This website is a gallery of computer-generated fractal art as well as a text to explain what it is and how it is created. Gallery 2D mainly shows two-dimensional (2D) images given by the basic programming routines described in Stories about Fractal Plotting, while Gallery 3D comprises images given by a variety of techniques based on college mathematics. The latter includes such 3D objects as fractal mountains and fractal forests and fractals painted on various nonplanar surfaces. Here are several examples:

  Symmetric Rocks in Desert

 Pyramid Escher-like Fern Mountains 

 Newton's Apple Seahorse Egg Broken Taiko Drum 

 Copper Bowl Antique Vase  

  Tamarack Forest

Mandelbrot Moon  
Copyright © 1997-2022 Junpei Sekino 
The website was last updated on January 20, 2022 

Digital Artist (Author's Profile): When Junpei Sekino was 10 years old he won first prize for the junior division in a national printmaking contest in Japan.
He now combines art and mathematics to create fractal art. ...from MathThematics, Book 3, Houghton Mifflin, 1998, 2008.
Stories about Fractal Plotting

Contents   Preparations: Canvases, Dynamical Systems and Iterations
The Divergence Scheme The Mandelbrot Set The Convergence Scheme
Mandelbrot Fractals Julia Fractals and Julia Sets Newton Fractals

Supplement  Fractal Coloring
Galleries  Gallery 2D Gallery 3D

Speaking loosely without using technical terms such as the Hausdorff-Besicovitch dimension, a fractal is an object that is self-similar, i.e., a large part of it contains smaller parts that resemble the large part in some way; see Figures 0.1-0.6 below. Mathematician Benoit Mandelbrot coined the term “fractal” in 1975 and created a branch of mathematics called fractal geometry seven years later. As an "IBM Fellow," he had access to some of the best computers available for his research at the time.

Our world has fractals everywhere exemplified by trees, mountains, blood vessels, mycelium strands, stock market graphs, weather patterns, seismic rhythms, ECG signals and brain waves. In its article entitled "How Mandelbrot's fractals changed the world," the BBC states that fractal geometry has practical applications in diverse areas including diagnosing some diseases, computer file compression systems and the architecture of the networks that make up the Internet.
 Figure 0.1. The Mandelbrot Set Figure 0.2. Julia Set Figure 0.3. Newton Fractal 

The idea of fractal was not particularly new in mathematics for Mandelbrot's time or the computer age, as Georg Cantor introduced the prototypical "Cantor Set" in 1883 almost 100 years before Mandelbrot published his book entitled "The Fractal Geometry of Nature." During the early 20th century, Pierre Fatou and Gaston Julia laid the foundations for fractals generated by "dynamical systems." It was in 1980 when Mandelbrot showed the famous fractal called the "Mandelbrot set" generated by a simple dynamical system and a computer. Almost immediately after that, the novelty and complexity of the Mandelbrot set reinvigorated the interests in fractals and stimulated mathematicians to develop further theories in fractal geometry.

Figure 0.4. Classical Fractals by Geometric Constructions

Koch Snowflake, 1904 Sierpinski Rectangle, 1916 Pythagorean Tree, 1942

On the other hand, chaos often associated with fractals, was basically born as a brand new subject in 1974 from biologist Robert May's computer simulations of population dynamics through the dynamical system called the
logistic equation. Like "fractal," the word "chaos" was used as a mathematical term for the first time in 1975 when the American Mathematical Monthly published "Period Three Implies Chaos" by T.Y. Li and James Yorke. The paper received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes. It soon became known that fractals and chaos are closely related and together they provide applications in sciences as well as in art.

NOVA broadcast on PBS in 2008: Largely because of its haunting beauty, the Mandelbrot set has become the most famous object in modern mathematics. It is also the breeding ground for the world's most famous fractals. Since 1980, the set has provided an inspiration for artists, a source of wonder for schoolchildren, and a fertile testing ground for the science of linear dynamics.

Figure 0.5.  From the Boundary of the Mandelbrot Set

Google, we find numerous websites that display stunningly beautiful computer-generated fractal art images. It indicates that a large population not only appreciates the digital art form but also participates in the eye-opening creative activity. Written below is a guide on how to program a computer and plot popular types of fractals generated by simple dynamical systems. It is not a text on computer programming or coding. Instead it tells the general principles needed for fractal plotting without going into too many specifics. It assumes the readers' basic programming experience and encourages them to be creative and engage in frequent computer experiments based on the essentials.

Particularly exciting is the moment the fractal image generated by our personal program emerges in our computer screen because of its rather unpredictable nature. The reader who may be merely intrigued by the general idea behind fractal plotting is encouraged to try it. Many of the images will stir our imaginations in the part of mathematics that is in fact quite deep and still filled with unknowns. Best of all, though, it is plain fun.

§ 1.  Preparations

The readers who want to try it out need to know (a) fundamental algebra and geometry of complex numbers, (b) beginning calculus, and (c) basic computer programming in such languages as C++, Delphi (Pascal) and Java.

(a) includes the practice of writing a complex number z as a point (x, y) in the xy-plane as well as the standard algebraic expression z = x + yi and ability to do basic arithmetic of complex numbers such as multiplication, division and exponentiation. The complex plane means the set of all complex numbers z = (x, y) which coincides with the Cartesian xy-plane. For each complex number z = (x, y), the absolute value of z means |z| = √(x2 + y2) and it represents the distance of z from the origin O of the complex plane. More generally, if z and w are complex numbers, |z - w| represents the distance between the complex numbers.

(b) includes the basic ideas about the derivative and a critical point of a function where the derivative vanishes. Particularly important in (c) is a two-dimensional (2D) array of real numbers (usually nonnegative integers), which will be used every time a fractal is generated.

We now introduce several preliminary ideas needed for fractal plotting.

Dynamical Systems and Iterations: When we solve a mathematical problem using a computer, we usually do it by exploiting what the machine does best, namely an iteration, which means repeating a certain process over and over, often for thousands or even millions of times, at a blinding speed. As an example of iterations, consider the equation called the Mandelbrot equation

(1.1)     zn+1 = zn2 + p ,

which involves two indexed variables zn+1 and zn and the third variable p called a parameter. All variables vary through complex numbers. The iteration index n is especially important for fractal plotting and it is there for us to iterate the equation to generate a sequence of complex numbers once the value of p and initial value z0 are given. For instance, let p = -2 and z0 = 0. Then setting the index n = 0, 1, 2, · · ·in (1.1), our properly programmed computer iterates (1.1) and calculates the sequence of numbers

     z1 = z02 + p = 02 - 2 = -2 ,
     z2 = z12 + p = (-2)2 - 2 = 2 ,
     z3 = z22 + p = 22 - 2 = 2 ,

and similarly,   zn = 2  for n = 4, 5, 6, · · · .  If we hold the value of z0 at z0 = 0 and change the value of p from p = -2 to p = -1.9 in (1.1) then the computer again iterates (1.1) and calculates thousands of terms within a fraction of a second to give us the sequence of numbers

     z0 = 0,  z1 = -1.9,  z2 = 1.71,  z3 = 1.0241, · · · ,  z30 = -1.1626,  z31 = -0.5483, · · · .

Similarly, if we leave the value of p fixed at p = -2 and change the value of z0 from z0 = 0 to z0 = 0.1, we get

     z0 = 0.1,  z1 = -1.99,  z2 = -1.9601,  z3 = 1.842, · · · ,  z30 = 0.7157,  z31 = -1.4877, · · · .

