Mandelbrot set

The Mandelbrot set has its origin in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups. On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first visualized the set. Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard (1985), who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in fractal geometry. The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books (1986), and an internationally touring exhibit of the German Goethe-Institut (1985). The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set. The cover was created by Peitgen, Richter and Saupe at the University of Bremen. The Mandelbrot set became prominent in the mid-1980s as a computer-graphics demo, when personal computers became powerful enough to plot and display the set in high resolution. The work of Douady and Hubbard occurred during an increase in interest in complex dynamics and abstract mathematics, and the topological and geometric study of the Mandelbrot set remains a key topic in the field of complex dynamics. ==Formal definition==

The Mandelbrot set is the uncountable set of values of c in the complex plane for which the orbit of the critical point z = 0 under iteration of the quadratic map :z \mapsto z^2 + c remains bounded. Thus, a complex number c is a member of the Mandelbrot set if, when starting with z_0 = 0 and applying the iteration repeatedly, the absolute value of z_n remains bounded for all n \in \Z^+. For example, for c = 1, the sequence is 0, 1, 2, 5, 26, ..., which tends to infinity, so 1 is not an element of the Mandelbrot set. On the other hand, for 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 the connectedness locus of the family of quadratic polynomials f(z) = z^2 + c, the subset of the space of parameters c for which the Julia set of the corresponding polynomial forms a connected set. In the same way, the boundary of the Mandelbrot set can be defined as the bifurcation locus of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it is iterated repeatedly) changes drastically. ==Basic properties==

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 centred on zero. A point c belongs to the Mandelbrot set if and only if |z_n|\leq 2 for all n\geq 0. In other words, the absolute value of z_n must remain at or below 2 for c to be in the Mandelbrot set, M, and if that absolute value exceeds 2, the sequence will escape to infinity. Since c=z_1, it follows that |c|\leq 2, establishing that c will always be in the closed disk of radius 2 around the origin. of the quadratic map The intersection of M with the real axis is the interval \left[-2,\frac{1}{4}\right]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family, :x_{n+1} = r x_n(1-x_n),\quad r\in[1,4]. The correspondence is given by :r = 1+\sqrt{1- 4 c}, \quad c = \frac{r}{2}\left(1-\frac{r}{2}\right), \quad z_n = r\left(\frac{1}{2} - x_n\right). This gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set. Douady and Hubbard showed that the Mandelbrot set is connected. They constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of M. Upon further experiments, he revised his conjecture, deciding that M should be connected. A topological proof of the connectedness was discovered in 2001 by Jeremy Kahn. The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of M, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle. The boundary of the Mandelbrot set is the bifurcation locus of the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters c for which the dynamics of the quadratic map z_n=z_{n-1}^2+c exhibits sensitive dependence on c, i.e. changes abruptly under arbitrarily small changes of c. It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p_0=z,\ p_{n+1}=p_n^2+z, and then interpreting the set of points |p_n(z)| = 2 in the complex plane as a curve in the real Cartesian plane of degree 2^{n+1}in x and y. Each curve n > 0 is the mapping of an initial circle of radius 2 under p_n. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below. ==Other properties==

