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 generated by a variety of techniques based on college mathematics. The latter includes such 3D objects as fractal mountains and forests and fractals painted on various nonplanar surfaces. Here are examples:

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.

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.

Through 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 chaotic and 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.

Roughly speaking, we use the dynamical system (1.1) and plot a fractal on a p-canvas as follows: Choose a value of z_{0}, say z_{0} = 0. For each pixel (i, j) on the p-canvasR, use its representative parameter p and (1.1) to generate the orbit of p with the fixed initial value z_{0} = 0. We then pick on certain behavior of the orbit and use it to color the pixel (i, j). As we have seen, the orbits from adjacent pixels on the p-canvas may have drastically different behaviors, which generally cause a dramatic color change in the image on the canvas; see, e.g., Figure 1.3 shown below.

Plotting a fractal on a z-canvas is similar: Choose a value of p, say p = -2. For each pixel (i, j) on the z-canvasR, use its representative parameter z_{0} and (1.1) to generate the orbit of z_{0} with the fixed parameter value p = -2. We then pick on certain behavior of the orbit and use it to color the pixel (i, j).

Preview of Upcoming Sections: In § 2 and § 4, we have detailed discussions of the plotting method on a p-canvas outlined above. In § 5, we extend the method to a more general dynamical system and call the resulting fractal a Mandelbrot fractal on a p-canvas. In § 6, we discuss a Julia fractal, which is painted on a z-canvas instead. There is a special subset of the Julia fractals consisting of fractals generated by so-called "Newton's rootfinding method." The fractals in the subset are called Newton fractals and are covered in § 7. Here are some of the sample fractals:

We say that a sequence z_{n} of complex numbers diverges to ∞ if the real sequence |z_{n}| diverges to ∞, i.e., if z_{n} 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, called the Divergence Scheme, associated with the notion of divergence of orbits of complex parameters p using the Mandelbrot set as an example.

Repeating the argument n times, we get |z_{n+1}| ≥ α^{n}|p|. Since α > 1, it follows that the orbit z_{n} of p given by (2.1) diverges to ∞; hence, p does not belong to the Mandelbrot set. Thus, if p is in the Mandelbrot set, then p must be in the circle of radius 2, or equivalently, the Mandelbrot set is bounded by the circle, as was to be shown.

If we run a computer program based on the above red-black divergence scheme, we get Figure 0.1 as its (resized) output image, in which the circle of radius θ = 2 is visible. The portion that retains the white canvas color and resembles a "snowman" figure is precisely the approximation of the Mandelbrot set plotted on the canvas by replacing "forever" (or ∞) in the alternative definition by "up until M = 1000."

The first of the two images in Figure 2.1 shows a closeup of the Mandelbrot set. We generally omit the word "approximation" for simplicity, understanding that almost all computer plots in mathematics are approximations of some objects.

The part of the Mandelbrot set shown in the image comprises the pixels (i, j) with d(i, j) = M and is painted black and the rest in multiple colors using our coloring routine. Figure 2.3 shows several (deformed) replicas of the "snowman," which we call mini Mandelbrot sets. They look like small isolated islands but as we'll find out in the next section, they are actually connected to the Mandelbrot set by razor-thin "filaments" belonging to the boundary of the Mandelbrot set.

(2) We have seen that a larger threshold θ generally provides a similar or slightly better output image provided that the maximum number of iterations M is sufficiently large. This fact can be used for the following situations. When we use a dynamical system other than the Mandelbrot equation (2.1) or a noncritical point for the initial value to explore wider fractal plotting possibilities, we may not know what threshold to use in the divergence scheme and a larger threshold is more likely to provide a solution. But be careful: Increasing the threshold also increases the computing time without notably improving the output image. There is a good reason why Mandelbrot used the smallest threshold θ = 2 in plotting the Mandelbrot set in the 1970s when the computers were much slower.

The Mandelbrot set was introduced to the world in 1980 by Mandelbrot through his computer-generated images, and since then, numerous articles about it have appeared in major journals, magazines and newspapers. In 2008, NOVA said on PBS that the Mandelbrot set had become the most famous object in modern mathematics. We will see some of the reasons for the fame in § 3, § 4 and § 6, including the fact that it is one of the most complex objects on the two-dimensional plane.
Recall that the Mandelbrot set is closed so it contains its boundary as its subset. 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 2, which is the (ordinary) dimension of the plane.

