This website is a gallery of computer-generated fractal art as well as a text that explains what it is and how it is created. Central to the website are a myriad of fractal images shown in "Stories about Fractal Art" augmented by the text as optional references. In addition, there are two galleries of fractal art: Gallery 2D is a collection of two-dimensional (2D) images made recently, while Gallery 3D comprises such 3D objects as fractal mountains (some are realistic and others imaginative), fractal 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.

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.

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 and various others appeared as shown in Figure 0.4. In the 1910s, Pierre Fatou and Gaston Julia independently laid the foundations for fractals generated by "dynamical systems" of complex numbers. The Julia sets, representing their fractals and named after Gaston Julia, were largely ignored, however, before the Mandelbrot set was born on a computer screen. The Cantor sets, the Julia sets and the Mandelbrot set are all subsets of the complex plane.

It is given by lighting up the invisible filaments of the boundary of the Mandelbrot set and reveals its startlingly complex line art in goldish color; see Shishikura's Theorem.

Figure 0.5(H). "Gold Lion"

A Julia set in color found on the "Map" of Figure 0.5(G)
"Elephants," "Lions" and "Seahorses," etc. are star players in the Julia sets.

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 inspired by May's discovery received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes.

The idea of chaos quickly evolved into comprehensive chaos theory in science and at the same time its affinity with fractal art became evident. As we'll see, it is the aforementioned sensitivity of chaos that causes drastic changes of colors and patterns, and almost surely, the boundary of the Mandelbrot set is exactly where the set is chaotic, revealing its dazzling details and complexities seen in the likes of Figures 0.5(A) and 0.5(G).

Mathematically, a Julia set is much simpler than the Mandelbrot set and known to be chaotic. Furthermore, its chaotic tendency gets stronger if it is born from near the boundary of the Mandelbrot set. In fractal art, chaos is highly desirable and the boundary of the Mandelbrot set plays a role of the utmost importance.

Googling we can find a host of websites displaying numerous computer-generated fractal art images, some of which are stunningly beautiful. It indicates that a large population not only appreciates the digital art form but also participates in the eye-opening creative activities. A fairly large part of this article is devoted to show how to program a computer and plot popular types of fractals generated by simple dynamical systems. It is not a text on computer programming but instead tells the general principles for fractal plotting in everyday language so as to entice more people into trying it.

Particularly exciting is the moment the fractal image generated by our personal program emerges in our computer screen, because of its potentially "astounding" artistry and built-in chaos-related details and unpredictability. Many of the fractal art images will stir our imaginations in the part of mathematics that is in fact quite deep and still filled with unknowns. It is plain fun.

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 x^{2} + y^{2} ≤ 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(A) 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

p-Canvases and z-Canvases: Consider any dynamical system like the Mandelbrot equation (1.1) comprising infinitely many orbits z_{n} of complex numbers, one for each choice of values of z_{0} and p, both varying through the complex plane. Recall that a canvas is a rectangle R in the complex plane consisting of pixels (i, j), each of which has a representative complex number (x, y). In fractal plotting, we view the complex numbers representing the pixels as values of p and call the canvas R a p-canvas or view these complex numbers as values of z_{0} and call the canvas R a z-canvas.
Plotting a fractal on a p-canvas is, roughly speaking, as follows: Choose a value of z_{0}, say z_{0} = 0, and an appropriate p-canvas following the direction in § 2 or § 4. For each pixel (i, j) on the p-canvasR, use its representative parameter p and the dynamical system to generate the orbit of p with the initial value z_{0} = 0. We then use a certain property of the orbit to color the pixel (i, j). As we have seen, the orbits from adjacent pixels on the p-canvas may have drastically different behaviors, possibly causing dramatic color changes in the image painted on the p-canvas.

Plotting a fractal on a z-Canvas is similar except that here we use orbits of z_{0} (with a fixed parameter value p) corresponding to all z_{0} on the z-canvas instead of orbits of varying p with a fixed initial value z_{0}. We call a fractal plotted on a z-canvas a Julia fractal, as it is typically a colorful fractal art image depicting a Julia set; see Figure 0.6(B). Because Julia sets are colorless subsets of the complex plane belonging to fractal geometry, there is a clear difference between a Julia set and a Julia fractal, but after a while, we'll follow the common practice of identifying them and calling the colorful version a Julia set rather than a Julia fractal.

For the similar reason, a fractal plotted on a p-canvas is called a Mandelbrot fractal. It is normally an artistic view of the Mandelbrot set, but again, we will call the colorful version the Mandelbrot set rather than a Mandelbrot fractal. There are some exceptional cases, however, like when we discuss a "Mandelbrot-like" set generated by a dynamical system other than the Mandelbrot equation. In such a case, the image on the p-canvas will be called a Mandelbrot fractal instead of the Mandelbrot set.

As we'll see, the Mandelbrot set generated by the Mandelbrot equation is bounded so its global figure fits in a circle or a rectangle like in Figure 1.4(B), and from the shape of its black silhouette, it is often nicknamed "Warty Snowman" lying sideways. A part of the global image is called a local image, but a local image in fractal art is usually given by zooming in on a microcroscopic rectangular neighborhood of a point that is very near or on the border of the snowman's silhouette.

Because the Mandelbrot set is also known to be closed, its boundary, which coincides with the snowman's silhouette's border, is a part of the Mandelbrot set. It is also known that the boundary has the "topological dimension" of 1, meaning intuitively that it comprises razor thin "filaments" without thickness, just like the boundary of a circular disk. So, the boundary is mostly invisible in local images unless we "light up" their filaments like in a tungsten light bulb; see Figure 1.4(C) shown below.

As Figure 1.4(B) shows, the filaments are like branched hairs growing outwards from the warty snowman and known to carry infinitely many miniature copies of the snowman, which we call "mini-Mandelbrot sets." One of them is visible in Figure 1.4(C) and, yes, the boundary of the Mandelbrot set is incredibly intricate.