Main cardioid and period bulbs The main cardioid is the period 1 continent. It is the region of parameters c for which the map f_c(z) = z^2 + c has an attracting fixed point. The set comprises all parameters of the form c(\mu) := \frac\mu2\left(1-\frac\mu2\right) where \mu lies within the open unit disk. Attached to the left of the main cardioid at the point c=-3/4, the period-2 bulb is visible. The q periodic Fatou components containing the attracting cycle meet at the \alpha-fixed point. If these are labelled counterclockwise as U_0,\dots,U_{q-1}, then component U_j is mapped by f_c to the component U_{j+p\,(\operatorname{mod} q)}. 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 the Feigenbaum 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 a center, which is a point c such that the inner Fatou domain for 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, f_c^n(0) = 0 for some n. If we call this polynomial Q^{n}(c) (letting it depend on c instead of z), we have that Q^{n+1}(c) = Q^{n}(c)^{2} + c and that the degree of Q^{n}(c) is 2^{n-1}. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations Q^{n}(c) = 0, n = 1, 2, 3, .... The number of new centers produced in each step is given by Sloane's . Local connectivity It is conjectured that the Mandelbrot set is locally connected. This conjecture is known as MLC (for Mandelbrot locally connected). By the work of Adrien Douady and John H. Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important hyperbolicity conjecture mentioned above. The work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies. Since then, local connectivity has been proved at many other points of M, but the full conjecture is still open. Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction. The display center pans left from the fifth to the seventh round feature (−1.4002, 0) to (−1.4011, 0) while the view magnifies by a factor of 21.78 to approximate the square of the Feigenbaum ratio. The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points. It is also conjectured to be self-similar around generalized Feigenbaum 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. Further results The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura. In the Blum–Shub–Smale model of real computation, the Mandelbrot set is not computable, but its complement is computably 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 on computable 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. Relationship with Julia sets As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia 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 has Hausdorff dimension two, 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. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the '''p/q-limb'''. Computer experiments suggest that the diameter of the limb tends to zero like \tfrac{1}{q^2}. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like \tfrac{1}{q}. A period-q limb will have 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 to q-1 and finding which antenna is the shortest. This experiment was performed independently by many people in the early 1990s, if not before; for instance by David Boll. Analogous observations have also been made at the parameters c=-5/4 and c=1/4 (with a necessary modification in the latter case). In 2001, Aaron Klebanoff published a (non-conceptual) proof for this phenomenon at c=1/4 In 2023, Paul Siewert developed, in his Bachelor thesis, a conceptual proof also for the value c=1/4, explaining why the number pi occurs (geometrically as half the circumference of the unit circle). In 2025, the three high school students Thies Brockmöller, Oscar Scherz, and Nedim Srkalovic extended the theory and the conceptual proof to all the infinitely many bifurcation points in the Mandelbrot set. Fibonacci sequence in the Mandelbrot set 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 polynomial f_c(z) = z^2 + c, where 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 period q and a rotation number p/q. In this context, the attracting cycle of  exhibits rotational motion around a central fixed point, completing an average of 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, the 2/5 bulb is identified by its attracting cycle with a rotation number of 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 the 2/5 bulb, and the 'smallest' non-principal spoke is positioned approximately 2/5 of a turn counterclockwise from the principal spoke, providing a distinctive identification as a 2/5-bulb. This raises the question: how does one discern which among these spokes is the 'smallest'? 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 map \theta\mapsto 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. The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by the Farey tree, a structure encompassing all rationals between 0 and 1. This ordering positions the bulbs along the boundary of the main cardioid precisely according to the rational numbers in the unit interval. Intriguingly, the denominators of the periods of circular bulbs at sequential scales in the Mandelbrot Set conform to the Fibonacci 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. Image gallery of a zoom sequence The boundary of the Mandelbrot set shows more intricate detail the closer one looks or magnifies the image. The following is an example of an image sequence zooming to a selected c value. The area shown is known as the "seahorse valley", which is a region of the Mandelbrot set centred on the point −0.75 + 0.1i. The magnification of the last image relative to the first one is about 1010 to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment. Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley" Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right Mandel zoom 03 seehorse.jpg|"Seahorse" upside down 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 a Misiurewicz point. Between the "upper part of the body" and the "tail", there is a distorted copy of the Mandelbrot set, called a "satellite". File:Mandel zoom 04 seehorse tail.jpg|The central endpoint of the "seahorse tail" is also a Misiurewicz point. File:Mandel zoom 05 tail part.jpg|Part of the "tail" – there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a simply connected set, which means there are no islands and no loop roads around a hole. File:Mandel zoom 06 double hook.jpg|Satellite. The two "seahorse tails" (also called dendritic structures) are the beginning of a series of concentric crowns with the satellite in the center. File:Mandel zoom 07 satellite.jpg|Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head". File:Mandel zoom 08 satellite antenna.jpg|"Antenna" of the satellite. There are several satellites of second order. File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley" 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. File:Mandelbrot full gradient.gif|Animated gradient structure inside the Mandelbrot set File:Mandelbrot inner gradient.gif|Animated gradient structure inside the Mandelbrot set, detail File:Mandelbrot gradient iterations.gif|Rendering of progressive iterations from 285 to approximately 200,000 with corresponding bounded gradients animated File:Mandelbrot gradient iterations thumb.gif|Thumbnail for gradient in progressive iterations ==Generalizations==