Note that the behavior of the sequence may change drastically if we alter the value of p or z0 slightly. We exploit such changes to draw an intricate fractal with a variety of colors.

We have shown only real sequences for simplicity, but actual numbers involved in fractal plotting are complex numbers in the complex plane. Thus, the Mandelbrot equation (1.1) comprises infinitely many sequences of complex numbers, one sequence zn for each choice of values of p and z0. Because most of the infinitely many sequences dance around in the complex plane as the iteration index n increases, it is appropriate to call the Mandelbrot equation a dynamic mathematical system or dynamical system. As we shall see in § 5, there are infinitely many dynamical systems including (1.1) and the
logistic equation (5.3), each of which generates infinitely many fractals.

Figure 1.1.  Fractal Generated by the Logistic Equation

"Circus Elephants"

Canvases: We begin with a simple example. Let R be the rectangle in the complex plane defined by -2 ≤ x ≤ 2 and -1.28 ≤ y ≤ 1.28 and suppose we wish to plot the graph of the inequality x2 + y2 ≤ 1 on R using a computer. We first decompose R into, say, 50 × 32 miniature rectangles of equal size called picture elements or pixels and then represent the pixels by pixel coordinates (i, j) in such a way that the upper left and lower right pixels are (0, 0) and (49, 31), respectively. Thus, the i- and j-axes of the pixel coordinate system are the rays emanating from the upper left corner of R and pointing east and south, respectively; see the diagram in Figure 1.2 on the left.

Let imax = 50, jmax = 32, xmin = -2, xmax = 2, ymin = -1.28 and ymax = 1.28. Then for each i = 0, 1, 2, · · ·, imax-1 and j = 0, 1, 2, · · ·, jmax-1, the pixel (i, j), which is a rectangle, contains infinitely many complex numbers (x, y). For our computational purpose, we choose exactly one representative complex number (x, y) in the pixel (i, j) by setting

(1.2)    Δx = (xmax - xmin) / imax; Δy = (ymax - ymin) / jmax,

(1.3)    x = xmin + i Δx;  y = ymax - j Δy.

Consequently, we may view R as the rectangle comprising imax x jmax = 50 x 32 pixels, each of which has a unique representative complex number. The rectangle R with the pixel structure is called a canvas for plotting the output image with the image resolution of 50 × 32 pixels.

Plotting the graph of the inequality x2 + y2 ≤ 1 on the canvas is now easy. For each pixel (i, j), we examime its representative complex number (x, y) on the canvas R. If it satisfies the inequality, color the pixel red, and otherwise, color it white. Since the coloring process uses only finitely many pixels of the canvas R, the output image that resembles the Japanese flag is an approximation of the true graph. The greater the number of pixels, the higher the image resolution and the more accurate the output image.

Figure 1.2 shows two approximations of a fractal called "Goldfish in Love." The one on the left is painted on a canvas with 50 × 32 pixels and the other on a canvas with 500 × 320 pixels.

Figure 1.2. Fractal with Different Image Resolutions

Technically, we can define a canvas using any positive integers imax and jmax and any real numbers xmin, xmax, ymin and ymax with  xmin < xmax and ymin < ymax, but we normally impose the ratio equality

(1.4)    (ymax - ymin)/(xmax - xmin) = jmax/imax

on the input values xmin, xmax, ymin and ymax. Then (1.4) implies that Δx = Δy in (1.2) so each pixel is a square as shown in Figure 1.2 and the red circle in the aforementioned output would not look oval.

p-Canvases, z-Canvases, Orbits, and a Preview of § 2 - § 7:  As we have seen with the Mandelbrot equation (1.1), a dynamical system comprises infinitely many sequences of complex numbers, one for each choice of values of z0 and p. For example, if we fix the value of z0 at, say, z0 = 0, then each value of the parameter p gives rise to a sequence zn of complex numbers, which we call the orbit of p (with the fixed value of z0). "Sequence" and "orbit" generally have a subtle difference in mathematics but we won't distinguish them here for simplicity.

We have also seen that a
canvas is a rectangle in the pixel coordinate system comprising imax × jmax pixels for some positive integers imax and jmax and that each pixel on the canvas is identified with a unique representative complex number belonging to the pixel. Interpreting the complex numbers representing the pixels as values of p, we call the canvas a p-canvas for the dynamical system (with the fixed value of z0).

Each pixel, er parameter p, on the p-canvas corresponds to a unique orbit of p that usually dances around in the complex plane, and as we shall see in § 2 - § 5, we plot a fractal called a "Mandelbrot fractal" on the p-canvas by examining behaviors of the orbits. Figure 1.3 shows a couple of examples.

Figure 1.3. Mandelbrot Fractals Painted on p-Canvases

"Mini Mandelbrot Sets" "Jellyfish Couple"

Now, suppose p is a fixed constant in the dynamical system while z0 is a variable. Then by the symmetric argument, we can talk about a z-canvas comprising pixels whose representative complex numbers are interpreted as values of z0. Each pixel, er complex number z0, on the z-canvas gives rise to a unique sequence zn (with the fixed value of p) called the orbit of z0, and as we shall see in § 6, we plot a fractal called a "Julia fractal" on the z-canvas by picking on certain behaviors of these orbits.

Figure 1.4. Julia Fractal Painted on z-Canvas

"Running Corolla" (pun intended)

There is a special subset of the Julia fractals, consisting of fractals generated in essence by so-called "Newton's rootfinding algorithm." The fractals in the subset are called "Newton fractals," which we will discuss in § 7.

Figure 1.5.  Newton Fractal Painted on z-Canvas

"Barn Owl"

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§ 2.  The Divergence Scheme

We say that a sequence zn of complex numbers diverges to ∞ if the real sequence |zn| diverges to ∞, i.e., if zn gets further away from the origin of the complex plane without bound as n gets larger. Our goal of § 2 is to introduce a fractal plotting technique associated with the notion of divergence of orbits of p using the Mandelbrot equation (1.1) as an example.

Define a function fp of a complex variable z involving a complex parameter p by setting fp(z) = z2 + p and write the Mandelbrot equation as

(2.1)   zn+1 = fp(zn) = zn2 + p.

The derivative of the function fp is fp'(z) = 2 z  so the critical point of fp is z = 0.  Throughout § 2 - § 4, we set

(2.2)   z0 = 0,

which is the critical point, so all orbits of p given by (2.1) are critical orbits, the orbits having the critical point as their initial values z0.

Because of the squaring in (2.1), a whole lot of the (critical) orbits of p appear to diverge to ∞, so we are interested in the region in the complex plane comprising the parameters whose orbits do not diverge to ∞. One of the advantages in using the critical orbits (more of which will be seen in § 6) is that it allows us to prove the following theorem that turned out to be extremely useful.

The Divergence Criterion:|zm| > 2  for some m if and only if the orbit zn of p diverges to ∞.

There are a couple of immediate corollaries to the theorem: First, the divergence criterion remains true if we replace 2 by any real number θ ≥ 2.  Second, if |p| > 2, then |z1| = |p| > 2 by (2.1), and consequently, the orbit zn of p diverges to ∞; hence the region in the complex plane comprising the parameters whose orbits do not diverge to ∞ lies in the circle of radius 2 about the origin.