In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set had become "the most famous object in modern mathematics." Naturally then, we begin our stories with the Mandelbrot set and devote a large part of the article to a myriad of its attributes that fascinate mathematicians and artists alike. The Mandelbrot set popularized fractal plotting by computers and has been the gold standard for all types of fractals.

In § 2, we state the notion of an orbit that diverges to ∞ and define the Mandelbrot set, denoted by ℳ, using the Mandelbrot equation (2.1) and its critical point. In § 4 we also introduce the notion of an orbit that converges to a cycle of period k for some positive integer k and in conjunction with § 2 explain how to plot ℳ globally and locally on p-canvases using algorithms called the divergence and convergence schemes. Figure 1.4(B) shows global images of ℳ contrasting the two distinct algorithms, while Figure 1.4(C) shows a local image of ℳ together with its complex boundary. As we have seen, ℳ contains its boundary as its subset.

The Mandelbrot set ℳ conceals a greater number of varying local images than the number of stars in the universe and even amateurs can find many of them and turn them into beautiful art pieces because of the simplicity of the divergence scheme, aka the "escape time algorithm." Consequently, its appeals have extended way beyond mathematical communities, contributing ℳ to gain its monumental popularity.

In § 3, we discuss some of the important mathematical properties of ℳ, focusing on its boundary. Note that in Figure 1.4(C) the goldish boundary comprising razor thin filaments gets increasingly more intricate and space-filling in many areas making them appear fuzzy and two-dimensional. Shishikura's theorem formally explains the phenomenon in terms of "fractal dimensions," and shows, in effect, that no figures in the plane are more complex than the boundary of ℳ. It boosts the Mandelbrot set ℳ to be one of the most complex figures ever plotted on a plane.

In § 4, we turn our attention to the interior of ℳ and use Mandelbrot's idea as a cue to define an atom to be a connected component of the interior. Then atoms include all of the disks and cardioids visible in ℳ, and as per the "density of hyperbolicity" conjecture, all atoms are associated with periods of certain cycles. A part of the association between atoms and periods is illustrated by the meticulously aligned numerical structure shown above in the periodicity diagram.

In spite of all these astounding complexities of ℳ, Adrien Douady and John H. Hubbard established earlier that ℳ is topologically tidy as it is compact, connected in one piece and without holes. The connectedness of ℳ is particularly important and called the Douady-Hubbard Theorem. The next-step problems of whether ℳ is pathwise connected and whether ℳ is locally connected are conjectured affirmatively but still remain unsolved, despite the efforts by some of the world's brightest minds.

In § 5, we define, given a parameter p in the complex plane, the filled-in Julia set of p generated by the Mandelbrot equation and show how to plot it on a z-canvas using the divergence and possibly convergence schemes. If p belongs to an atom of ℳ, then the period of the atom affects the shape of the filled-in Julia set in an amazing and sometimes amusing way, although the exact cause is shrouded in mystery. It makes the periodicity diagram all the more important in plotting many of the filled-in Julia sets.

For example, the image shown below is the filled-in Julia set of a parameter chosen from an atom of period 9 near the "neck" of ℳ, while Figure 1.4(A) shows two filled-in Julia sets born from the same atom of period 17 × 5 near the cusp of ℳ. As there are infinitely many atoms, there are infinitely many varieties of filled-in Julia sets.

The boundary of a filled-in Julia set is called a Julia set. It shapes the filled-in Julia set, and as it borders between the filled-in Julia set and its complement with totally different characters, it becomes chaotic just like in human activities. Julia sets are, therefore, important objects in fractal art, fractal geometry and chaos theory.

The Fatou-Julia theorem implies that (†) is equivalent to the computer-friendly definition of § 2, providing Mandelbrot with the convenient algorithm to compute ℳ. ℳ was born, therefore, as a map on the complex plane to illustrate the dichotomy, but it has grown to be much more complete and provides more specific information about the shapes of the Julia sets. Without ℳ, we can never plot images like Figures 1.6(A)(B)(C), Figure 0.5(B) and Figure 0.6(F). Wikipedia shows very roughly what the map looks like.

In § 6, we consider more general dynamical systems, and in particular, the dynamical system (6.2) generated by a simple cubic polynomial with a conjugate pair of critical points z_{0} = ± i / √3 = ± (0, 1 / √3). Thus, (6.2) differs from the Mandelbrot equation (2.1) in the important fact that it comes with two critical points and hence without the dichotomy. It turned out that (6.2) gives rise to an even greater variety of Julia sets and other fractal images than (2.1). Now we ask: Can we find the "Mandelbrot set" of the dynamical system (6.2) that works as a map guiding us to find these Julia sets?

If we repeat the process of § 2 and § 4 and generate the "Mandelbrot sets" of the two critical points, we get rather comical fractals nicknamed "Speared M Sets" ℳ_{1} and ℳ_{2} shown in Figure 1.7(A). To be precise, ℳ_{1}, for instance, corresponds to z_{0} = i / √3 and is the portion not including the dark green background. It does include, however, its hairy boundary and the mini Mandelbrot set seen near the bottom of Figure 6.1, but it is omitted from Figure 1.7(A) for simplicity. The tip of the "spearhead" turned out to be the origin (0, 0) of the complex plane and ℳ_{2} is the mirror image of ℳ_{1} through the real axis of the complex plane.

We again define atoms of say, ℳ_{1}, to be connected components of the interior of ℳ_{1}. Thus, atoms are in a wide variety of shapes that include the interior of a purple disk as well as the interior of the red "spearhead" with jagged edges. Both ℳ_{1} and ℳ_{2} contain local images not seen in the Mandelbrot set ℳ of (2.1) including the "Burst M Set" of Figure 1.7(B). It contains numerous mini Mandelbrot sets and atoms looking like flying fragments.

