Fractals/Iterations in the complex plane/Parameter plane

Parameter plane structure algorithms models Structure of parameter plane 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 Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity 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. Parts of parameter plane components ( incuding islands) curves points curves or paths 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).Generalizations of Douady's magic formula by Adam Epstein, Giulio Tiozzo boundary of period 2 hyperbolic componenet, which is a circle Paths: external rays internal rays escape route real slice " each period-doubling cascade superstable orbit ... generate the corresponding chaotic band and the Misiurewicz point that separates the chaotic bands and "[Pastor97a] : Harmonic structure of one-dimensional quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini where: B is a chaotic region from -2 to MF = Feigenbaum point = the Myrberg-Feigenbaum A is a periodic region : from MF to 1/4 is the precedes symbol. The binary relationship: "x precedes y" is written: . It is used to distinguish other orders from total orders. chaotic region where: is chaotic band 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." The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1 named parts of Mandelbrot set Valley: double spiral spindle seahorse valley/coast ( or Seahorse Valley East ) mandelmap - A detailed map of the Mandelbrot Set, in a beautiful vintage style by Bill Tavis main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part.seahorsevalley From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022 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 elephantsmandelbrot-locations by woofractal 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.Mandelbrot Buds and Branches by Timothy Chase Map of the Mandelbrot Set. Copyright © 2005-2011 Janet Parke 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 = limiting 1/n limb of main cardioid as n tends to infinity. See demo 2 page 10 from program mandel by Wolf Jung Mini Mandelbrot sets Names: midget mini mandelbrot set baby Mandelbrots island = island mu-molecule = island mu-unit the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018 primitive component small copies of itself babies Mandelbrot sets = BMSsM. Romera, G. Pastor, A. B. Orue, D. Arroyo, F. Montoya, "Coupling Patterns of External Arguments in the Multiple-Spiral Medallions of the Mandelbrot Set", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 135637, 14 pages, 2009. https://doi.org/10.1155/2009/135637 Parameter rays of mini Mandelbrot sets 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 Devaney In Global Analysis of Dynamical Systems, ed.: H. Broer, B. Krauskopf, G. Vegter. IOP Publishing (2001), 329-338 or Homoclinic Points in Complex Dynamical Systems 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 accessible from main cardioid ( M0) by a finite number of boundary crossing ? distorted islands Julia island or Embedded Julia Set or virtual Julia Sets or medallions "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 Munafoembedded julia set from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017. "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot." https://www.flickr.com/photos/nonnameavailable/28654921940/ Medallions: carrot = non-spiral medallion (-1.6898799090349986e-01 1.0423707254693195e+00 3.0218142193747843e-07 ) type: long double cauliflower = single spiral medallion (-0.1543869 1.0308295 3.0218142193747843e-07 ) type: long double double spiral medallion (-0.16092059 1.03663239 0.000001 ) type: long double triple spiral medallion ( -1.5403777941777627e-01 1.0369221371305641e+00 6.5186720162412668e-07 ) type: long double 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 1999https://www.mrob.com/pub/muency/r2namingsystem.html orbit portrait by Milnor 2000 shrubs by M Romero et al 2004 M. Romera et al, Int. J. Bifurcation Chaos 13, 2279 (2003). https://doi.org/10.1142/S0218127403007941 Shrubs in the Mandelbrot Set Ordering 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.:SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya 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 of parameter plane: wake / limb migdet / island / mu-moleculemumolecule From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020. distortedfractalforums.com : how-distorted-can-a-minibrot-be = warpeddistribution From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020 hyperbolic component of Mandelbrot set Wake is a part of parameter plane between 2 parameter rays landing on the same parabolic point ( root point, bond) Limb is a part of Mandelbrot set between 2 parameter rays landing on the same parabolic point ( root point, bond) Strict ( math ) definition from the paper Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account by John W. Milnor Theorem 1.2. The Wake . The two corresponding parameter rays land at a single point of the parameter plane. These rays, together with their landing point, cut the plane into two open subsets and with the following property: A quadratic map has a repelling orbit with portrait if and only if , and has a parabolic orbit with portrait if and only if . Definitions: the Mandelbrot set This open set will be called the -wake in parameter space (compare Atela [A]), will be called the root point of this wake The intersectionIntersection_(set_theory) in wikipedia will be called the -limb of the Mandelbrot set The open arc consisting of all angles of dynamic rays which are contained in the interior of S1, or all angles of parameter rays which are contained in , will be called the characteristic arc for the orbit portrait Nomenclature types (source of the names): mu-ency Mu-Ency - The Encyclopedia of the Mandelbrot Set by R Munafo Mandelbrot papers and books How to move on parameter plane ? 