Suppose R is the square
canvas bounded by xmin = -2, xmax = 2, ymin = -2 and ymax = 2 containing the circle of radius 2 and comprising 2,000 × 2,000 = 4,000,000 pixels. Regard R as a p-canvas so each pixel (i. j) in the canvas is identified with a unique representative parameter p belonging to it.

Let M = 1000 and θ = 2 and for every pixel (i. j), er parameter p, in the p-canvas R, iterate (2.1) at most M times and compute the nonnegative integer

(2.3)  d(i, j) = the least (or first) iteration index m < M such that |zm| > θ,

if such m exists, and d(i, j) = M, otherwise. Here, M is the maximum number of iterations which is intended for thwarting the computer to get trapped in an infinite loop. Also, in actual computation, we use |zm|2 > θ2 instead of |zm| > θ to avoid the "hidden" square root operation so as to shorten the computing time. Thus, each d(i, j) represents, unless d(i, j) = M, the smallest number of iterations m or "time" it takes for the orbit zn of z0 to escape from the circle of radius θ before taking a long journey toward ∞.

The formula (2.3) generates an array d(i, j) of the least iteration indices, or an iteration array for short, over the canvas R, which can be converted to a visual image on the canvas. For example, if we paint the whole canvas white initially and then color the pixels (i, j) red if d(i, j) < M and is even, and color it black if d(i, j) < M and is odd, we get
Figure 0.1. The image in Figure 2.1 on the left shows a cropped and resized version of the figure. The image on the right is similarly generated except that it uses θ = 10 instead of θ = 2 in (2.3).

Figure 2.1. The Mandelbrot Set

zm > θ, θ = 2 zm > θ, θ = 10

The Mandelbrot Set:  By definition, the famous Mandelbrot set is the set of all complex parameters p in the complex plane whose critical orbits do not diverge to ∞, or equivalently, the set of all parameters p whose critical orbits stay within the circle of radius 2 forever. The "snowman" of Figure 2.1 on the left, which is left untouched by the above red-black painting scheme and retains the initial white canvas color, depicts an approximation of the Mandelbrot set on the p-canvas given by replacing "forever" by "up until M = 1000 ".

The Divergence Scheme:  We call the aforementioned process of computing an iteration array d(i, j) over a canvas R comprising pixels (i, j) the divergence scheme for fractal plotting. In § 4, we'll see iteration arrays generated by what we call the "convergence scheme" that deals with the orbits that "converge" to various "cycles." These iteration arrays can be converted to colorful fractals by our
fractal coloring techniques.

Zooming in on Local Images:  Figure 2.1 shows that the red-black pattern in either image keeps getting more complex without bound as we keep zooming in on an area nearer the boundary of the Mandelbrot set and provides us with a valuable clue as to how to find attractive fractal images. We often call an image such as the ones shown above a global image of the Mandelbrot set so as to contrast it with a local image captured by zooming in on a small neighborhood around the Mandelbrot set.

Example 1:  The image shown below on the left is a local image of the Mandelbrot set given by zooming in on a small rectangular neighborhood of the point (-0.6884971875, 0.27988465625), which is just outside of the Mandelbrot set but very near its boundary. It is given by the divergence scheme followed by a fairly simple coloring scheme.

People who are familiar with multivariable calculus may extend the project of plotting a fractal on a plane to another surface such as a sphere.

Figure 2.2. A Local Image of the Mandelbrot Set Generated by the Divergence Scheme

The Mandelbrot set has an amazing property that allows us to zoom in further on a part of a fractal image to find another image. During the zooming process, we can easily find the pixel coordinates (i, j) of the point we are interested in and convert it to the Cartesian coordinates (x, y) using the
conversion formulas and a computer. So, a point that looks like (-0.6884971875, 0.27988465625) should not intimidate the readers.

Example 2:  The image shown below is a cropped and resized image from a figure on a large p-canvas with 6,400 × 3,200 pixels centered at the complex number (0.2820607, 0.011014375) with the horizontal radius 0.0000011. The part of the Mandelbrot set shown in the image is painted black and the exterior of the Mandelbrot set in multiple colors using the techniques described in
Fractal Coloring. Note that the figure shows several replicas of the "snowman," which we call mini Mandelbrot sets, painted black that look like small isolated islands.

We used the large canvas and M = 100,000 as the maximum number of iterations to generate the image so as to get a possibility for zooming in further and capturing an additional image like Figure 0.6. A large canvas also provides an option for making a high resolution printout of the image.

Figure 2.3.  Another Local Image of the Mandelbrot Set

Threshold Values:  We call θ ≥ 2 that appears in (2.3) a threshold value for the orbit to begin its journey toward ∞.
Figure 2.1 shows that when θ is greater, the red-black pattern is more detailed and it is likely to produce better "local" images as well as a better approximation for the Mandelbrot set. On the other hand, it is intuitively clear that the computing time is longer when θ is greater, and it explains why Mandelbrot used the smallest threshold value available, namely θ = 2, when the computers were incomparably slower than today's counterparts.

There is another merit in using a larger threshold value. When we use a different dynamical system or a noncritical point for the initial value to explore wider fractal plotting possibilities, we often don't know what threshold value to use in the divergence scheme and a large threshold value is likely to provide a solution for it. But be careful: increasing the size of a threshold value quickly reaches a point of diminishing returns like increasing the maximum number of iterations for the accuracy of the approximation.


§ 3.  The Mandelbrot Set and Its Complexity

The Mandelbrot set is famous for a reason and it turned out to be one of the most complex figures ever plotted on a plane. Although it may not sound obvious unless we know something about
fractal dimensions, the following celebrated theorem guarantees that no figures on the plane are more complex than the Mandelbrot set.

Shishikura's Theorem: The fractal dimension of the boundary of the Mandelbrot set is the same as the dimension of the plane, namely 2.

But we wonder. Take the image in Figure 2.3, for example. Its colorful and intricate patterns surely look impressive, but does it have the maximum complexity implied by the theorem? We know that the boundary of the Mandelbrot set is somehow responsible for the striking patterns but it is colored black and mostly invisible in Figure 2.3.

It turned out that we can actually see what the boundary of the Mandelbrot set is like by lighting up its razor-thin filaments:

Figure 3.1.  The Boundary of the Mandelbrot Set in Figure 2.3

It shows that the Mandelbrot set in the rectangular area is vividly self-similar and composed of replicas of a large part of the image. Through the self-similarity, we can see it contains infinitely many mini Mandelbrot sets painted black, although most of them are too small to be visible. The luminous boundary of the Mandelbrot set appears to be an elaborate network connecting the mini Mandelbrot sets, and its filaments get so dense near them, it fills infinitely many areas of the plane like a space-filling curve. The observation provides us with an intuitive idea as to why the "fractal dimension" of the boundary of the Mandelbrot set is equal to the topological dimension of the plane and why it is equated with the complexity of the Mandelbrot set.

The aforementioned self-similarity in Figure 3.1 seems to show that the Mandelbrot set with its complex boundary is, quite surprisingly, connected as one piece, and it turns the complement (outside) of the Mandelbrot set into an extremely convoluted maze. If we are shrunk to a pixel size and trapped in the maze painted in darker colors (not including black), can we get out of the maze? The following important theorem in fractal geometry answers these questions definitively. Here, a set is "simply connected" if it has no loop to trap anybody in it.