Let's pause for a moment and look at the fractal in Figure 2.3, for example, in which a part of the Mandelbrot set is visible. The intricate image surely looks impressive, but exactly where is the boundary of the Mandelbrot set and what does it have to do with the colorful patterns?

It turned out that the boundary of the Mandelbrot set is all over the image as we can see in Figure 3.1 given by darkening the entire Figure 2.3 and lighting up its razor-thin filaments:

The boundary of 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 filaments of the boundary of the Mandelbrot set get so dense around them, it fills 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 dimension of the plane and why it is equated with the complexity of the Mandelbrot set.

The self-similarity in Figure 3.1 seems to show that the Mandelbrot set with its complex boundary is 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, can we get out of the maze? The following theorem 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 "Mandelbrot Set" by Wikipedia, Mandelbrot initially conjectured (but later revised) 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 also says: The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard, who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in fractal geometry.

We are not done yet with the complex nature of the Mandelbrot set and still stay with it. In § 3, we discussed the boundary of the Mandelbrot set, but we now turn our attention to its interior, which is, by definition, the Mandelbrot set minus its boundary.

The Mandelbrot set has become so illustrious, everybody interested in fractals knows its "snowman" shape by heart. To its main body, which is a heart-shaped "cardioid," a bunch of circular disks are attached, and to each of these disks another bunch of disks are attached. The fractal pattern repeats as if the cardioid has children, grandchildren, great grandchildren and so on and so forth; see e.g., Figure 2.1.

This already shows an amazing structure of the interior of the Mandelbrot set, but unfortunately, the divergence scheme cannot distinguish its basic components—the interiors of the disks and cardioids—and paints them in a single color like black or white as we saw in § 2. Our current goal is to find their distinct mathematical properties and paint them in various colors, like in Figure 4.1 below, by developing a different algorithm called the "convergence scheme."
Atoms and Molecules: Before proceeding further, let's use the Mandelbrot's idea shown in his article as a cue and call the interior of each disk or cardioid an atom and anything made up of atoms a molecule. For example, in Figure 4.2, we see three molecules composed of atoms, each of which is painted red, orange, green, purple or blue. Calling the basic objects "atoms" turns out to be especially convenient, as we will see objects made up of a wide variety of shapes in addition to disks and cardioids in the upcoming sections. Like in chemistry, we will also see that atoms are associated with "orbits" and "periods" (but in different ways). For example, each of the red "rose petals" shown in Figure 5.0 is an atom of period 21.

To make the algorithm a little more convenient, we follow the path in § 2 and define a new iteration array on the p-canvasR comprising pixels (i, j) by

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

if such m exists, and d(i, j) = M, otherwise; compare (4.1) with (2.3).

Example 1: Figure 4.2 shows three resized images cropped from the output images of various convergence schemes. Here, the p-canvasR is the rectangle in the complex plane with center (-0.52, 0) and horizontal radius 1.65 and comprising 3,000 × 2,500 pixels. For the large canvas size, we use M = 20000 and ε = 10^{-8}.

The first image is given by the convergence scheme with period index k = 1, which shows that the interior of the cardioid is an atom of period 1 and paints it (by subtle shades of) red.

The second image is given by the convergence scheme with period indices k = 1, 2, 3, 4, which is basically defined by the convergence scheme with period index (cswpi) k = 1 followed by the cswpi k = 2 followed by the cswpi k = 3 followed by the cswpi k = 4. It shows that the interior of the largest disk is an atom of period 2 and paints it orange. Similarly, the green and purple atoms are of periods 3 and 4, respectively.

The third image is given by a straightforward extension of the scheme described in the preceding paragraph. Because there aren't enough colors that are easily distinguishable, the correspondence between the periods and colors of the atoms is not one-to-one. For example, the atoms of periods 2 and 5 are painted orange in the third image.

Example 1 shows that the convergence scheme may mean the convergence scheme with a single period index or multiple period indices.

