Mandelbrot set

TheMandelbrot set(/ˈmændəlbroʊt,-brɒt/)is a two-dimensionalsetwith a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal andfractalstructures. The set is defined in thecomplex planeas thecomplex numbersc{\displaystyle c}for which the functionfc(z)=z2+c{\displaystyle f_{c}(z)=z^{2}+c}does notdivergeto infinity wheniteratedstarting atz=0{\displaystyle z=0}, i.e., for which the sequencefc(0){\displaystyle f_{c}(0)},fc(fc(0)){\displaystyle f_{c}(f_{c}(0))}, etc., remains bounded inabsolute value.
This set was first defined and drawn byRobert W. Brooksand Peter Matelski in 1978, as part of a study ofKleinian groups.Afterwards, in 1980,Benoit Mandelbrotobtained high-quality visualizations of the set while working atIBM'sThomas J. Watson Research CenterinYorktown Heights, New York.
Images of the Mandelbrot set exhibit an infinitely complicatedboundarythat reveals progressively ever-finerrecursivedetail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is afractal curve. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample pointc{\displaystyle c}, whether the sequencefc(0),fc(fc(0)),…{\displaystyle f_{c}(0),f_{c}(f_{c}(0)),\dotsc }goes to infinity. Treating therealandimaginary partsofc{\displaystyle c}asimage coordinateson thecomplex plane, pixels may then be colored according to how soon the sequence|fc(0)|,|fc(fc(0))|,…{\displaystyle |f_{c}(0)|,|f_{c}(f_{c}(0))|,\dotsc }crosses an arbitrarily chosen threshold (the threshold must be at least 2, as −2 is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary). Ifc{\displaystyle c}is held constant and the initial value ofz{\displaystyle z}is varied instead, the correspondingJulia setfor the pointc{\displaystyle c}is obtained.
The Mandelbrot set has become popular outsidemathematicsboth for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples ofmathematical visualization,mathematical beauty, andmotif.
The Mandelbrot set has its origin incomplex dynamics, a field first investigated by theFrench mathematiciansPierre FatouandGaston Juliaat the beginning of the 20th century. The fractal was first defined and drawn in 1978 byRobert W. Brooksand Peter Matelski as part of a study ofKleinian groups.On 1 March 1980, atIBM'sThomas J. Watson Research CenterinYorktown Heights,New York,Benoit Mandelbrotfirst visualized the set.
Mandelbrot studied theparameter spaceofquadratic polynomialsin an article that appeared in 1980.The mathematical study of the Mandelbrot set really began with work by the mathematiciansAdrien DouadyandJohn H. Hubbard(1985),who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work infractal geometry.
The mathematiciansHeinz-Otto PeitgenandPeter Richterbecame well known for promoting the set with photographs, books (1986),and an internationally touring exhibit of the GermanGoethe-Institut(1985).
The cover article of the August 1985Scientific Americanintroduced thealgorithmfor computing the Mandelbrot set. The cover was created by Peitgen, Richter andSaupeat theUniversity of Bremen.The Mandelbrot set became prominent in the mid-1980s as acomputer-graphics demo, whenpersonal computersbecame powerful enough to plot and display the set in high resolution.
The work of Douady and Hubbard occurred during an increase in interest incomplex dynamicsandabstract mathematics,and the study of the Mandelbrot set has been a centerpiece of this field ever since.
The Mandelbrot set is theuncountable setof values ofcin thecomplex planefor which theorbitof thecritical pointz=0{\textstyle z=0}underiterationof thequadratic map
remainsbounded.Thus, acomplex numbercis a member of the Mandelbrot set if, when starting withz0=0{\displaystyle z_{0}=0}and applying the iteration repeatedly, theabsolute valueofzn{\displaystyle z_{n}}remains bounded for alln>0{\displaystyle n>0}.
For example, forc= 1, thesequenceis 0, 1, 2, 5, 26, ..., which tends toinfinity, so 1 is not an element of the Mandelbrot set. On the other hand, forc=−1{\displaystyle c=-1}, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.