Interestingly, all of the fragments and jagged edges of the spearhead atoms, which are the irregularities not seen in ℳ, disappear in the third image of Figure 1.7(A), which we call "Atomic Fusion" given by superimposing ℳ_{1} and ℳ_{2}. Figure 1.7(C) shows a part of the fusion in different colors.

As shown in § 6, the "Atomic Fusion" comprising ℳ_{1} ∪ ℳ_{2} and ℳ_{1} ∩ ℳ_{2} provides a pictorial interpretation of the Fatou-Julia theorem with the two critical points and seems to work impeccably as a map guiding us to various Julia sets. For that reason, we call ℳ_{1} ∪ ℳ_{2} together with ℳ_{1} ∩ ℳ_{2} the Mandelbrot set of the dynamical system (6.2).

Technical Description: The Julia Set of p = (-1.022, 0.14846) On a z-Canvas Centered at z_{0} = (0, 0) Generated by the Dynamical System z_{n+1} = z_{n}(z_{n}^{2} + p)
The Julia set is connected.

In § 7, we go back to the Mandelbrot set ℳ and discuss possibly the most charming feature it possesses, which is the striking simplicity of the Mandelbrot equation (2.1) generating all the wonders of ℳ we have witnessed. It turned out that the simple form does not diminish the power of the Mandelbrot set in the sense that the Julia sets of any quadratic dynamical systems are geometrically similar to the Julia sets of the Mandelbrot equation.

Rather than showing this fact in full generality (which can be done by high school algebra), we will verify it using a special quadratic dynamical system called the logistic equation. The process involves the important idea of "conjugacy" between two dynamical systems. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.

So, we will show several fractals found by the logistic equation in § 7, even though we may encounter similar fractals using the Mandelbrot equation if we are lucky. In creating fractal art, we can deviate from standard procedures of mathematics such as using critical points and freely conduct computer experiments. Here are examples:

There is a special subset of the Julia fractals consisting of fractals generated by so-called "Newton's rootfinding method." We call them Newton fractals and discuss them in § 8. Here are sample fractals:

People who are familiar with multivariable calculus can venture into plotting fractals in a 3D space. One of the possibilities is to map a fractal from the plane to various surfaces such as a sphere and a torus. We will throw in 3D examples here and there in the upcoming sections.

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. The object 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 generated by the Mandelbrot equation (1.1).

We now use the divergence criterion and a computer to plot the Mandelbrot set ℳ. Let R be a square canvas comprising 2,000 × 2,000= 4,000,000 pixels centered at the origin (0, 0) of the complex plane with radius 2, i.e., R is bounded by xmin = -2,xmax = 2,ymin = -2 and ymax = 2. Defining a canvas is always the first step of fractal plotting.

We call the plotting process given by the if-statement the divergence scheme, so as to contrast it with the convergence scheme, which we will introduce in § 4.

Of course, an actual computer program based on the divergence scheme can be streamlined in many ways. Probably the most important is to use |z_{m}|^{2} > θ^{2} instead of |z_{m}| > θ to avoid using the hidden square root in |z_{m}| and shorten the computing time as it is used millions, if not billions, of times while running the program.

Figure 0.1 shown at the outset of this article is the output image of the computer program 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 an approximation of ℳ plotted on the canvas with finitely many pixels and by replacing ∞ in the definition of ℳ by "up until M = 1000."

Recall that the Mandelbrot set is denoted by ℳ and is closed so it contains its boundary as its subset. It is also known that the topological dimension of the boundary is 1 like the boundary of a circular disk, so we intuitively picture it as an object made of "razor-thin filaments" without thickness. Does it mean that the area of the boundary is zero? Nobody can find the answer, and we suddenly realize that it is considerably more complex than it appears in a global image like Figure 2.1.
Although it may not sound obvious unless we know something about fractal dimensions, the following celebrated theorem implies that no figures on the plane are more complex than the boundary of the Mandelbrot set, boosting the Mandelbrot set to be one of the most complex objects ever plotted on a plane.

Shishikura's Theorem (1998): The fractal dimension of the boundary of the Mandelbrot set is 2 (which is the topological dimension of the plane).

Let's pause for a moment and look at its local image in, say, Figure 2.3, in which a part of ℳ is visible. The intricate image surely looks impressive, but exactly where is the boundary of ℳ and what does it have to do with the colorful patterns? It turned out that the boundary of ℳ 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 image shows that the boundary of ℳ in the rectangular area is vividly self-similar, making it a fractal as per our informal definition. Shishikura's theorem also makes it a fractal according to Mandelbrot's definition: A fractal means a set for which the Hausdorff-Besicovitch dimension (aka the fractal dimension) strictly exceeds the topological dimension.
Through the self-similarity of indefinitely repetitive patterns, we observe that the luminous filaments of the boundary of ℳ get so dense they work like space-filling curves in infinitely many areas of the plane. It provides us with an intuitive idea as to why the "fractal dimension" of the boundary of ℳ is the same as the topological dimension of the plane.

Fractal dimensions in fact measure complexities and space-filling capacities of any curves (i.e., objecs of topological dimension 1) by fractional scales between 1and 2, and the boundary of ℳ attains the maximum value of 2 per Shishikura's theorem. By comparison, the fractal dimension of the boundary of the "Koch Snowflake" shown in Figure 0.4 is 1.2619. It should be noted in the aforementioned Mandelbrot's definition of fractal that the fractal dimension of the boundary of ℳ also exceeds its topological dimension by the largest possible value.
The boundary image of Figure 3.1 also shows that it works like the basic monochromatic line art of the multicolored ukiyo-e woodblock prints, which are completed by the additional steps of coloring between the lines. The finer the line art, the more elaborate the final ukiyo-e print; see Figure 0.6(A).