To avoid making it cluttered, the descriptions in Example 1 highlight only the important ideas and do not include bits and pieces of technical details used in the computer program. For example, the convergence scheme is applied only on the upper half of the Mandelbrot set, which is symmetric about the real axis of the complex plane, so as to shorten the computing time. The omissions also include fairly easy subroutines used to get around the problems described below:

Figure 4.3. The Eyeball Effect (Right) Given by the Convergence Scheme

Mini Mandelbrot Sets: The images displayed below are parts of the Mandelbrot set of Figure 4.2 painted on the large canvas, and they show not only elaborately aligned circular atoms but also several mini Mandelbrot sets that look like little flying insects. Looking at his 1980's black and white computer printout of the Mandelbrot set, Mandelbrot initially thought they were "dirt."

rather than a constant like M = 20000. It is a waste of computer time to allow M = 20000 for a tiny atom like the one with period k = 60.

Figure 4.4. The Interior and Boundary of Mini Mandelbrot Set

Artist's Rendering: We saw earlier an approximation of the Mandelbrot set in an image like Figure 2.1 and a more colorful version in Figure 4.1. The latter is an artist's rendering of the Mandelbrot set rather than its approximation, because its definition says nothing about atoms and periods. In fact, almost all images in this website, especially those that involve the convergence scheme, are artist's renderings of certain mathematical objects. Shown below are additional artist's renderings of the Mandelbrot set.

Periodicity Diagram: If we label the atoms of the Mandelbrot set in Figure 4.2 by their periods instead of coloring, we get Wikipedia's Periodicity Diagram. 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.

Soon after Mandelbrot published its computer plot 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 posters, book covers, T-shirts, coffee mugs and webpages. Although the hidden beauty of the Mandelbrot set is inexhaustible, it has become quite a challenge to unearth newer patterns from the Mandelbrot equation (2.1) using available computers and software. Consequently, a creative work such as the image shown below calls for a modification of the Mandelbrot equation (2.1), and there are infinitely many formulas available for it.

where the initial value z_{0} is one of the two conjugate critical points ± i / √3 of f_{p}. We also note that an atom may no longer resemble a disk or a cardioid and instead may look like an arrowhead from the stone age with jagged edges. It has a variety of molecules as well and they include the interior of "Giant Mandelbrot Set" (who was shot by the arrow) and the interior of "Toddler Mandelbrot Set" (who released the arrow at the giant) seen near the right edge of Figure 5.1 like an isolated island. We name some of the molecules and areas partly for fun but mainly for necessity as we will be talking about them several times on the website.

The origin (0, 0) of the complex plane is at the tip of the arrowhead in Figure 5.1, which is actually given by rotating the original output 90^{o} counterclockwise about the origin to have it better fit in the webpage. In the image called "Arrowhead Bay" shown below, the origin coincides with the upper left corner and the red zones adjacent to the upper edge and the left edge belong to the "Giant Mandelbrot Set" and the arrowhead, respectively. Thus, the upper and left edges are parts of the real and imaginary axes in the standard orientation of the complex plane, as illustrated in the image.

"Arrowhead Bay"

Like the Mandelbrot set, the "Giant Mandelbrot Set" contains infinitely many circular atoms that satisfy the numerical pattern of the periodicity diagram, and the circular atoms include the largest and the second largest blue atoms in "Arrowhead Bay" whose periods happened to be 7 and 8, respectively. Note that the "seaweed" around the blue atom of period 7 contains "seven-finger" branches, and likewise for the "seaweed" around the atom of period 8.

We also observe that the boundary of the "Giant Mandelbrot Set" shown in "Arrowhead Bay" contains infinitely many extra atoms that look like tropical fish tangled with the "seaweed." These atoms all have period 1 and are apparently connected to the "Giant Mandelbrot Set" by its boundary (called "seaweed").

Interestingly, the fish-like atoms begin to disintegrate near the circular atom of period 6, which is at the mouth of "Arrowhead Bay," and they become extinct near the circular atom of period 5, which is located just outside of the bay. The boundary of the "Giant Mandelbrot Set" near the circular atom of period 5 is depicted in the image shown below. It shows no signs of fish but it still encloses numerous mini Mandelbrot sets. Unlike the Mandelbrot set, however, the boundary now appears to be disconnected.