The Mandelbrot set can also be defined as theconnectedness locusof the family ofquadraticpolynomialsf(z)=z2+c{\displaystyle f(z)=z^{2}+c}, the subset of the space of parametersc{\displaystyle c}for which theJulia setof the corresponding polynomial forms aconnected set. In the same way, theboundaryof the Mandelbrot set can be defined as thebifurcation locusof this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it isiteratedrepeatedly) changes drastically.
The Mandelbrot set is acompact set, since it isclosedand contained in theclosed diskof radius 2 centred onzero. A pointc{\displaystyle c}belongs to the Mandelbrot set if and only if|zn|≤2{\displaystyle |z_{n}|\leq 2}for alln≥0{\displaystyle n\geq 0}. In other words, theabsolute valueofzn{\displaystyle z_{n}}must remain at or below 2 forc{\displaystyle c}to be in the Mandelbrot set,M{\displaystyle M}, and if that absolute value exceeds 2, the sequence will escape to infinity. Sincec=z1{\displaystyle c=z_{1}}, it follows that|c|≤2{\displaystyle |c|\leq 2}, establishing thatc{\displaystyle c}will always be in the closed disk of radius 2 around the origin.
TheintersectionofM{\displaystyle M}with the real axis is the interval[−2,14]{\displaystyle \left[-2,{\frac {1}{4}}\right]}. The parameters along this interval can be put inone-to-one correspondencewith those of the reallogistic family,
The correspondence is given by
This gives a correspondence between the entireparameter spaceof the logistic family and that of the Mandelbrot set.
Douady and Hubbard showed that the Mandelbrot set isconnected. They constructed an explicitconformal isomorphismbetween the complement of the Mandelbrot set and the complement of theclosed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set isdisconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts ofM{\displaystyle M}. Upon further experiments, he revised his conjecture, deciding thatM{\displaystyle M}should be connected. Atopologicalproof of the connectedness was discovered in 2001 byJeremy Kahn.
The dynamical formula for theuniformisationof the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness ofM{\displaystyle M}, gives rise toexternal raysof the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of theYoccoz parapuzzle.
Theboundaryof the Mandelbrot set is thebifurcation locusof the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parametersc{\displaystyle c}for which the dynamics of the quadratic mapzn=zn−12+c{\displaystyle z_{n}=z_{n-1}^{2}+c}exhibits sensitive dependence onc,{\displaystyle c,}i.e. changes abruptly under arbitrarily small changes ofc.{\displaystyle c.}It can be constructed as the limit set of a sequence ofplane algebraic curves, theMandelbrot curves, of the general type known aspolynomial lemniscates. The Mandelbrot curves are defined by settingp0=z,pn+1=pn2+z{\displaystyle p_{0}=z,\ p_{n+1}=p_{n}^{2}+z}, and then interpreting the set of points|pn(z)|=2{\displaystyle |p_{n}(z)|=2}in the complex plane as a curve in the realCartesian planeof degree2n+1{\displaystyle 2^{n+1}}inxandy.Each curven>0{\displaystyle n>0}is the mapping of an initial circle of radius 2 underpn{\displaystyle p_{n}}. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.
Themaincardioidis the period 1 continent. It is the region of parametersc{\displaystyle c}for which the map
has anattracting fixed point. It consists of all parameters of the form
for someμ{\displaystyle \mu }in theopen unit disk.
To the left of the main cardioid, attached to it at the pointc=−3/4{\displaystyle c=-3/4}, a circular bulb, theperiod-2 bulbis visible. The bulb consists ofc{\displaystyle c}for whichfc{\displaystyle f_{c}}has anattracting cycle of period 2. It is the filled circle of radius 1/4 centered around −1.