One of the most important topological properties in fractal geometry is "connectedness" of a set and Figure 3.1 appears to show that the Mandelbrot set ℳ with its complex boundary is "connected" as "one piece." To give precision to the intuitive concept involving "one piece," R. C. Buck adopts the following formal definition in his classical textbook for Advanced Calculus: Suppose S is a nonempty set of points in the xy-plane. S is said to be connected if it is impossible to split S into two disjoint sets, neither one empty, without having one of the sets contain a boundary point of the other.

For example, it is known that the "neck" of the "snowman" in Figure 2.1 is the point (-3/4, 0), and if we cut the head off the body of the snowman with the vertical line x = -3/4, then either the head or the body contains the boundary point of the other, namely (-3/4, 0). Thus, the particular attempt fails to show that ℳ is disconnected. Because of the complexity of its boundary, proving whether or not ℳ is connected is by no means a simple task, as evidenced by the fact that Mandelbrot initially conjectured ℳ to be disconnected and reversed it later without substantiation―before Adrien Douady and John H. Hubbard settled it:
The Douady-Hubbard Theorem (1982): The Mandelbrot set is connected. They also proved that ℳ is "simply connected," which means ℳ has no holes. Topologically speaking therefore, ℳ is well-behaving as a compact set in one piece without a hole. As described by Wikipedia, Douady and Hubbard established many of the fundamental properties of ℳ at an early stage and created the name "Mandelbrot set" in honor of Mandelbrot. They were the pioneers of the mathematical study of ℳ.

"Who Discovered the Mandelbrot Set?" is the title of an interesting read that appeared in Scientific American in 2009. It writes: Douady now says, however, that he and other mathematicians began to think that Mandelbrot took too much credit for work done by others on the set and in related areas of chaos. "He loves to quote himself," Douady says, "and he is very reluctant to quote others who aren't dead."

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. 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 view of the fractal on the left that reveals the boundary of the mini-Mandelbrot set.

Topological Properties (continued): We stated earlier the precise definition of a set being "connected" as "one piece" and now wish to dig into the notion of "pieces" as a preparation for the upcoming sections. We showed, while discussing the definition by Buck, that the "snowman" of Figure 2.1 cannot be split into "two pieces," the head and body, without having either one of them contain a boundary point of the other.

If we restrict our attention to the interior of ℳ which does not contain any of the boundary points, the situation changes completely. Not only can we split the head from the body without worrying about the boundary points, we can actually decompose the snowman into numerous disjoint connected body parts including all those (circular) disks attached to the cardioid body. Note that each of the disks is an open set without a boundary point and it is maximal in the sense that it is not a proper subset of a larger connected subset of the interior of ℳ.

In general, if S is any nonempty set of points in the complex plane, a nonempty maximal connected subset of S is called a connected component of S. It is easy for people familiar with elementary set theory to use the idea of an "equivalence relation" and prove that S can be partitioned into the disjoint union of its connected components. Thus, S is connected if and only if it consists of exactly one connected component (or "piece"). By virtue of the Douady-Hubbard theorem, ℳ has exactly one connected component, but its interior is disconnected and has infinitely many connected components including the aforementioned open disks.

The set S is said to be totally disconnected if it is disconnected and every connected component of S comprises just one point. As we'll see, many fractals are totally disconnected, but the interior of ℳ is not one of them.

Compactness, connectedness, the number of connected components, being simply connected without a hole and being totally disconnected are all topological properties. Topologists generally identify homeomorphic objects and use topological properties to distinguish objects. In the 3D space, for example, a donut and a coffee cup with a handle are the same to topologists but the "broken taiko drum" shown below and a ping pong ball are different.

"Broken Taiko Drum"

Here, we have the mini-Mandelbrot set of Figure 3.3 flipped vertically and painted in different colors and its application in multivariable calculus.

We are not done yet with the complex nature of the Mandelbrot set ℳ and still stay with it. In § 2 and § 3, we discussed the complement and the boundary of ℳ; see "Daytime and Nighttime Views" of a Fractal. We now turn our attention to its interior, namely, ℳ 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 tangentially attached, and to each of these disks another bunch of disks are tangentially attached; see "Mandelbrot Set" by Wikipedia for detail. The fractal pattern repeats as if the cardioid has children, grandchildren, great grandchildren and so on and so forth. Here, a "cardioid" means, instead of the familiar curve, the curve together with all the points inside the curve.
As Figure 3.1 shows, ℳ also contains infinitely many mini-Mandelbrot sets, each of which is a smaller copy of ℳ, again comprising a cardioid (which may be distorted) with infinite generations of disks (which may be distorted) and even smaller mini-Mandelbrot sets. If we remove the boundary of ℳ from ℳ, we are left with the interior of ℳ comprising the interiors of these disks and cardioids, etc., which are, as we have discussed, the connected components of the interior of ℳ.

Atoms and Molecules: Let's use Mandelbrot's idea shown in his article as a cue and call each connected component of the interior of ℳ an atom of ℳ and a (disjoint) union of one or more atoms a molecule. Thus, atoms include the interiors of all those disks and cardioids with various degrees of distortion and possibly other shapes we have not recognized. An atom and the interior of a mini-Mandelbrot set are examples of molecules.

As we saw in § 2, the divergence scheme cannot distinguish these atoms and paints them in a single color like black or white. Our current goal is to develop another simple algorithm called the convergence scheme which will be used to color ℳ like in Figure 4.1 and many other fractals in upcoming sections. Along the way, we will see that the atoms are associated with "periods" like in chemistry (but in a totally different way).

Example 1 (The Mandelbrot Set): Start with the p-canvasR, which is the rectangle in the complex plane with center (-0.52, 0) and horizontal radius 1.65 and comprises 3,000 × 2,500 pixels.

We first apply the divergence scheme with M = 20000 and θ = 2 on R and extract the Mandelbrot set ℳ comprising the pixels p whose orbits do not diverge to ∞. Then apply various convergence schemes with ε = 10^{-8} on ℳ. Figure 4.2 shows the (resized) output images of three molecules.