We now examine some of the molecules found in Figure 5.1 starting with an area near its right edge. Here is a closeup of the "Toddler Mandelbrot Set" shown in the standard orientation of the complex plane.

Compared to the Mandelbrot set, the "Toddler Mandelbrot Set" has a proportionately larger head (like a toddler) and its boundary gets broken into pieces that look like stars in a christmas tree. The interior of the "Toddler Mandelbrot Set" is painted by the convergence scheme with period indices k = 2, 4, 6, ..., 50 and the colors matched with the colors of the Mandelbrot set. Thus, the red atom has period 2, while the largest circular atom painted orange has period 4. The period of an atom will play an important role when we plot "Julia fractals" in § 6. "Green Monster" is such an example born from the cardioid atom near its cusp.

Not surprisingly, fractals we find around the boundary of the "Toddler Mandelbrot Set" are similar to those found near the Mandelbrot set. Here's one of them, which can be used as a night sky of 3D landscapes such as "Mandelbrot Moon" in Gallery 3D.

Another area in Figure 5.1 that provides a rich fishing ground for attractive fractals is in and around the blue molecule located between the arrowhead and "Toddler Mandelbrot Set" that looks like a pair of balloons. We call it "Broken Balloons" because of its "bursted lips" with jagged edges and small fragments; see the image shown below. It is generated by the convergence scheme with period indices k = 3, 6, 9, ..., 60. Like the cardioid body of the Mandelbrot set, we again painted the atoms of the smallest period 3 red.

Figure 5.1 also contains two "Squished Mandelbrot Sets," each of which has a "bursted" cardioid. The molecule can be seen near the left edge of Figure 5.1, but its magnified image shown below uses different colors. It is generated by the convergence scheme with period indices k = 3, 6, 9, ..., 60 with k = 3 corresponding to the red atoms, just like in the "Broken Balloons."

"Squished Mandelbrot Sets" break down near the "bursted" cardioids and the flying fragments provide interesting fractals. For example, Figure 5.0 shown at the outset of this section is given by zooming in on microscopic rose-shaped fragments in a p-canvas centered at p = (0.04978, 1.094143) and by the convergence scheme with period index k = 21 = 7 × 3.
Recall that Figure 5.1 is a Mandelbrot fractal of z_{0} = i / √3, which is a critical point of f_{p} in (5.2), and it turned out that the Mandelbrot fractal of the conjugate critical point z_{0} = - i / √3 given by the same fractal plotting process is the mirror image of Figure 5.1 through the real axis. If we superimpose the two mirror images, we get a surprising results as shown in Figure 5.1(D): The big arrowhead in one image fits perfectly in the cardioid body of the "Giant Mandelbrot Set" in the mirror image and the lips of the "Broken Balloons" in Figure 5.1(B) are beautifully repaired by the "Squished Mandelbrot Sets" of Figure 5.1(C).

Figure 5.1(D). The "Giant Mandelbrot Set" with the "Mandelbrot Balloons"

In § 6 where we deal with the "Julia fractals," we will talk more about the superimposed Mandelbrot fractals by the dynamical system (5.2) as it represents an ideal example in analyzing and applying the important "Fatou-Julia Theorem."

Daytime and Nighttime Mini Mandelbrot Set by the Logistic Equation

In 1974, while conducting a computer simulation of certain population changes, biologist Robert May discovered "very complicated orbits" of z_{0} 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 z_{0} = 0.1 (10% of the sustainable population of, say, an insect species) as the fixed initial value for the orbits of p.