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." marcm200fractalforums.org : julia-sets-true-shape-and-escape-time 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."marcm200fractalforums.org : julia-sets-true-shape-and-escape-time 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: From Cantor to parabolic Parameter along External RayFrom Cantor to Semi-hyperbolic Parameter along External Rays by Yi-Chiuan Chen and Tomoki Kawahira parabolic point = root point = landing point of above external ray internal ray: From Hyperbolic to Parabolic Parameters along Internal RaysFrom Hyperbolic to Parabolic Parameters along Internal Rays by Yi-Chiuan Chen and Tomoki Kawahira along circular curves : equipotentials boundaries of hyperbolic components internal circular curves loops math.stackexchange question: parameter-plane-dynamics-of-fixed-points-and-their-preimages-for-standard-quadra The dynamics of the polynomials of moving along curves is : Reading escaping trees from Hubbard trees in Sn by Matthieu Arfeux 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 External ray From Cantor to Semi-hyperbolic Parameters along External RaysFrom Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839 Yi-Chiuan ChenTomoki KawahiraTomoki Kawahira 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 Cantor to Misiurewicz: along the parameter ray of angle 15/56 by Tomoki Kawahira Cantor to Parabolic along the parameter ray of angle 1/3: by Tomoki Kawahira 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 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 + contious 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: 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 + 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 Plane types Criteria for plane classifications fractal formula ( function): c-plane, lambda-plane plane transformations There are many different types of the parameter planeAlternate Parameter Planes by David E. Joyce exponentialmap by Robert Munafo by function plain c-plane: plain lambda plane where by transformations Twisted Mandelbrot Sets by Eric C. HillOn quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I by Luna Lomonaco, Carsten Lunde Petersen inverted c-plane = 1/c plane: exponential plane ( map) mu-ency : exponential map by R MunafoExponential mapping and OpenMP by Claude Heiland-Allenexponential_mapping_with_kalles_fraktaler by Claude Heiland-Allen unrolled plain (flatten' the cardiod = unroll ) Linas Vepstas : Self Similar?the flattened cardioid of a Mandelbrot by Tom Rathborne = "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)Stretching cusps by Claude Heiland-Allen See also Mandelbrot Sets: Alternate Parameter Planes 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. Points realTotally real points in the Mandelbrot Set by Xavier Buff, Sarah Koch 2022 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 relation: Julia set interior / boundary / exterior How to save parameters of the point? parameter files: files with saved parameter values image files with saved parameters examples of important points Examples from the parameter plane and Mandelbrot set: The northwest external angle is 3/8 The north external angle is 1/4North by  Robert P. Munafo, 2010 Sep 20. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022. The northeast external angle is 1/8Northeast by Robert P. Munafo, 2010 Sep 20. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022. The west external angle is 1/2 The east external angle is 0 The southwest external angle is 5/8 The south external angle is 3/4 The southeast external angle is 7/8, Point Types point pixel of parameter plane c parameter of complex quadratic polynomial complex number point coordinate 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), biaccessible, triaccessible, .... 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 setFrom Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839 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 :stackexchange : classification-of-points-in-the-mandelbrot-set 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 = biaccessible ). 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 ... 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)fractalforums : parameter-adjustment-art-or-luck ? 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 interesting c points by Owen Maresh see also known regins in Visual Guide To Patterns In The Mandelbrot Set by Miqel morphing interesting areasfractalforums : deep-zooming-to-interesting-areas zoom autozoom math.stackexchange question: value-to-use-as-center-of-mandelbrot-set-zoom Mandelbrot Examples with parameters by David J. Eck examples 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" marcm200fractalforums.org : parameter-adjustment-art-or-luck 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." marcm200fractalforums.org: julia-sets-true-shape-and-escape-time Curves Curves on the parameter plane rays Parameter External Ray internal rays equipotential curves boundary of whole Mandelbrot set of hyperbolic components Algorithms general or representation functions Boolean_escape_time BDM/M Complex potential atom domains bof60 The_Lyapunov_exponent True shape Discrete Langrangian Descriptors 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 Models 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 How to to determine the ideal number of maximum iterations for an arbitrary zoom level in a Mandelbrot fractal a-way-to-determine-the-ideal-number-of-maximum-iterations-for-an-arbitrary-zoom Automatic Dwell Limit From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023. stackoverflow question: how-many-iterations-of-the-mandelbrot-set-for-an-accurate-picture-at-a-certain-z stackoverflow question: calculate-a-dynamic-iteration-value-when-zooming-into-a-mandelbrot Adaptive maxiter depending on the inverse square root of the magnification 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