The first image is generated by the convergence scheme with period index k = 1 and shows that the interior of the cardioid is an atom of period 1. Painting in subtle shades of red is done by a basic technique included in the Fractal Coloring site.

The second image is given by the convergence scheme with period indices k = 1, 2, 3, 4, which is basically defined as the natural sequence of the four convergence schemes, the one with period index k = 1 followed by the one with period index k = 2, etc. It shows that the interior of the largest disk is an atom of period 2 and painted in subtle shades of 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.

where k is the period of the corresponding atom of ℳ. We call λ the period of ℳ '.

For example, the most visible mini-Mandelbrot set of Figure 4.3 happens to have period λ = 4 and is shown in Figure 4.4 and the inset of the periodicity diagram. There, it is painted by the convergence scheme with period indices λ k = 4, 8, 12, ..., 100 and with the colors of the Mandelbrot set of Figure 4.2 so as to emphasize the one-to-one correspondence between the atoms of ℳ and the atoms of the mini-Mandelbrot set.

Here's an additional technical detail: For the convergence scheme with period indices λ k = 4, 8, 12, ..., 100, we used the variable maximum number of iterations

(4.2) M = 20000 - 150k,

rather than a constant like M = 20000 to speed up the computation without notably sacrificing the appearance of the output image.

Figure 4.4. The the mini-Mandelbrot Set of Period λ = 4 and Its Boundary

Periodicity Diagram: If we label the atoms of the Mandelbrot set in Figures 4.2 and 4.4 by their periods instead of colors, we get the following periodicity diagram. The periods in the diagram show meticulously aligned numerical patterns that are easy to recognize and will play an important role in plotting many of the "Julia sets" in the next section. The numerical patterns are yet another amazing property of the Mandelbrot set ℳ.

As mentioned in Introduction, the Julia sets precede the Mandelbrot set by about sixty years and were created by Pierre Fatou and Gaston Julia before computers became available. To show the main ingredients of their work, we begin with a polynomial function of a complex variable z,

consisting of all orbits of z_{0} in the complex plane. For a technicality, we may view any of the coefficients c_{j} to be the constant parameter p but p plays no roles at this stage.

Define the filled-in Julia set of f denoted by 𝒦(f) to be the set of all initial values z_{0} in the complex plane whose orbits do not diverge to ∞ and define the Julia set of f denoted by 𝒥(f) to be the boundary of 𝒦(f). Then it can be shown that 𝒦(f) and 𝒥(f) are both compact and contain infinitely many points. In particular, 𝒦(f) is bounded and contains 𝒥(f) as its subset. The Julia set 𝒥(f) turned out to be a fractal "almost" always, and if it is totally disconnected, we call it a Cantor set or Cantor dust. The Julia set 𝒥(f) that bounds 𝒦(f) also bounds the complement of 𝒦(f) and becomes chaotic.

Julia and Fatou independently proved the following powerful theorem in 1918-1919:

The Fatou-Julia Theorem: (1) If all of the critical points of f belong to 𝒦(f), then 𝒥(f) is connected as one piece, and (2) if none of the critical points of f belong to 𝒦(f), then 𝒥(f) is totally disconnected and forms a Cantor set.

As in the case of ℳ, therefore, the image shows the complement of 𝒦(p) with its boundary 𝒥(p) painted by the goldish color. The image shows that the critical point at its center does not belong to 𝒦(p); hence, the Julia set 𝒥(p) is a Cantor set by the Fatou-Julia theorem.

Recall that the center of the z-canvas used in the example is the critical point z_{0} = 0 of the function f_{p}, whose orbit coincides with the critical orbit of p. Because the period of p is 11, the critical orbit converges to a cycle of period 11 at the center of the canvas. Therefore, the convergence scheme with period index k = 11 is a natural choice in decorating the filled-in Julia set 𝒥_{p}. Note that the critical point belongs to 𝒦(p): hence, 𝒥(p) is connected by the Fatou-Julia theorem.

It is another fascinating fact about the Mandelbrot set that the period of the parameter p is always reflected in the shape of the filled-in Julia set of p, as in the number of "Hydra's heads," although why it is so is not completely understood.

Example 3: The filled-in Julia set of p = (-0.6891, 0.27896) called "Hydra's Ash" in Figure 1.6(C) is given by the parameter near the aforementioned atom of period 11 but lies outside of ℳ. We'll see a quick way to tell why 𝒥(p) is a Cantor set. Interestingly, 𝒦(p) still retains a hydra's shape even though the parameter is detached from the atom.

Here are additional examples of filled-in Julia sets 𝒦(p) born from circular atoms attached to the red cardioid near its cusp:

The Fundamental Dichotomy: For any parameter p in the complex plane, the Julia set of p is either connected or totally disconnected. For example, the Julia set of Figure 1.1(A) is neither connected nor totally disconnected and the dichotomy assures us that a Julia set like this never arises from the dynamical system (5.3). Mandelbrot, who once studied under Gaston Julia and later became an "IBM fellow," used a computer to visualize the fundamental dichotomy that divides up the complex plane into two parts. He initially defined the Mandelbrot set to be

(†) the set ℳ comprising all parameters p in the complex plane whose Julia sets are connected.

He knew how to compute ℳ as the Fatou-Julia theorem clearly implies that the Julia set of p is connected if and only if the critical orbit of p does not diverge to ∞; see the computer-friendly definition of § 2. Thus, the famed Mandelbrot set ℳ was born from the dichotomy of the Julia sets. It also explains why the critical point of (5.3) is indispensable in computing ℳ.

Henceforth, we call (†) the alternative definition of ℳ. It shows that the Julia sets 𝒥(p) of Figures 5.1(B)(C) (and hence 𝒦(p) as well) are connected, while "Hydra's Ash" mentioned in Example 3 is a Cantor set.