Recall that a Mandelbrot fractal is generated by a dynamical system of the form (5.1) with a fixed initial value z_{0} and painted on a p-canvas, where each pixel is identified with a parameter p. A fractal is called a Julia fractal of p if it is given by a dynamical system of the same form with a fixed parameter p and painted on a z-canvas, where each pixel is identified with an initial value z_{0}. As we saw in § 1, each initial value z_{0} on the z-canvas generates a sequence of complex numbers called the orbit of z_{0} (with the fixed parameter p) belonging to (5.1), and the orbits play a central role in generating a Julia fractal of p.
Julia Fractals by the Mandelbrot Equation: Just as we did earlier, we begin our discussion with the familiar Mandelbrot equation
(6.1) z_{n+1 }= f_{p}(z_{n}) = z_{n}^{2} + p,

Example 1(A) Figure 6.1(A) shows a Julia fractal, which we call "Hydra with Eleven Heads," depicting (an artist's rendering of) the filled-in Julia set of the parameter p = (-0.68938, 0.27896) belonging to an atom of period11 in the Mandelbrot set; see the periodicity diagram. The green/blue "Hydra" is painted on a z-canvas centered at the origin of the complex plane by a straightforward application of the convergence scheme with period index k = 11 and its background by the divergence scheme with the threshold θ = 2; we will explain the threshold value a little later.

It is another fascinating fact about the Mandelbrot set that the period of (the atom containing) the parameter p is always reflected in the shape of the filled-in Julia set of p, like the number of Hydra's heads, but why it is so is not completely understood. If the parameter p is chosen from an atom not directly attached to the main caridoid of the Mandelbrot set, or if it is chosen from an atom in the mini Mandelbrot set, the outcome can get much more interesting.

Figure 6.1(A). The Filled-in Julia Set of p = (-0.68938, 0.27896)

"Hydra with Eleven Heads" born from an atom of period 11

where α = |z_{m}| - 1 > θ - 1 ≥ 1. In particular, we have |z_{m+1}| ≥ α|z_{m}| > |z_{m}| > θ, allowing us to repeat the above argument once again to get

Example 2 The Julia fractal shown below is (an artist's rendering of) the filled-in Julia set of p = (0.281215625, 0.0113825) belonging to an atom of period 68 = 17 × 4. The atom is attached to an atom of period 17 which is attached to the cardioid atom of the Mandelbrot set near its cusp; see the periodicity diagram. The filled-in Julia set named "Red Lion" is painted by the convergence scheme with period index 68 = 17 × 4 and its colorful background by the divergence scheme with the threshold θ = 2.

The Julia set shown on the right is known to be a non-self-intersecting continuous loop in the plane called a Jordan curve. The Jordan curve theorem states that a Jordan curve divides the plane into two parts, a bounded region called "interior" and an unbounded region called "exterior." The theorem seems utterly obvious from a typical image like the one shown above, but the Julia set as a Jordan curve can get extremely convoluted if the parameter gets arbitrarily close to the boundary of the cardioid. In fact, the proof of the Jordan curve theorem is far from obvious and provides one of the fascinating topics in mathematics.

Recall that a mini Mandelbrot set we discussed earlier comprises atoms of periods λ × 1, λ × 2, λ × 3, ... for some positive integer λ ≥ 2and λ represents the period of its main cardioid atom. Note that λ = 1 is the period of the main cardioid atom of the Mandelbrot set. If p is a parameter of period λ ≥ 2 belonging to the cardioid atom of the mini Mandelbrot set, we call the filled-in Julia set of p "Devil's Stepping Stones" of period λ.

For example, let p = (-0.1581, 1.03545), which belongs to the main cardioid atom of period λ = 4 of the Mini Mandelbrot set in Figure 4.4. The image shown below illustrates "Devil's Stepping Stones" of period 4. The greenish "stepping stones" are plotted by the convergence scheme with period index k = 4 and the rest by the divergence scheme with the threshold θ = 2.

For example, let p_{0} = (0.25000316374967, -0.00000000895902), which belongs to the interior of the cardioid of the mini-Mandelbrot set in the image shown below on the left. In particular, p_{0} does not belong to the boundary of the Mandelbrot set; hence, it is not a Misiurewicz point. Also, because of the dichotomy theorem, the Julia set of p_{0}, which is the boundary of "Devil's stepping stones" shown on the right, is connected.

The image on the left shown below is a Mandelbrot fractal given by the critical orbits and plotted on a p-canvas centered at the parameter p_{0} while the image on the right is a Julia fractal of p_{0} painted on a z-canvas centered at the origin. Their striking resemblance shows that a Julia set can be as complex as the boundary of the Mandelbrot set locally.