More generally, for every positive integerq>2{\displaystyle q>2}, there areϕ(q){\displaystyle \phi (q)}circular bulbs tangent to the main cardioid calledperiod-q bulbs(whereϕ{\displaystyle \phi }denotes theEuler phi function), which consist of parametersc{\displaystyle c}for whichfc{\displaystyle f_{c}}has an attracting cycle of periodq{\displaystyle q}. More specifically, for each primitiveq{\displaystyle q}th root of unityr=e2πipq{\displaystyle r=e^{2\pi i{\frac {p}{q}}}}(where0<pq<1{\displaystyle 0<{\frac {p}{q}}<1}), there is one period-q bulb called thepq{\displaystyle {\frac {p}{q}}}bulb, which is tangent to the main cardioid at the parameter
and which contains parameters withq{\displaystyle q}-cycles having combinatorial rotation numberpq{\displaystyle {\frac {p}{q}}}. More precisely, theq{\displaystyle q}periodicFatou componentscontaining the attracting cycle all touch at a common point (commonly called theα{\displaystyle \alpha }-fixed point). If we label these componentsU0,…,Uq−1{\displaystyle U_{0},\dots ,U_{q-1}}in counterclockwise orientation, thenfc{\displaystyle f_{c}}maps the componentUj{\displaystyle U_{j}}to the componentUj+p(mod⁡q){\displaystyle U_{j+p\,(\operatorname {mod} q)}}.
The change of behavior occurring atcpq{\displaystyle c_{\frac {p}{q}}}is known as abifurcation: the attracting fixed point "collides" with a repelling period-qcycle. As we pass through the bifurcation parameter into thepq{\displaystyle {\tfrac {p}{q}}}-bulb, the attracting fixed point turns into a repelling fixed point (theα{\displaystyle \alpha }-fixed point), and the period-qcycle becomes attracting.
Bulbs that are interior components of the Mandelbrot set in which the mapsfc{\displaystyle f_{c}}have an attracting periodic cycle are calledhyperbolic components.
It is conjectured that these are theonlyinterior regionsofM{\displaystyle M}and that they aredenseinM{\displaystyle M}. This problem, known asdensity of hyperbolicity, is one of the most important open problems incomplex dynamics.Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.For real quadratic polynomials, this question was proved in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in theFeigenbaum diagram. So this result states that such windows exist near every parameter in the diagram.)
Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).
Each of the hyperbolic components has acenter, which is a pointcsuch that the inner Fatou domain forfc(z){\displaystyle f_{c}(z)}has a super-attracting cycle—that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. Therefore,fcn(0)=0{\displaystyle f_{c}^{n}(0)=0}for somen. If we call this polynomialQn(c){\displaystyle Q^{n}(c)}(letting it depend oncinstead ofz), we have thatQn+1(c)=Qn(c)2+c{\displaystyle Q^{n+1}(c)=Q^{n}(c)^{2}+c}and that the degree ofQn(c){\displaystyle Q^{n}(c)}is2n−1{\displaystyle 2^{n-1}}. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equationsQn(c)=0,n=1,2,3,...{\displaystyle Q^{n}(c)=0,n=1,2,3,...}.The number of new centers produced in each step is given by Sloane'sOEIS:A000740.
It is conjectured that the Mandelbrot set islocally connected. This conjecture is known asMLC(forMandelbrot locally connected). By the work ofAdrien DouadyandJohn H. Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the importanthyperbolicity conjecturementioned above.
The work ofJean-Christophe Yoccozestablished local connectivity of the Mandelbrot set at all finitelyrenormalizableparameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.Since then, local connectivity has been proved at many other points ofM{\displaystyle M}, but the full conjecture is still open.
The Mandelbrot set isself-similarunder magnification in the neighborhoods of theMisiurewicz points. It is also conjectured to be self-similar around generalizedFeigenbaum points(e.g., −1.401155 or −0.1528 + 1.0397i), in the sense of converging to a limit set.The Mandelbrot set in general is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.
TheHausdorff dimensionof theboundaryof the Mandelbrot set equals 2 as determined by a result ofMitsuhiro Shishikura.The fact that this is greater by a whole integer than its topological dimension, which is 1, reflects the extremefractalnature of the Mandelbrot set boundary. Roughly speaking, Shishikura's result states that the Mandelbrot set boundary is so "wiggly" that it locallyfills spaceas efficiently as a two-dimensional planar region. Curves with Hausdorff dimension 2, despite being (topologically) 1-dimensional, are oftentimes capable of having nonzero area (more formally, a nonzero planarLebesgue measure). Whether this is the case for the Mandelbrot set boundary is an unsolved problem.