The Julia Set of p = (0.25000316374967, -0.00000000895972)

Example 4: The parameter p used for Figure 5.2(A)(B) shown above belongs to the interior of the cardioid atom of the mini-Mandelbrot set of Figure 0.5(A) whose period is large and unknown. So, Figure 5.2(A) is painted by the divergence scheme alone and shows the complement of 𝒦(p) with Figure 5.2(B) showing its boundary 𝒥(p). Because p belongs to ℳ, the Julia set 𝒥(p) is connected by the alternative definition of ℳ.

The two "lions" are painted by the convergence scheme with period index 85 and the background by the divergence scheme with the threshold θ = 2. The curling directions of the mane of the "Twin Lions" are opposite to each other and depend on the locations of the parameters in the atom.

Figures 5.3 & 5.4. "Twin Lions" born from the same atom of period 17 × 5

"Esmeralda Lion" shown in Gallery 2D is an enlarged version of the filled-in Julia set shown above on the left. "Ruby Lion" shown below is an enlarged version of the filled-in Julia set shown above on the right.

The Jordan curve theorem states that a Jordan curve divides the plane into two parts, a bounded region called "inside" and an unbounded region called "outside." 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 geometrically 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 involving algebra, analysis and topology and provides one of the fascinating topics in mathematics.
Example 7: The image shown below is the filled-in Julia set of the parameter p = (-1.0073, 0.2552) chosen from a circular atom of period 2 × 4. The atom is the leftmost blue disk shown in the first image of Figure 4.3 and is attached to the orange atom of period 2. Both factors 2 and 4 are visible in the Julia set.

Figure 5.7. "Run for the Sun"

A Filled-in Julia Set Born from an Atom of Period 2 × 4

It is a local image of the Julia fractal shown in Figure 5.2(A) but is painted by using different colors and the eyeball effect. The eyeball effect makes it easier to identify the numerous "cuttlefish" swimming in the global image, in which their eyes are closed. Note that one of the cuttlefish is at the center of the image.

Figure 5.2(A) is a global Julia set of the parameter p = (0.25000316374967, -0.00000000895972) and if we zoom in on the center of the figure between the eyes of the central cuttlefish, we get another local image revealing the cuttlefish's mouth painted black:

Recall that the Mandelbrot set and a (filled-in) Julia set belong to two different complex planes, one comprising parameters p and the other initial values z_{0} of the Mandelbrot equation (1.1). The Mandelbrot set is by definition the set of all parameters p whose critical orbits do not diverge to ∞ and a filled-in Julia set is similarly defined in the other complex plane.

A parameter p is called a Misiurewicz point if the critical orbit of p is not a cycle but becomes a cycle after finitely many iterations. For example, while discussing (1.1), we saw that the critical orbit of p = -2 is

Because it is not a cycle but becomes a 1-cycle after two iterations, the parameter p = -2 is a Misiurewicz point.

Some of the known facts are: (1) Misiurewicz points belong to the boundary of the Mandelbrot set. (2) If p is a Misiurewicz point, then the filled-in Julia set of p has no interior points, hence, coincides with the Julia set of p.
(3) Misiurewicz points are "dense" in the boundary of the Mandelbrot set, i.e., every open disk about a point on the boundary of the Mandelbrot set contains a Misiurewicz point.

Tan Lei's Theorem (1990): If p is a Misiurewicz point, the Julia set of p centered at z_{0} = 0 and a local image of the Mandelbrot set centered at p are asymptotically similar through uniform scaling (enlarging and reducing) and rotation; see Wikipedia and geometric similarity.

At first glance, the scope of Tan Lei's theorem seems to be rather limited because of the aforementioned properties (1) and (2), but (3) boosts the theorem to be enormously powerful: Let p be a parameter on or near the boundary of the Mandelbrot set. Then it is either a Misiurewicz point or near a Misiurewicz point, and consequently, in a local image of the Mandelbrot set centered at p, we are likely to see a shape resembling the Julia set of p near its center z_{0} = 0. For this reason, the Mandelbrot set is sometimes called an "index" to all Julia sets.

This probably explains why the local images like Figures 5.9 and 5.10 are strikingly similar even though the parameter p belonging to the interior of the mini-Mandelbrot set is not a Misiurewicz point. The sidenote to Figure 3.3 shows that the distance between p and a nearby Misiurewicz point is much less than 10^{-13}. Figure 5.11 shows we can zoom out from Figures 5.9 and 5.10 while retaining some degree of similarity.

Figure 5.11. "Cuttlefish" Swimming in the Mandelbrot Set (Left) and in the Julia Set (Right)

Soon after Mandelbrot published its computer plot generated by the simple process of § 2 in 1980, the Mandelbrot set became so popular that a great many computer hobbyists, digital artists, mathematicians and scientists have explored around it and shown their fractal art 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 local images of the Mandelbrot set or Julia sets that look drastically different from what have been published by using available computers and software. An easy way to find a new pattern such as the one shown below is to use a dynamical system other than the Mandelbrot equation and there are infinitely many of them.

The origin (0. 0) of the complex plane is at the tip of the spearhead, which coincides with the upper left corner of the closeup image shown below. We call the area "Spearhead Bay." Like the Mandelbrot set, the Giant M Set contains infinitely many circular atoms that satisfy the numerical pattern of the periodicity diagram. These circular atoms include the largest and the second largest blue atoms shown in Spearhead Bay, whose periods happened to be 7 and 8, respectively. Note that the "seaweed" growing out of the blue atom of period 7 contains seven-way junctions and likewise for the "seaweed" around the atom of period 8.

We also note that in Spearhead Bay, the seaweed grows only on the side of the Giant Mandelbrot Set and tangles with infinitely many extra atoms that look like tropical fish.
Interestingly, the fish-like atoms begin to disintegrate near the circular atom of period 6, which is painted purple at the mouth of Spearhead Bay, and they become extinct near the circular atom of period 5, which is located just outside of the bay.