The Douady-Hubbard Theorem: The Mandelbrot set is connected and simply connected.

According to Wikipedia, Mandelbrot initially thought (but later conjectured otherwise) that the Mandelbrot set was disconnected as he could not detect the thin filaments connecting different parts of the set in his computer-generated images. It shows the delicate nature of the boundary of the Mandelbrot set and at the same time warns us not to rely too much on computer-generated images, which generally provide us with powerful research tools.

Daytime and Nighttime Fractals:  Shown below on the left is another fractal generated by the Mandelbrot equation and the divergence scheme, where the Mandelbrot set is colored black and invisible. When its boundary's thin filaments are turned on, we get the nighttime image on the right, vividly showing the presence of the Mandelbrot set in the daytime image on the left.

Figure 3.2.  Daytime and Nighttime Fractals

Figure 3.2 again shows that fractals given by the Mandelbrot equation we normally see are daytime images like the image on the left and painted on the complement of the Mandelbrot set. The nighttime images are not as colorful but they make it easier for us to visualize the two aforementioned theorems about the Mandelbrot set.

Plotting the complex boundary of the Mandelbrot set with reasonable accuracy may demand weeks of computing time even with a fast modern computer. Figure 3.3 shown below is a resized image from a fractal on the p-canvas with 4,000 × 4,000 pixels centered at the point

p = (0.25000316374967, -0.00000000895902)

with a microscopically small radius ≈ 0.0000000000001 = 10-13.  We note that p is very near the cusp (0.25, 0) of the cardioid in
Figure 2.1.

Figure 3.3.  A Mini Mandelbrot Set under the Microscope

M = 1,500,000 M = 500,000

For the above image on the left, we used whopping 1,500,000 iterations of the Mandelbrot equation for each black pixel because of ∞ involved in the definition of the Mandelbrot set. If we use M = 500,000 (still a large number) instead, the outline of the mini Mandelbrot set becomes blurry as shown in the above picture on the right. Fortunately, computers (especially used ones) are inexpensive nowadays and we can easily afford a second or third computer to do tedious jobs. Programming carefully so as to minimize computing time is not as important as it used to be.

Shown below is a nighttime fractal that reveals the boundary of the mini Mandelbrot set of Figure 3.3. The Mandelbrot set certainly appears to be connected in the image but does it look simply connected?

Figure 3.4.  The Boundary of the Mini Mandelbrot Set

Errors in Computation:  Aside from programming bugs and other human errors, fractal plotting entails two unavoidable errors, each of which contributes to a loss of mathematical precision in the output. One is the truncation error resulting from "truncating" the infinite process after the finite number of steps given by M and the other is the roundoff error caused by our "imperfect" computer that has to round almost every real number involved. For the images in this website, M is usually between 500 and 100,000 but it occasionally gets as large as 1,500,000 as shown in Figure 3.3. Using larger M is better in theory as it reduces the truncation error but is worse in practice as it makes the computing time longer and at the same time causes more roundoff errors to propagate. Balancing the good and the bad to find an optimum number M is a difficult problem in computing. Although it is not an issue if our goal is in art, it is something we should keep in our mind. Computers are great tools for mathematical research but they mislead us from time to time.

§ 4.  The Convergence Scheme

We are not done yet with the complexity of the Mandelbrot set and still stay with it. The Mandelbrot set has become so illustrious, everybody with at least some interest in fractals knows its "snowman" shape in
Figure 2.1 by heart: The main body is in the form of a heart-shaped "cardioid" with a bunch of circular disks attached, and to each of the disks another bunch of disks are attached. The pattern repeats as if the cardioid has children, grandchildren, great grandchildren and so on and so forth, but beyond that nobody knows exactly what's happening. Here, the cardioid means the bounding curve together with its interior.

As we have seen, the divergence scheme paints the interiors of the cardioid and all of the disks in a single color like white as it is incapable of distinguishing them. Our current goal is to find their distinguished mathematical properties and paint them in various colors, like Figure 4.1 below, by developing a scheme that is different from the divergence scheme.

Figure 4.1. The Interior and Boundary of the Mandelbrot Set

A sequence cn of complex numbers is called a cycle if there is a positive integer k satisfying cn = cn+k for any index n. The smallest such integer k is called the period of the cycle, and a cycle with period k is called a k-cycle for short. For example, the sequence

    1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, · · ·

is a 3-cycle but not a 6-cycle or a 9-cycle. The sequence  0, 0, 0, 0, · · ·  is a 1-cycle, which we identify with the constant 0.

A sequence zn is said to converge to a k-cycle cn if it gets closer and closer to cn or behaves more and more like cn as n gets bigger.  If a sequence converges to a k-cycle, we also say that the sequence is attracted to the k-cycle.

Example 1: The sequences  1/2, 1/3, 1/4, 1/5, 1/6, · · ·  and  1/2, 2/3, 3/4, 4/5, 5/6, · · ·  converge to the constants 0 and 1, or equivalently, to the 1-cycles 0, 0, 0, 0, · · ·  and  1, 1, 1, 1, · · ·,  respectively. Therefore, the sequence

    1/2, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 4/5, 1/6, 5/6, · · ·, 1/1000, 999/1000, 1/1001, 1000/1001, · · ·

converges to the 2-cycle 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, · · ·, 0, 1, 0, 1, · · ·.

Suppose ε is a small real number and zn the orbit of a parameter p as in § 2. We note that if zn gets arbitrarily close to a k-cycle cn as n gets bigger, then because |cn+k - cn| = 0, we surely have |zn+k - zn| < ε for large n. Thus, we have:

Proposition: If the orbit zn converges to a k-cycle then |zn+k - zn| < ε for sufficiently large n.

The proposition provides us with our convergence scheme with period index k, which is given by replacing |zm| > θ of the
divergence scheme by the inequality |zm+k - zm| < ε.  Thus, the convergence scheme generates the iteration array

(4.1)  d(i, j) = the least (or first) iteration index m < M such that |zm+k - zm| < ε

over the canvas R comprising pixels (i, j), which in turn can be converted to a visual image by fractal coloring. In (4.1), zn is the orbit of p which is the parameter representing the pixel (i, j), M is a maximum number of iterations (just like in the divergence scheme) and ε is a small positive real number such as 10-6 or min{Δx, Δy}; see (1.2). There are no set formulas for M and ε so we find them by educated guesses through our experience and computer experiments.

The convergence scheme with period index k is intended not only to compute the iteration array d(i, j) but also to detect every parameter whose orbit converges to a k-cycle. We will talk about "a parameter whose orbit converges to a k-cycle" frequently, so let's call it a parameter of period k for short. In particular, a parameter of period 1 is the same as a parameter whose orbit converges to a single point (1-cycle).

Figure 4.2.  The Mandelbrot Set by the Divergence and Convergence Schemes

The convergence scheme may mean the convergence scheme with a single period index or the scheme with multiple period indices as the following example shows.

Example 2: The images in Figure 4.2 are given by applying the divergence scheme and the convergence scheme with the period indices k = 1, 2, · · ·, 96 to the Mandelbrot equation. The part of the convergence scheme that deals with period index k = 1 detects that the interior of the cardioid of the Mandelbrot set comprises parameters of period 1 and paints it in various shades of red, and the part with period index k = 2 detects that the interior of the largest disk comprises parameters of period 2 and paints it in orangish colors. The repetitive use of the convergence scheme up to k = 96 with various colors generates the image on the right.