Multibrot sets Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions :z \mapsto z^d + c. For an integer d, 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 recursion z \mapsto z^3 + 3kz + c, whose two critical points are the complex square roots of the parameter k. A parameter is in the cubic connectedness locus if both critical points are stable. For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus. The Multibrot set is obtained by varying the value of the exponent d. The article has a video that shows the development from d = 0 to 7, at which point there are 6 i.e. (d-1) lobes around the perimeter. In general, when d is a positive integer, the central region in each of these sets is always an epicycloid of (d-1) cusps. A similar development with negative integral exponents results in (1-d) clefts on the inside of a ring, where the main central region of the set is a hypocycloid of (1-d) cusps. Higher dimensions 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, the quaternions, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions. Taking a 3-dimensional cross section at d = 0\ (q = a + bi +cj + dk) results in a solid of revolution of the 2-dimensional Mandelbrot set around the real axis. Other non-analytic mappings The tricorn fractal, also called the Mandelbar set, is the connectedness locus of the anti-holomorphic family z \mapsto \bar{z}^2 + c. It was encountered by Milnor in his study of parameter slices of real cubic polynomials. It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials. Another non-analytic generalization is the Burning Ship fractal, which is obtained by iterating the following: :z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c. ==Computer drawings==

There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the naïve "escape time algorithm" will be shown, since it is the most popular and one of the simplest algorithms. In the escape time algorithm, a repeating calculation is performed for each x, y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel. The x and y locations 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 next x, y point 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 of pixels. To color any such pixel, let c be the midpoint of that pixel. Iterate the critical point 0 under f_c, checking at each step whether the orbit point has a radius larger than 2. When this is the case, 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. In pseudocode, 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 a complex data type. The program may be simplified if the programming language includes complex-data-type operations. 0 y0), not (x,y). --> for each pixel (Px, Py) on the screen do x0 := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.00, 0.47)) y0 := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1.12, 1.12)) x := 0.0 y := 0.0 iteration := 0 max_iteration := 1000 while (x^2 + y^2 ≤ 2^2 AND iteration color := palette[iteration] plot(Px, Py, color) Here, relating the pseudocode to c, z and f_c: • z = x + iy • z^2 = x^2 +i2xy - y^2 • c = x_0 + i y_0 and so, as can be seen in the pseudocode in the computation of x and y: • x = \mathop{\mathrm{Re}} \left(z^2+c \right) = x^2-y^2 + x_0 and y = \mathop{\mathrm{Im}} \left(z^2+c \right) = 2xy + y_0. 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.). Python code Here is the code implementing the above algorithm in Python: import numpy as np import matplotlib.pyplot as plt • Setting parameters (these values can be changed) x_domain, y_domain = np.linspace(-2, 2, 500), np.linspace(-2, 2, 500) bound = 2 max_iterations = 50 # any positive integer value colormap = "nipy_spectral" # set to any matplotlib valid colormap func = lambda z, p, c: z**p + c • Computing 2D array to represent the Mandelbrot set iteration_array = [] for y in y_domain: row = [] for x in x_domain: z = 0 p = 2 c = complex(x, y) for iteration_number in range(max_iterations): if abs(z) >= bound: row.append(iteration_number) break else: try: z = func(z, p, c) except (ValueError, ZeroDivisionError): z = c else: row.append(0) iteration_array.append(row) • Plotting the data ax = plt.axes() ax.set_aspect("equal") graph = ax.pcolormesh(x_domain, y_domain, iteration_array, cmap=colormap) plt.colorbar(graph) plt.xlabel("Real-Axis") plt.ylabel("Imaginary-Axis") plt.show() The value of power variable can be modified to generate an image of equivalent multibrot set (z = z^{\text{power}}+c). For example, setting p = 2 produces the associated image. ==See also==