It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the powerα{\displaystyle \alpha }of the iterated variablez{\displaystyle z}tends to infinity) is convergent to the unit (α{\displaystyle \alpha }-1)-sphere.
In theBlum–Shub–Smalemodel ofreal computation, the Mandelbrot set is not computable, but its complement iscomputably enumerable. Many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based oncomputable analysis, which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between thegeometryof the Mandelbrot set at a given point and the structure of the correspondingJulia set. For instance, a value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected. Thus, the Mandelbrot set may be seen as a map of the connected Julia sets.
This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set hasHausdorff dimensiontwo, and then transfers this information to the parameter plane.Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.
For every rational numberpq{\displaystyle {\tfrac {p}{q}}}, wherepandqarecoprime, a hyperbolic component of periodqbifurcates from the main cardioid at a point on the edge of the cardioid corresponding to aninternal angleof2πpq{\displaystyle {\tfrac {2\pi p}{q}}}.The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called thep/q-limb. Computer experiments suggest that thediameterof the limb tends to zero like1q2{\displaystyle {\tfrac {1}{q^{2}}}}. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like1q{\displaystyle {\tfrac {1}{q}}}.
A period-qlimb will haveq−1{\displaystyle q-1}"antennae" at the top of its limb. The period of a given bulb is determined by counting these antennas. The numerator of the rotation number,p, is found by numbering each antenna counterclockwise from the limb from 1 toq−1{\displaystyle q-1}and finding which antenna is the shortest.
In an attempt to demonstrate that the thickness of thep/q-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for theseriesto diverge forz=−34+iε{\displaystyle z=-{\tfrac {3}{4}}+i\varepsilon }(−34{\displaystyle -{\tfrac {3}{4}}}being the location thereof). As the series does not diverge for the exact value ofz=−34{\displaystyle z=-{\tfrac {3}{4}}}, the number of iterations required increases with a smallε{\displaystyle \varepsilon }. It turns out that multiplying the value ofε{\displaystyle \varepsilon }with the number of iterations required yields an approximation ofπ{\displaystyle \pi }that becomes better for smallerε{\displaystyle \varepsilon }. For example, forε{\displaystyle \varepsilon }= 0.0000001, the number of iterations is 31415928 and the product is 3.1415928.In 2001, Aaron Klebanoff proved Boll's discovery.
The Mandelbrot Set features a fundamental cardioid shape adorned with numerous bulbs directly attached to it.Understanding the arrangement of these bulbs requires a detailed examination of the Mandelbrot Set's boundary. As one zooms into specific portions with a geometric perspective, precise deducible information about the location within the boundary and the corresponding dynamical behavior for parameters drawn from associated bulbs emerges.
The iteration of the quadratic polynomialfc(z)=z2+c{\displaystyle f_{c}(z)=z^{2}+c}, wherec{\displaystyle c}is a parameter drawn from one of the bulbs attached to the main cardioid within the Mandelbrot Set, gives rise to maps featuring attracting cycles of a specified periodq{\displaystyle q}and a rotation numberp/q{\displaystyle p/q}. In this context, the attracting cycle of  exhibits rotational motion around a central fixed point, completing an average ofp/q{\displaystyle p/q}revolutions at each iteration.