Figure 4.3.  Closeups of the Mandelbrot set by the Convergence Scheme

Note that the small replicas of the snowman that look like isolated islands are actually connected by networks belonging to the boundary of the Mandelbrot set; see
Figure 4.1.

Periodicity Diagram: If the interior of, say, a disk in the Mandelbrot set comprises a parameter of period k, let's say the disk has period k. For example, we have seen that the largest disk has period 2. If we label the cardioid and disks of the Mandelbrot set with its periods 1, 2, etc., instead of coloring red and orange, etc., we get Wikipedia's Periodicity Diagram instead of the colorful image shown above. The periods in the diagram have interesting numerical patterns that are easy to recognize and will play an important role when we plot colorful "Julia fractals" in § 6. The numerical patterns are yet another amazing property of the Mandelbrot set.

Weakness of the Convergence Scheme: The convergence scheme is based on the aforementioned proposition whose converse is unfortunately not necessarily true. For example, the convergence scheme with k = 1 may be fooled by an orbit that diverges to ∞ very slowly and treat it like a convergent orbit. Also, the convergence scheme with, say, period index 6 cannot distinguish parameters of periods 1, 2, 3 and 6 and we can locate parameters of period 6 only after identifying all parameters of periods 1, 2, 3. For most of the practical problems, however, we can overcome these shortcomings by carefully programming a computer to do the right thing.

Brute Force Algorithm: Because of the aforementioned weakness, we can compute the iteration array for an image like Figure 4.2 on the right only by the time-consuming brute force algorithm that goes like this: For each pixel (er parameter) on the canvas, check if its orbit diverges using the divergence scheme; if it does, compute d(i, j), but if not, check if it has period 1 using the convergence scheme; if it does, compute d(i, j), but if not, check if it has period 2, etc. etc. Repeat the process until it comes to the last period index k like k = 96 for the image of Figure 4.2 on the right. The algorithm is simple and straightforward, but it is often called an "exponential time algorithm" as its computing time grows almost exponentially with k.

The Eyeball Effect: Weaknesses and shortcomings are not all that bad and sometimes they can give us unexpected artistic effects. Even a programming bug can give us beautiful output sometimes. The picture below on the left is essentially the same as
Figure 3.3 and is given by the divergence scheme alone while the one on the right is painted by the divergence scheme (with different colors) and the convergence scheme with period index k = 1. The "eyeballs" painted by the convergence scheme are caused by its weakness of confusing some of the slowly divergent orbits as convergent. The images show which parameters are affected. Figure 6.5 illustrates the "eyeballs" more vividly.

Figure 4.4. The Eyeball Effect (Right) Given by the Convergence Scheme

§ 5.  Mandelbrot Fractals

Since it was published in 1980, the Mandelbrot set became so popular that a great many digital artists, mathematicians and computer programmers have explored around it and shown their fractal images on a variety of objects including webpages, posters, book covers, T-shirts, and coffee mugs. Although the complexity of the Mandelbrot set shown on a two-dimensional canvas is boundless and the hidden beauty inexhaustible, it has become quite a challenge to unearth new patterns from the Mandelbrot equation (2.1) using available computers and software. Consequently, a creative work calls for a modification of the formula, and there are infinitely many formulas available for it.

We call a fractal given by a dynamical system of the form

(5.1)   zn+1 = fp(zn)

a Mandelbrot fractal if it is painted on a p-canvas, where fp is any nonlinear elementary function of a complex variable containing a parameter p. The initial value for the orbits is often a critical point of fp as in the case of the Mandelbrot set but it is not a requirement. There are functions fp without convenient critical points, and even if fp has one, a noncritical point often produces interesting fractals from fp. We always have the advantage of having a fast computer that allows us to satisfy our curiosity by engaging in a computer experiment.

Figure 5.1. The Mandelbrot Set Shot by Arrow

The comical image shown above is a Mandelbrot fractal given by

(5.2)   zn+1 = fp(zn) = zn3 + zn + p

with z0 = i / √3 which is a critical point of fp. The tip of the arrowhead is at the origin (0, 0) of the complex plane and the tiny isolated figure on the right is a mini Mandelbrot set (who shot the arrow). Here, the picture is actually given by rotating the original output 90o counterclockwise to have it better fit in the webpage. In this article, we use rotations and horizontal/vertical flippings of fractal images freely for artistic effects.

The image shown below is a Mandelbrot fractal given by zooming in on the boundary of the mini Mandelbrot set, which can be used as a night sky of 3D landscapes.

"Mandelbrot Island" "Desert Butte"

The Logistic Equation: In 1838 Pierre Verhulst introduced a differential equation called the "logistic equation," which became a widely used mathematical model for population dynamics. If we replace the derivative in the equation by its approximating difference quotient and do some algebra, we get the following formula that is more suitable for computer applications:

(5.3)   zn+1 = fp(zn) = p(1 - zn) zn .

It is again called the logistic equation (or logistic map) and is equally applicable in the population dynamics when the variables are restricted to real numbers.

If we expand its variable and parameter to complex numbers and apply the divergence and convergence schemes on the critical orbits of p with the fixed initial values z0 = 0.5, we get the following Mandelbrot fractal. For convenience, we call the portion without the dark background the logistic set, which is defined just like the Mandelbrot set.

Figure 5.2. The Interior and Boundary of the Logistic Set

The origin (0, 0) of the complex plane coincides with the center of the red disk on the left but not the intersection point of the figure 8, which is the point (1, 0). The real axis of the complex plane is the horizontal line through the two straight antennas of the logistic set and the right-hand antenna coincides with the closed interval [α, 4] on the real axis with α ≈ 3.57.

Like the cubic dynamical system (5.2), we can find numerous mini Mandelbrot sets including the one shown below by zooming in on the boundary of the logistic set, .

Figure 5.3.  Daytime and Nighttime Fractals by the Logistic Equation

The image shown below on the left, "Partying Elephants," is given by zooming in on a small neighborhood in the nighttime fractal in Figure 5.3 and the image on the right, "Flock of Owls," by a rotation of the "elephant." If we zoom in on the midpoint between the two eyes of any "owl," we find a mini Mandelbrot set.

"Partying Elephants" and "Flock of Owls"

In 1974, while conducting a computer simulation of certain population changes, biologist Robert May discovered "very complicated orbits" of z0 given by the logistic equation and p belonging to the aforementioned closed interval [α, 4]. It led us to the concept of chaos, which began to develop in 1975, the year in which Mandelbrot coincidentally coined the term fractal. In 1993, a chaotician appeared in Steven Spielberg's hit movie, "Jurassic Park," tacitly suggesting a possibility of chaos in the controlled dinosaur populations.

So, it is natural that we plot Mandelbrot fractals of the dynamical system (5.3) by zooming in on the interval [α, 4]. Figure 5.3 is one of them and uses a noncritical point z0 = 0.1 (10% of the sustainable population of, say, an insect species) as the fixed initial value for the orbits of p.

Figure 5.4.  "Spring Reflection"

By the way, people familiar with multivariable calculus might find a fun project in mapping an image like Figure 5.4 on surfaces like the ones shown below. We can see more examples of this sort in
Gallery 3D.