"Spearhead Bay"

The boundary of the Giant M Set near the circular atom of period 5 is depicted in the image shown below. It shows no signs of fish but, like in the Mandelbrot set, it contains five-way junctions and encloses numerous mini-Mandelbrot sets. Unlike the Mandelbrot set however, the boundary now appears to be disconnected.

"Seaweed with Five-Way Junctions"

Another area in Figure 6.1 that provides a rich fishing ground for attractive fractals is in and around the blue molecule located between the spearhead and the Toddler M 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 6.1 also contains two "Squished M Sets," each of which has a "bursted" cardioid. The molecule can be seen near the top of Figure 6.1, 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."

Let ℳ_{1} = ℳ (i / √3) and ℳ_{2} = ℳ (-i / √3) so ℳ_{1} and ℳ_{2} are the Mandelbrot sets of the conjugate critical points i / √3 and -i / √3 of f_{p} in (6.2), respectively. Thus, ℳ_{1} comprises the parameters p whose orbits with the initial value z_{0} = i / √3 do not diverge to ∞ and ℳ_{2} likewise. As mentioned earlier, ℳ_{1} is given by Figure 6.1, and as shown in Figure 1.7(A), ℳ_{2} turned out to be symmetric to ℳ_{1} through the real axis of the complex plane.

As shown in Figure 1.7(A), if we superimpose ℳ_{1} and ℳ_{2}, we get a surprising result illustrated by what we call "Atomic Fusion" in fractal geometry. For example, "Big Spearhead" atom in ℳ_{2} fits perfectly in the cardioid atom of ℳ_{1} and the lips of the "Broken Balloons" in ℳ_{2} are beautifully repaired by the "Squished M Sets" of ℳ_{1}; see Figure 6.6 shown above.

Figure 6.7 shows a simplified version of "Atomic Fusion," where most of the boundary curves in ℳ_{1} and ℳ_{2} are omitted for the ease of its interpretation. Here, the union ℳ_{1}∪ℳ_{2} is painted by colors other than black and the intersection ℳ_{1}∩ℳ_{2} by colors other than black and green. Thus, the black zone is the complement of ℳ_{1}∪ℳ_{2} denoted by [ℳ_{1}∪ℳ_{2}]^{c} and the green zone is the symmetric difference

ℳ_{1}Δℳ_{2} = ℳ_{1}∪ℳ_{2} - ℳ_{1}∩ℳ_{2}.

The concept of Julia set naturally extends from (2.1) to our dynamical system (6.2). Thus, the filled-in Julia set of a parameter p, denoted by 𝒦 (p), is the set of all possible initial values z_{0} of (6.2) in the complex plane whose orbits with the fixed value of p do not diverge to ∞ and the Julia set of p, denoted by 𝒥 (p), is the boundary of the filled-in Julia set.

Therefore, if z_{0} is a critical point of (6.2), then z_{0} belongs to 𝒦 (p) if and only if the orbit of p with the initial value z_{0} (which is the same as the orbit of z_{0} with the fixed parameter p) does not diverge to ∞, i.e., p belongs to ℳ_{1} if z_{0} = i / √3 and p belongs to ℳ_{2} if z_{0} = -i / √3.

It is now straightforward to prove that the Fatou-Julia Theorem applied on (6.2) can be written as follows:

(1) If p belongs to ℳ_{1}∩ℳ_{2} then the Julia set of p is connected;
(2) if p belongs to [ℳ_{1}∪ℳ_{2}]^{c} then the Julia set of p is a Cantor set.

(1) and (2) imply:

(3) If the Julia set of p is disconnected but not a Cantor set, then p belongs to ℳ_{1}Δℳ_{2}.

All of our relevant computer output seems to show that the converse of (3) is true, but the Fatou-Julia Theorem does not confirm it.

Recall that the Mandelbrot set of (2.1) is initially defined so as to visualize the fundamental dichotomy, which is the Fatou-Julia theorem applied on (2.1) with exactly one critical point. Note also that we have just shown that the "Atomic Fusion" comprising ℳ_{1}∪ℳ_{2} and ℳ_{1}∩ℳ_{2} illustrates the Fatou-Julia theorem applied on (6.2) with exactly two critical points. It is natural, therefore, that we call ℳ_{1}∪ℳ_{2} together with ℳ_{1}∩ℳ_{2} the Mandelbrot set of the dynamical system (6.2).

In general, the Mandelbrot set of the dynamical system (6.1) comprises the union and the intersection of all of the Mandelbrot sets of the critical points of (6.1) satisfying the properties comparable to (1), (2) and (3). In particular, if (6.1) has exactly one critical point, then the Mandelbrot set of (6.1) coincides with the Mandelbrot set of the critical point as is the case in (2.1).

Just like the Mandelbrot set of (2.1), the Mandelbrot set of (6.2), er the "Atomic Fusion" diagram, is indispensable in finding the Julia sets of (6.2) with infinitely many varieties, although we have to be a little careful with its hairy boundary not shown in the diagram.

Example 1: The Julia set of Figure 0.1(B) called "Twin Dragons" and shown at the outset of this website is given by the parameter p = (0.185, 0.00007666) belonging to ℳ_{1}∩ℳ_{2}; hence, it is connected. It is actually given by rotating the output image 90^{o} to better fit on the webpage; see geometric similarity. If we move the parameter to p = (0.185, 0) that lies on the real axis, the output image becomes symmetric about the center horizontal line providing us with "Identical Twin Dragons." Figure 6.8 shown below contains three topologically distinct "Twin Dragons."

It can be seen near the bottom of Figure 6.7 that it intersects with both ℳ_{1}∩ℳ_{2} and the symmetric difference ℳ_{1}Δℳ_{2}.

The connected "Roses" of Figure 6.9(A) is the Julia set of the parameter p = (0.02912, -1.093853) belonging to an atom of period 3 × 7 in "Broken Balloons." The parameter p belongs to ℳ_{1}∩ℳ_{2}, so the numerous "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 ?