Figure 6.6(B). The Filled-in Julia Set depicting "Circus Elephants"

For another example, let p = (3.0014564, 0.08), which belongs to "Elephant Bay" and outside of the logistic set. The image shown below is a Julia fractal painted by the divergence scheme that depicts the Julia set of p. As shown below, the dichotomy theorem applies on the Julia set given by the logistic equation and confirms that it is Cantor dust. The image emerges very quickly on a computer screen—within less than one thousandth of runtime used for the image in Figure 6.6(B).

Figure 6.6(C). Cantor Dust depicting "Circus Elephants"

given by the Mandelbrot equation (6.3). The images are painted by exactly the same coloring routine and exhibit geometrically the same Julia sets and filled-in Julia sets. As we can see in Figure 6.7, the artist's renderings of the filled-in Julia sets, called "Dancing Seahorses" turned out to be a little different and (inadvertently) provide us with two interesting options.

Figure 6.7. "Dancing Seahorses" by the Logistic and Mandelbrot Equations

The parameter q = (3.02382, 0.1) belongs to the orange atom of period 2 in the logistic set adjacent to "Elephant Bay" and is transformed by (6.5) to the parameter p belonging to the orange atom of the Mandelbrot set of period 2. Here is the Julia set of p of period 2, which is the same as the Julia set of q bounding "Dancing Seahorses." It is (topologically) more complex than the Jordan curve of Figure 6.6(A).

Figure 6.8. The Julia Set of a Parameter of Period 2

The transformation by the change of variables actually goes both ways: If we solve the quadratic equation (6.5) for q, we get

(6.6) q = 1 ± √ (1 - 4 p),

so the Mandelbrot equation (6.3) with parameter p can be written as a special form of the logistic equation (6.2) with q replaced by the right-hand side of (6.6) with either sign of ±. For example, the image shown below on the left is a Julia fractal of p = (-0.6069, 0.4147) given by the Mandelbrot equation (6.3) 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 (6.2). Again their artistic patterns painted by the same coloring routine are a little off, but they have geometrically the same Julia sets.

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

Because of the two-way transformations between the Mandelbrot and logistic equations, we often say that these equations are conjugate of each other. "Dancing Seahorses" and "Blue Roses" exemplify the fact that the conjugate dynamical systems share geometrically the same Julia fractals. In light of the similarities of Mandelbrot and Julia fractals by the Mandelbrot equation (and by the logistic equation), it is not surprising if many of the Mandelbrot fractals generated by two dynamical systems also have similar appearances.

A fractal artist should not be discouraged, however, from venturing into the logistic equation and other quadratic dynamical systems that are conjugate to the Mandelbrot equation. In fractal plotting, getting just the right shapes and colors is often a chance encounter due to its chaos-related sensitivity and we have better chances if we engage in wider computer experiments. If we lose our computer program or recipe for a certain fractal image, it would be nearly impossible to reproduce it again or produce its local images by zooming.

The Fatou-Julia Theorem: The concept of Julia set naturally extends from a quadratic dynamical system such as the Mandelbrot equation 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 a fascinating and useful fact however. Gaston Julia and Pierre Fatou independently proved the theorem shown below in 1918-1919 way before the computer era.
Theorem: Consider a dynamical system of the form

If f_{p} of (6.7) has two or more critical points, the situation may become much more interesting. Consider for example, the cubic dynamical system (5.2) with two critical points ± i/√3. For simplicity, let A be the set of parameters whose critical orbits with the initial value i/√3 stay within a finite bound and B the same with the initial value -i/√3. A is depicted by Figure 5.1 and B is the mirror image of A through the real axis. As we saw in Figure 5.1(D), A and B have interesting relations when they are superimposed.

For example, Figure 0.2 shown at the outset of this website is a Julia fractal of p = (0.185, 0.00007666) belonging to A∩B, and hence, the filled-in Julia set of p, called "Twin Dragons," is connected. The next three images show topologically different "twin dragons."