Critical Points vs Noncritical Point in Fractal Art:  As we have seen (and as we'll again see in § 6), a critical point of a dynamical system plays an important role in fractal geometry. In fractal art however, use of a noncritical point often provides interesting effects like deforming fractal images given by a critical point. For example, Figure 5.5 is a fractal given by the logistic equation and its critical point z0 = 0.5 on a small rectangular area with center (3.45028095, 0.01014955), while Figure 5.6 is given by the noncritical point z0 = 0.1 on a nearby vicinity. The noncritical point simplifies the complex boundary of the logistic set shown in Figure 5.5 which appears to be connected and breaks it down to disconnected fragments as shown in Figure 5.6.
Gallery 2D shows more colorful examples of Mandelbrot fractals generated by the logistic equation and noncritical points as the initial values.

Figure 5.5. "Fractal Swordfish" with Critical Point z0 = 0.5

Figure 5.6. "Fractal Swordfish" with Noncritical Point z0 = 0.1

Change of Variables:  Interestingly, the logistic equation (5.3) can be transformed into a special form of the Mandelbrot equation
(2.1) by a simple change of variables. To see it, we first copy the Mandelbrot equation (2.1) and the logistic equation (5.3) as

(5.4)   zn+1 = zn2 + p;

(5.5)  ζn+1 = q(1 - ζn) ζn .

Suppose   zn = a ζn + b for some constants a and b and write (5.4) as

   a ζn+1 + b = (a ζn + b)2 + p

Multiplying out the right-hand side and solving the equation for ζn+1 , we get

   ζn+1 = - a ζn(-2b/a  -   ζn) + (b2 - b + p)/a.

Comparing the last line with (5.5), we get  -a = q,   -2b/a = 1   and  b2 - b + p = 0,  which implies

(5.6)   p = q(2 - q)/4 .

Therefore, we can write the Mandelbrot equation (5.4) as

(5.7)   zn+1 = zn2 + q(2 - q)/4 ,

which is another way of writing the logistic equation (5.5) with parameter q. It means that we can get all fractals given by the logistic equation including the
logistic set of Figures 5.2 by the special form of the Mandelbrot equation (5.7) on q-canvases—but not by p-canvases and the Mandelbrot equation (5.4). The situation gets more interesting in the next section when we discuss "Julia fractals" painted on z-canvases which do not distinguish (5.7) and (5.4).

The transformation by the change of variables actually goes both ways: If we solve the quadratic equation (5.6) for q, we get

(5.8)   q = 1 ± √ (1 - 4 p),

so the Mandelbrot equation (5.4) with parameter p can be written as a special form of the logistic equation (5.5) with q replaced by the right-hand side of (5.8) (with either of ±). Because of the relation, we often say that the Mandelbrot and logistic equations are conjugates of each other. In fact, we can show, using similar algebra, that the Mandelbrot equation and any quadratic dynamical system with a single parameter are conjugate.

The next three images are Mandelbrot fractals given by the dynamical system shown below, which we call the third degree logistic equation, for convenience:

(5.9)   zn+1 = fp(zn) = p(1 - zn2) zn .

Figure 5.7. The Third Degree Logistic Set

Figure 5.8. "Moray Eels"

Figure 5.7, which is a global image, is given by using the critical point 1/√3 of the function fp as the initial value z0, while the two local images of Figure 5.8 are given by noncritical points 0.1 and 0.5, respectively. The little disks of the global image are deformed and cracked by the noncritical points and give birth to interesting figures like the ones shown in Figure 5.8. These figures often have strong resemblance to "Julia fractals" born from the disks, which we'll discuss in the next section.

§ 6.  Julia Fractals and Julia Sets

Recall that a Mandelbrot fractal is generated by a dynamical system of the form
(5.1) with a fixed initial value z0 and painted on a p-canvas comprising parameters p.  A fractal is called a Julia fractal if it is given by a dynamical system of the same form with a fixed parameter p and painted on a z-canvas comprising initial values z0 instead. It is named after Gaston Julia, who was one of the early pioneers of fractals generated by dynamical systems of complex numbers.

Thus, if we use the divergence scheme to generate a Julia fractal, then for each pixel (i, j) in the canvas R, we use the complex number representing the pixel as an initial value z0 and iterate (5.1) with the fixed parameter p until we get

(6.1)  d(i, j) = the least (or first) iteration index m < M such that |zm| > θ,

if such m exists, and d(i, j) = M, otherwise. As we have seen earlier, M is a maximum number of iterations and θ is a threshold value for the orbit to begin its journey toward ∞. If we use the convergence scheme with period index k, then instead of (6.1), we have

(6.2)  d(i, j) = the least (or first) iteration index m < M such that |zm+k - zm| < ε,

if such m exists, and d(i, j) = M, otherwise. Here, ε is a small real number such as 10-8.  d(i, j) collectively forms an array over the canvas R called an iteration array, which in turn can be converted to a visual fractal by Fractal Coloring or other means.

Let's follow various concepts related to Julia fractals with the familiar Mandelbrot equation:

(6.3)   zn+1 = fp(zn) = zn2 + p.

Example 1: Figure 6.1 shows two nearly equal parameters that are fixed and two Julia fractals given by (6.3) and the respective parameters. The first parameter belongs to the interior of the Mandelbrot set, or more precisely, the interior of a disk of periodicity 11 and the second parameter to the outside of the Mandelbrot set. We call the Julia fractals "Hydra with Eleven Heads" and "Hydra's Ash," respectively.  The green/blue "Hydra" on the left is given by the convergence scheme with period index k = 11 and its background by the divergence scheme. "Hydra's Ash" is given by the divergence scheme alone.

It is another fascinating fact about the Mandelbrot set that the periodicity of a disk where the parameter p resides always has a geometric implication in the Julia fractal such as the number of heads in the figure, but why it is so is not completely understood. If the period is a large composite number rather than a prime and the disk is not directly attached to the "big caridoid" of the Mandelbrot set, the outcome gets really interesting.

Figure 6.1.  "Hydra with Eleven Heads" and "Hydra's Ash"

Julia Sets:  Recall that the
Mandelbrot set is by definition the set of all parameters p in the complex plane whose critical orbits do not diverge to ∞. As we have seen, we can always view the complex plane as the set of parameters p or the set of initial values z0 for the orbits zn with a fixed parameter p.

If p is a fixed parameter, then by the filled-in Julia set of p, we mean the set of all initial values z0 in the complex plane whose orbits zn (with the fixed p) do not diverge to ∞. Thus, the definitions of a filled-in Julia set and the Mandelbrot set are similar, but unlike the Mandelbrot set, there are infinitely many filled-in Julia sets. Like the Mandelbrot set, we can approximate many of the filled-in Julia sets by using the divergence and/or convergence schemes and plotting them on a canvas with finitely many pixels. For example, "Hydra with Eleven Heads" of Figure 6.1 without the reddish background is a computer approximation of a filled-in Jula set.

By the Julia set of p, we mean the boundary of the filled-in Julia set. As the names suggest, the filled-in Julia set introduced earlier plays a supporting role for the Julia set in mathematics, and it is because a Julia set is where "chaos" occurs as well as it is a fractal whose fractal dimension varies and may equal to that of the boundary of the Mandelbrot set. Julia sets are among the most important in fractal geometry, but in fractal art, it is usually filled-in Julia sets we look at and appreciate.