The bulbs within the Mandelbrot Set are distinguishable by both their attracting cycles and the geometric features of their structure. Each bulb is characterized by an antenna attached to it, emanating from a junction point and displaying a certain number of spokes indicative of its period. For instance, the2/5{\displaystyle 2/5}bulb is identified by its attracting cycle with a rotation number of2/5{\displaystyle 2/5}. Its distinctive antenna-like structure comprises a junction point from which five spokes emanate. Among these spokes, called the principal spoke is directly attached to the2/5{\displaystyle 2/5}bulb, and the 'smallest' non-principal spoke is positioned approximately2/5{\displaystyle 2/5}of a turn counterclockwise from the principal spoke, providing a distinctive identification as a2/5{\displaystyle 2/5}-bulb.This raises the question: how does one discern which among these spokes is the 'smallest'?In the theory ofexternal raysdeveloped byDouadyandHubbard,there are precisely two external rays landing at the root point of a satellite hyperbolic component of the Mandelbrot Set. Each of these rays possesses an external angle that undergoes doubling under the angle doubling mapθ↦{\displaystyle \theta \mapsto }2θ{\displaystyle 2\theta }. According to this theorem, when two rays land at the same point, no other rays between them can intersect. Thus, the 'size' of this region is measured by determining the length of the arc between the two angles.
If the root point of the main cardioid is the cusp atc=1/4{\displaystyle c=1/4}, then the main cardioid is the0/1{\displaystyle 0/1}-bulb. The root point of any other bulb is just the point where this bulb is attached to the main cardioid. This prompts the inquiry: which is the largest bulb between the root points of the0/1{\displaystyle 0/1}and1/2{\displaystyle 1/2}-bulbs? It is clearly the1/3{\displaystyle 1/3}-bulb. And note that1/3{\displaystyle 1/3}is obtained from the previous two fractions byFarey addition, i.e., adding the numerators and adding the denominators
01{\displaystyle {\frac {0}{1}}}⊕{\displaystyle \oplus }12{\displaystyle {\frac {1}{2}}}={\displaystyle =}13{\displaystyle {\frac {1}{3}}}
Similarly, the largest bulb between the1/3{\displaystyle 1/3}and1/2{\displaystyle 1/2}-bulbs is the2/5{\displaystyle 2/5}-bulb, again given by Farey addition.
13{\displaystyle {\frac {1}{3}}}⊕{\displaystyle \oplus }12{\displaystyle {\frac {1}{2}}}={\displaystyle =}25{\displaystyle {\frac {2}{5}}}
The largest bulb between the2/5{\displaystyle 2/5}and1/2{\displaystyle 1/2}-bulb is the3/7{\displaystyle 3/7}-bulb, while the largest bulb between the2/5{\displaystyle 2/5}and1/3{\displaystyle 1/3}-bulbs is the3/8{\displaystyle 3/8}-bulb, and so on.The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by theFarey tree, a structure encompassing all rationals between0{\displaystyle 0}and1{\displaystyle 1}. This ordering positions the bulbs along the boundary of the main cardioid precisely according to therational numbersin theunit interval.
Starting with the1/3{\displaystyle 1/3}bulb at the top and progressing towards the1/2{\displaystyle 1/2}circle, the sequence unfolds systematically: the largest bulb between1/2{\displaystyle 1/2}and1/3{\displaystyle 1/3}is2/5{\displaystyle 2/5}, between1/3{\displaystyle 1/3}and2/5{\displaystyle 2/5}is3/8{\displaystyle 3/8}, and so forth.Intriguingly, the denominators of the periods of circular bulbs at sequential scales in the Mandelbrot Set conform to theFibonacci number sequence, the sequence that is made by adding the previous two terms – 1, 2, 3, 5, 8, 13, 21...
The Fibonacci sequence manifests in the number of spiral arms at a unique spot on the Mandelbrot set, mirrored both at the top and bottom. This distinctive location demands the highest number of iterations of  for a detailed fractal visual, with intricate details repeating as one zooms in.
The boundary of the Mandelbrot set shows more intricate detail the closer one looks ormagnifiesthe image. The following is an example of an image sequence zooming to a selectedcvalue.
The magnification of the last image relative to the first one is about 10to 1. Relating to an ordinarycomputer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers.
The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is aMisiurewicz point. Between the "upper part of the body" and the "tail", there is a distorted copy of the Mandelbrot set, called a "satellite".
The islands in the third-to-last step seem to consist of infinitely many parts, as is the case for the corresponding Julia setJc{\displaystyle J_{c}}. They are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value ofc{\displaystyle c}for the correspondingJc{\displaystyle J_{c}}is not the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th step.