Figure 6.11. "Twin Dragons" of p = (0.2011575, 0.00002) in A∩B

p = (0.21828, -0.00230) in [A∪B]^{c}

p = (0.2176, 0.0128) in AΔB

The last two "twin dragons" are, respectively, a Cantor set and a disconnected Julia set which is not a Cantor set. The imaginary components of the parameters show that all "twin dragons" come from an area of "Venn diagram" that is near the real axis. "Twin dragons" of parameters lying on the real axis are symmetric about the center vertical line.

Recall that "Broken Balloons" is a molecule comprising atoms of periods k = 3 × 1, 3 × 2, 3 × 3, · · · . It can be seen near the right edge of "Venn diagram" and intersects both A∩B and the symmetric difference AΔB.

"Blue Rose" shown below is a Julia fractal of a parameter p belonging to an atom of period 3 × 7 in "Broken Balloons." p also belongs to A∩B, so the numerous "blue roses" seen in the image are connected by the "stems." We can clearly see the number 7 in the picture but where do we see the number 3 ? Hint: "Devil's Steppingstones."

Connected "Blue Rose" of p = (0.02912, -1.093853) in A∩B

"Blue Rose" shown below is a Julia fractal of p belonging to an atom of period 3 × 4 in "Broken Balloons." p also belongs to AΔB, so the Julia set is disconnected, which we can see in the broken "stems." Note that the Julia set is not a Cantor set. Where in the picture do we see the number 3 ?

Disconnected "Blue Rose" of p = (0.07761, -1.12427) in AΔB

"Lions" and "Elephants" also pop up along with many other shapes in and around "Broken Balloons." The next three images are the Julia sets of parameters belonging to [A∪B]^{c} near "Broken Balloons, so they are all Cantor sets.

The Julia Set of p = (0.0144, -1.192) in [A∪B]^{c}

p = (0.087, -1.1848) in [A∪B]^{c}

p = (0.092, -1.1728) in [A∪B]^{c}

"Toddler Mandelbrot Set" seen near the right edge of Figure 5.1 belongs to AΔB but it is omitted from "Venn diagram." Recall that it comprises atoms of periods k = 2 × 1, 2 × 2, 2 × 3, · · · . It also produces a great many attractive fractals but they are naturally very similar to the fractals coming out from the Mandelbrot set—except that they are all disconnected as "Toddler Mandelbrot Set" belongs to AΔB. For example, the image which is shown below and resembles the "Hydra" of Figure 6.1 is a Julia fractal of p = (0.00399109,-1.98545775) belonging to an atom of period 2 × 13. It contains numerous dots in its background, each of which contains a "baby hydra."

"Hydra with Thirteen Heads" born from an Atom of "Toddler Mandelbrot Set"

Here's another monster, this time born from the cardioid atom of period 2 × 1 in "Toddler Mandelbrot Set" near its cusp.

Figure 6.12. "Dancing Seahorses" by the Third Degree Logistic Equation

p = (1.18, 0.376)

p = (1.1565, 0.3688)

Here is another dancer from the fifth degree logistic equation, z_{n+1 }= f_{p}(z_{n}) = p(1 - z_{n}^{4}) z_{n}. The Julia set is emphasized in the nighttime fractal on the right.

Figure 6.13. "Dancing Bouquet" by the Fifth Degree Logistic Equation

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

(7.1)
z_{n+1 }= z_{n} - g(z_{n})/g'(z_{n})

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 z_{0}, 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."

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"

So, what should we do if we don't know the roots for the input polynomial? One way is to use Müller's Method instead of Newton's method. Although Müller's method lacks the simplicity of Newton's method, it generally works well and automatically finds all roots of the polynomial fairly quickly. All computer programs for the Newton fractals shown in this website (including Gallery 2D) use Müller's method—even for Example 1 whose roots are well known (so we don't have to write down the roots in the program).

Once our computer program starts running smoothly, plotting Newton fractals provides us with great entertainment. It is easy to pick an input polynomial (or non-polynomial) from infinitely many choices with anticipation from not knowing what to expect in the output and a high-res output image generally emerges within minutes rather than hours and days of runtime. Figures 7.3through 7.6 shown below are among numerous Newton fractals for which we randomly chose the input polynomials.

Here's an example given by a fifth degree polynomial. Just for fun, we painted it on a sphere and a torus as well as on a plane.