The concept of Julia set naturally extends to a more general dynamical system (5.1), but a lot of things about it are still in mystery and belong to experimental mathematics by the use of computers. Here's one fascinating and useful fact however. The following theorem was established about a hundred years ago before the computer era by Gaston Julia and Pierre Fatou independently and explains why the critical orbits are important.

The Fatou-Julia Theorem:  Consider a dynamical system of the form

(6.4)   zn+1 = fp(zn) =   cm znm +  cm-1 znm-1· · · + c2 zn2 +  c1 zn +  p ,

where m is an integer ≥ 2 and cm, cm-1, · · ·, c2, c1 are complex constants.  Then the Julia set of p is connected if and only if every critical orbit of p stays within a finite bound.

Note that if m > 2 in (6.4), p may have multiple critical orbits as fp may have multiple critical points. In case of the Mandelbrot equation (6.3), each p has a single unique critical orbit; hence the Fatou-Julia theorem is stated beautifully as:

    The Julia set of p is connected if and only if p belongs to the Mandelbrot set.

For the Mandelbrot equation, it is also known that if the Julia set of p is disconnected then it must be a set of "totally disconnected" points called Cantor dust or a Cantor set (named after Georg Cantor, the pioneer of set theory) and cannot be the disjoint union of, say, three connected pieces. For example, because of our choice of the parameters in Figure 6.1, the Julia set (hence the filled-in Julia set) in "Hydra with Eleven Heads" is connected (as one piece) while the Julia set in "Hydra's Ash" is a Cantor set. The boundaries of the "Gold Lions" in Figure 6.2 shown below are also examples of connected Julia sets. Note that the connectedness is not entirely obvious if we rely only on our eyes.

Figure 6.2.  "Gold Lions" of Periods 14 and 42 by the Mandelbrot Equation

Plotting Colorful Filled-In Julia Sets of the Mandelbrot Equation: In the following, we'll plot a few connected filled-in Julia set using parameters p that are inside the Mandelbrot set. Since the Mandelbrot set lies in the circle of radius 2 about the origin in the complex plane, suppose |p| ≤ 2. Then it is not particularly difficult to prove that any θ ≥ 2 works as a threshold value for our divergence scheme (6.1). If we have a computer plot of the Mandelbrot set and its computer program (which is handy for Step 3 described below), we can draw a colorful connected filled-in Julia set as follows:

Step 1: Use Wikipedia's
Periodicity Diagram to choose a disk D of periodicity k of our interest.
Step 2: Identify the disk D in our own computer plot of the Mandelbrot set and choose a pixel (parameter) p in the interior of D.
Step 3: Convert the pixel coordinates p = (i, j) to the Cartesian coordinates p = (x, y) using (1.3).
Step 4: Use the divergence scheme and the convergence scheme with period index k to plot the filled-in Julia set of p on a z-canvas centered at z0 = 0. The divergence scheme takes care of the complement (or the background) of the filled-in Julia set.

Example 2: For the "Gold Lions" of Figure 6.2 shown above, note that there is a disk, call it D14, of period 14 attached to the cardioid near its cusp and a disk D42 of period 42 = 14 × 3 attached to D14 in the Periodicity Diagram.

For the "Gold Lion" on the right, choose
p = (0.296555, 0.020525) in the interior of D42 using Steps 2 and 3 and plot a Julia fractal using Steps 4 and 5. The "Gold Lion" is the filled-in Julia set of p, which is connected as its boundary, the Julia set of p, is connected according to the Fatou-Julia Theorem, and it clearly shows a geometric pattern associated with the numbers 14 and 3. The left "Gold Lion" is given by p = (0.296498, 0.020525) chosen from the interior of D14. Because the two parameters are almost equal, the two figures are similar, but they are not quite "topologically equivalent," as the left "Gold Lion" does not show the number 3 as clearly as the other "Gold Lion." Recall that we use rotations and horizontal/vertical flippings of fractals freely for artistic effects.

Example 3: The "Red Lion," the Julia fractal of p = (0.281215625, 0.0113825) and period 68 = 17 × 4, shown below is similarly generated, where we can again see the numbers 17 and 4 clearly.

Figure 6.3.  "Red Lion" of Period 68 = 17 × 4

The Filled-In Julia Set Complement of the Filled-In Julia Set

The filled-in Julia set and its complement shown above constitute the "Red Lion" and share the Julia set as their common boundary. The former is given by the convergence scheme with period index 68 and the latter by the divergence scheme.

Unfortunately, even after looking at all these images, it is still difficult to fathom how complex the Julia set can be. As in the case for the boundary of the Mandelbrot set, it is generally challenging to plot the razor-thin filaments of a Julia set clearly and efforts have been made in this area; see e.g., Plotting the Julia set by Wikipedia. If we manage to light up its filaments however, we get an image like this:

Figure 6.4.  The Julia Set of "Red Lion" in Figure 6.3

Similarities of Mandelbrot and Julia Fractals: Although the precise reason is unknown except at so-called
Misiurewicz points, we note that in many cases Mandelbrot and Julia fractals from the Mandelbrot equation have striking similarities locally if they are generated by equal parameters.

For example, let p0 = (0.25000316374967, -0.00000000895902). The image on the left in Figure 6.5 shown below is a Mandelbrot fractal given by the Mandelbrot equation (6.3) on a p-canvas centered at the parameter p0 that belongs to the mini Mandelbrot set. It is also depicted in Figure 3.3. The image on the right is a Julia fractal of p0 painted on a z-canvas centered at the origin.

Both images contain infinitely many "cuttlefish" with their eyeballs given by the eyeball effect of the convergence scheme. The divergence scheme painted all other areas. The Mandelbrot set on the left and the filled-in Julia set on the right, both colored black, are connected and their complex (and invisible) boundaries shape the strikingly similar patterns shown in the images. The example shows that a Julia set can be as complex as the boundary of the Mandelbrot set locally; see Figure 3.4.

Figure 6.5.  "Partying Cuttlefish" by the Mandelbrot Equation

Closeup of "Partying Cuttlefish"

Julia Fractals by Other Dynamical Systems:
Figure 0.2 shown at the outset of this website is a Julia fractal generated by the dynamical system (5.2), for which the Fatou-Julia Theorem is applicable. There p = (0.185, 0.00007666), and the gold "Twin Dragons" is painted on a z-canvas centered at the origin by the convergence scheme with period index k = 2. If we alter the value of p and find its period, we get a variety of "Twin Dragons" such as:

Figure 6.6.  "Twin Dragons" by the Dynamical System (5.2)

And here's a Julia set from the logistic equation given by p = (2.994915, 0.1). As the readers probably recognize, the Julia set has a period of just 1 and hence the boundary of the "elephants" is (topologically) among the simplest (and in fact "topologically equivalent" to a circle) even though the fractal appears to be extremely intricate.

Figure 6.7.  "Circus Elephants" by the Logistic Equation

We have shown that the logistic and the Mandelbrot equations are conjugate dynamical systems and as such they have similar Julia fractals. For example, the image shown below on the left is a Julia fractal of the parameter q = (3.02382, 0.1) given by the logistic equation (5.5) and the image on the right is a Julia fractal of the parameter

    p = q(2 - q)/4 ≈ (-0.771462, -0.101191)

given by the Mandelbrot equation through the conjugacy; see (5.6). Although their artistic patterns painted by exactly the same coloring routine came out differently, the filled-in Julia sets and their boundaries are essentially the same mathematically—i.e., "topologically equivalent."

