Mandelbrot set
  • boundary of Mandelbrot set
  • interior of Mandelbrot set ( hyperbolic parameter)
    • centers,
    • other internal points ( points of internal rays )

Definitions

• a parameter c in the boundary of Mandelbrot set ∂M is semi-hyperbolic if the critical point is non-recurrent and belongs to the Julia set
  • A typical example of semi-hyperbolic parameter is a Misiurewicz point: We say a parameter ˆc is Misiurewicz if the critical point of fcˆ is a pre-periodic point.

partial classification of boundary points

Classification :

• Boundaries of primitive and satellite hyperbolic components:
  • Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccesible ).
  • Siegel ( a unique parameter ray landing with irrational external angle)
  • Cremer ( a unique parameter ray landing with irrational external angle)
• Boundary of M without boundaries of hyperbolic components:
  • non-renormalizable (Misiurewicz with rational external angle and other).
  • renormalizable
    • finitely renormalizable (Misiurewicz and other).
    • infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers
• 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 critical point may be rcurrent or not, the number of branches at branch points may be bounded or not ...

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
• 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:

• An effective algorithm to compute Mandelbrot sets in parameter planes by A. Garijo, X. Jarque, J. Villadelprat Published 2017
• 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

Structure of the Mandelbrot set

• Sharkovsky ordering

Size or area

• mandelbrot-area by Kerry Mitchell ( 2001)
• The size of Mandelbrot bulbs by A.C. Fowler Mark J. Mcguinness

See also

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