While the Mandelbrot set is typically rendered showing outside boundary detail, structure within the bounded set can also be revealed. For example, while calculating whether or not a given c value is bound or unbound, while it remains bound, the maximum value that this number reaches can be compared to the c value at that location. If the sum of squares method is used, the calculated number would be max:(real^2 + imaginary^2) - c:(real^2 + imaginary^2). The magnitude of this calculation can be rendered as a value on a gradient.
This produces results like the following, gradients with distinct edges and contours as the boundaries are approached. The animations serve to highlight the gradient boundaries.
Multibrot setsare bounded sets found in thecomplex planefor members of the general monic univariatepolynomialfamily of recursions
For anintegerd, these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness locus has also been studied; here one considers the two-parameter recursionz↦z3+3kz+c{\displaystyle z\mapsto z^{3}+3kz+c}, whose twocritical pointsare thecomplex square rootsof the parameterk. A parameter is in the cubic connectedness locus if both critical points are stable.For general families ofholomorphic functions, theboundaryof the Mandelbrot set generalizes to thebifurcation locus.
TheMultibrot setis obtained by varying the value of the exponentd. The article has a video that shows the development fromd= 0 to 7, at which point there are 6 i.e.(d−1){\displaystyle (d-1)}lobes around theperimeter. In general, whendis a positive integer, the central region in each of these sets is always anepicycloidof(d−1){\displaystyle (d-1)}cusps. A similar development with negative integral exponents results in(1−d){\displaystyle (1-d)}clefts on the inside of a ring, where the main central region of the set is ahypocycloidof(1−d){\displaystyle (1-d)}cusps.
There is no perfect extension of the Mandelbrot set into 3D, because there is no 3D analogue of the complex numbers for it to iterate on. There is an extension of the complex numbers into 4 dimensions, thequaternions, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.These can then be eithercross-sectionedorprojectedinto a 3D structure. The quaternion (4-dimensional) Mandelbrot set is simply asolid of revolutionof the 2-dimensional Mandelbrot set (in the j-k plane), and is therefore uninteresting to look at.Taking a 3-dimensional cross section atd=0(q=a+bi+cj+dk){\displaystyle d=0\ (q=a+bi+cj+dk)}results in a solid of revolution of the 2-dimensional Mandelbrot set around the real axis.
Of particular interest is thetricornfractal, the connectedness locus of theanti-holomorphicfamily
The tricorn (also sometimes called theMandelbar) was encountered byMilnorin his study of parameter slices of realcubic polynomials. It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials.
Another non-analytic generalization is theBurning Ship fractal, which is obtained by iterating the following:
There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the most widely used and simplest algorithm will be demonstrated, namely, the naïve "escape time algorithm". In the escape time algorithm, a repeating calculation is performed for eachx,ypoint in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.
Thexandylocations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the nextx,ypoint is examined.
The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition.
To render such an image, the region of the complex plane we are considering is subdivided into a certain number ofpixels. To color any such pixel, letc{\displaystyle c}be the midpoint of that pixel. Iterate the critical point 0 underfc{\displaystyle f_{c}}, checking at each step whether the orbit point has a radius larger than 2. When this is the case,c{\displaystyle c}does not belong to the Mandelbrot set, and color the pixel according to the number of iterations used to find out. Otherwise, keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.
Inpseudocode, this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have acomplex data type. The program may be simplified if the programming language includescomplex-data-typeoperations.
Here, relating the pseudocode toc{\displaystyle c},z{\displaystyle z}andfc{\displaystyle f_{c}}:
and so, as can be seen in the pseudocode in the computation ofxandy:
To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.).
Here is the code implementing the above algorithm inPython:
The value ofpowervariable can be modified to generate an image of equivalentmultibrot set(z=zpower+c{\displaystyle z=z^{\text{power}}+c}). For example, settingp = 2produces the associated image.
The Mandelbrot set is widely considered the most popularfractal,and has been referenced several times inpopular culture.