Figure 6.8.  "Dancing Seahorses" by the Logistic and Mandelbrot Equations

We now start with a Julia set given by the Mandelbrot equation and reproduce it using the logistic equation and the conjugacy. The image shown below on the left is a Julia fractal of the parameter p = (-0.6069; 0.4147) given by the Mandelbrot equation and the image on the right is a Julia fractal of the parameter

    q = 1 + √ (1 - 4 p) ≈ (2.902038, -0.436059)

given by the logistic equation (5.5); see (5.8). Again their patterns painted by the same coloring routine are a little different, but they have essentially the same Julia sets. In light of the similarities of Mandelbrot and Julia fractals, therefore, it is not at all surprising if we see analogous Mandelbrot fractals generated by these conjugate dynamical systems.

Figure 6.9.  "Blue Roses" by the Mandelbrot and Logistic Equations

Figure 6.10.  The Julia sets of "Dancing Seahorses" and "Blue Roses"

In "Dancing Seahorses" of Figure 6.9, the parameters p and q both belong to disks of periodicity 2 respectively in the Mandelbrot and logistic sets, hence the period of "Dancing Seahorses" is 2, while "Blue Roses" of Figure 6.10 have period 7. Consequently, the Julia set on the left shown above is simpler (topologically) than the Julia set on the right. The difference gets more pronounced if we compare them with the Julia set of "Red Lion" that has period 68; see Figure 6.4.

The next image shows a Julia set generated by the third degree logistic equation (5.9).

Figure 6.11.  "Dancing Seahorses" by the Third Degree Logistic Equation

Here is another from the fifth degree logistic equation, zn+1 = fp(zn) = p(1 - zn4) zn; compare it with (5.9). The Julia set is emphasized in the nighttime fractal on the right.

Figure 6.12.  "Dancing Bouquet" by the Fifth Degree Logistic Equation

§ 7.  Newton Fractals

A Julia fractal is called a Newton fractal if it is given by a dynamical system of the form

(7.1)   zn+1 = zn - g(zn)/g'(zn)

where the parameter p = 0 is invisible and g is an elementary function with its derivative g'. Although g is a function of a complex variable, the familiar rules of differentiation in high school calculus hold for g. In this website, g is almost always a polynomial which allows us to take advantage of the time-saving scheme called
Horner's Method to efficiently evaluate both g and g' that appear in the dynamical system. Horner's method is nothing but "synthetic division" taught in high school algebra, and it should be interesting for the reader to see how (differently) it is applied in computer programming.

The reader may have noted already that the dynamical system (7.1) is nothing but the Newton-Raphson Root-Finding Algorithm, aka Newton's Method. Hence, each orbit of (7.1) converges to a root of g quickly more often than doing something else, and it allows us to plot most of the Newton fractals by the convergence scheme (with period index k = 1) alone with a relatively small maximum number of iterations like M = 500.

Furthermore, if we know all the roots of g prior to the fractal plotting, we can modify the convergence scheme fairly easily so as to add more colors to Newton fractals of g; see Example 1 below. Because a Newton fractal is a Julia fractal, a "canvas" and an "orbit" always mean a z-canvas and an orbit of z0, respectively, in this section.

Example 1 (Roots of Unity): Among all attractive Newton fractals, probably the simplest to plot are given by a polynomial of the form

    g(z)  =  z d -  1

as its roots are readily available by hand calculations or Googling "roots of unity."

Figure 7.1.  Newton Fractals of g(z)  =  z 5 -  1

"Crab Queue"    

The leftmost image of Figure 7.1 is a Newton fractal for g(z)  =  z 5 -  1 painted on a square canvas centered at the origin with radius 1.1. It uses five essentially different colors, sky blue, purple, red, amber, and blue, associated with the five roots of g. The sky blue region, e.g., comprises the initial values z0 in the canvas whose orbits converge to the root r = (1, 0) and is called the basin of attraction of Newton's method for the root.

Thus, there are five basins of attraction in the leftmost fractal and they are divided by the basin boundary. The basin boundary is precisely the Julia set of the Newton fractal, and that is where Newton's rootfinding algorithm behaves in a "chaotic" fashion and fails to provide a root. It is known that the Julia set is a Cantor dust.

The second image of Figure 7.1 is a variation of the first and the third, called "Crab Queue," is given by zooming in on one of the "bands" in the second image. And here we have the Julia set of the "Crab Queue."

Example 2 (Cyclotomic Polynomials): Another interesting example with known roots is a
Cyclotomic Polynomial. The picture on the left in Figure 7.2 is a Newton fractal of the "30th cyclotomic polynomial"

    g(z)  =  z 8 +  z 7 -  z 5 -  z 4 -  z 3 +  z  +  1

with the unit disk highlighted. Since g happens to be a factor of  z 30 -  1,  its roots are among the 30th roots of unity that lie on the unit circle. In the picture, the thirty dots on the unit circle show where the roots of unity are located and eight of them colored yellow show the whereabouts of the roots of g. The picture on the right is a Newton fractal of the "20th cyclotomic polynomial"

    g(z)  =  z 8 -  z 6 +  z 4 -  z 2 +  1.

Figure 7.2.  Cyclotomic Polynomials with Eight Roots

Finding the Roots: As we have seen, a Newton fractal of a polynomial with colorful basins of attraction requires its roots to be known. So, if we don't know the roots, how can we find them? A natural choice seems to be the use of Newton's method, but unfortunately, its chaotic nature makes it difficult to program a computer and consistently find the roots. Another way is to use
Müller's Method instead. Although Müller's method lacks the impressive simplicity and speed of Newton's method, it generally works well and automatically finds all roots of the polynomial. All Newton fractals shown in Gallery 2D use Müller's method even when the polynomials have known roots.

Figure 7.3 is a Newton fractal of a fifth degree polynomial whose roots are given by Müller's method. Just for fun, we painted it on a plane, a sphere and a torus.

Figure 7.3.  Newton Fractal on Plane, Sphere and Torus

Figure 7.4 shows a Newton fractal of a 12th degree polynomial painted on a plane and a sphere. The second image which is on a sphere is intended to give a 3D appearance. All of the twelve roots are again found by Müller's method quickly.

Figure 7.4.   "Dragonfly" on Plane and Sphere

Part of the Julia Set of "Dragonfly"

Figure 7.5 illustrates two Newton fractals of a fifth degree polynomial. We painted them by highlighting different parts of essentially the same image. Note that we can find numerous replicas of the "fish" in the "crab" and vice versa.

Figure 7.5.   "Newton Fish" and "Newton Crab"

Shown below is a similar fractal with a better view of the Julia set. The Julia set appears to be bounded but it is in fact a part of an unbounded Julia set highlighted like the "Fish" and "Crab" of Figure 7.5.

Figure 7.6.  "Stag Beetle"

Newton Fractals of Rational Functions: Through the quotient rule of differentiation and the aforementioned Horner's method, it is straightforward to extend the fast plotting methods we have seen from the polynomials to the rational functions. Figure 7.7 shows a simple example, called "Five Crabs in a Circle," where the "huddling crabs" are the basins of attraction for the fifth roots of unity and the Julia set is, unlike in the earlier Newton fractals, bounded.

Figure 7.7.  Newton Fractals of g(z)  =  (z 5 -  1) / (z 5 +  1)


Pacific Northwest Section, Mathematical Association of America

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