The disconnected "Roses" of Figure 6.9(B) shows the Julia set of the parameter p = (0.07761, -1.12427) belonging to an atom of period 3 × 4 in "Broken Balloons." The parameter p is chosen from the symmetric difference ℳ_{1}Δℳ_{2}, so the Julia set is disconnected, which we can see in the broken "stems." Note that the Julia set is not a Cantor set.

Example 3: The "Toddler M Set" seen near the bottom edge of Figure 6.1 belongs to ℳ_{1}Δℳ_{2} but it is omitted from Figure 6.7. Recall that it comprises atoms of periods k = 2 × 1, 2 × 2, 2 × 3, · · · . It produces a great many attractive fractals but they are naturally similar to the fractals coming out from the Mandelbrot set for the quadratic system—except that they are all disconnected. For example, the image which is shown below and resembles the "Hydra" of Figure 5.1(A) is the Julia set of p = (0.00399109,-1.98545775) belonging to an atom of period 2 × 13. It contains numerous dots in its background each of which is a baby hydra.

Figure 6.11. "Lernaean Hydra with Thirteen Heads and Offsprings"

Example 4: While the "Toddler M Set" generate Julia sets that resemble Julia sets of the Mandelbrot set seen in § 5, the "Giant M Set" produces Julia sets that do not resemble anything from the Mandelbrot set, apparently affected by the "Spearhead." "Twin Dragons" of Figure 6.8 are such examples. Here is another, this time from near the neck of the giant.

As we have seen, the Mandelbrot equation (1.1) generates infinitely many Julia sets through the iterations of (1.1), which in turn give rise to the enormously complex Mandelbrot set through the fundamental dichotomy of the Julia sets. In the whole process, the first thing we note is probably the striking simplicity of the formula (1.1).

When we paint a fractal, we use a canvas comprising millions of pixels. Since coloring each pixel easily requires thousands of orbit evaluations, one extra addition or multiplication of complex numbers in a formula like (1.1) can make a significant difference in a computer's runtime. So, the simplicity is not only aesthetically pleasing but also important from a practical viewpoint.

Fortunately, it turned out that (1.1) with its simple form is not as constraining as it first appears. The reason is that if (6.3) is any quadratic dynamical system, then we use high school algebra of "completing the square" to show that it is "conjugate" to (1.1), guaranteeing that any Julia set generated by (6.3) is geometrically similar to a Julia set generated by (1.1) and vice versa. Thus, by knowing the Julia sets of (1.1), we effectively know the Julia sets of (6.3).

Rather than showing the "conjugacy" in full generality, we will verify it using a special quadratic dynamical system called the logistic equation. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.

What is the logistic equation? In 1838 Pierre Verhulst introduced a differential equation called the "logistic equation," which became widely used to describe the population dynamics with self-limiting growth. If we replace the derivative in the differential equation by its approximating difference quotient and do some algebra, we get the following "difference equation," which is more suitable for computer applications and again called the logistic equation:

Expand its variables and parameters to complex numbers and let ℳ ' be the aforementioned "Mandelbrot set" defined by (7.1). Then by the reason shown in the remark, we get ℳ ' shown below by applying the divergence and convergence schemes on the critical orbits of p with the fixed initial value z_{0} = 0.5. For convenience, we call the entire moleculeℳ ' comprising the atoms together with its hairy boundary the logistic set, short for the "logistic equation's Mandelbrot set."

As we'll see later, there is a two-to-one function F mapping the logistic set onto the Mandelbrot set such that for any parameter p, the Julia sets of p and F(p) are geometrically similar; see Figure 7.4. The function F becomes a one-to-one correspondence if it is restricted to the left half or the right half of the logistic set. Thus, F actually fails to be two-to-one at the intersection point (1, 0), which works as a "double cusp" of the Mandelbrot set.

Julia Sets by the Logistic Equation: We now show a few (filled-in) Julia sets generated by the logistic equation. Figure 7.8 shows the filled-in Julia set of the parameter p = (2.994915, 0.1) belonging to the red atom of period 1 in the logistic set at Elephant Bay. Hence, it is decorated by the convergence scheme with period index 1 and its complement by the divergence scheme. The second image shown below is the boundary of the filled-in Julia set, namely, the Julia set of p = (2.994915, 0.1). It is a Jordan curve which is homeomorphic to a circle.

Julia Fractal of q = (3.0014564, 0.08) by the Logistic Equation (7.2)

The parameter p = (3.0237615, 0.1) that generates "Dancing Seahorses" shown below belongs to the orange atom of period 2 in the logistic set at Elephant Bay, hence the filled-in Julia set is painted by the convergence scheme with period index 2. Elephant bay is sandwiched by a red atom of period 1 and an orange atom of period 2, and interestingly, a parameter from the orange shore generates "seahorses" instead of "elephants."

and reserve z_{n} and p for the Mandelbrot equation

(7.3) z_{n+1 } = z_{n}^{2} + p ,

where p and q ≠ 0 are constant parameters, while the initial values ζ_{0} and z_{0} vary through the entire complex plane. It is important to remember that the filled-in Julia set of p by (7.3) is by definition the set of all z_{0} in the complex plane whose orbits z_{n} do not diverge to ∞ and likewise for the filled-in Julia set of q by (7.2). Also, the Julia set of q means the boundary of the filled-in Julia set of q.

Secondly, applying the triangle inequality on the transformation (7.4) and its inverse, it is easy to show that ζ_{n} diverges to ∞ if and only if z_{n} diverges to ∞; hence, the transformation (7.4) with n = 0 maps the (filled-in) Julia set of q onto the (filled-in) Julia set of p in a one-to-one fashion.

It is not particularly difficult to show that the transformation (7.4) with n = 0 is not only a homeomorphism but also a "similarity transformation" from the complex plane as the set of ζ_{0} to the complex plane as the set of z_{0} so that the aforementioned Julia sets are geometrically similar.

Now, without assuming conjugacy, we wish to show that (7.2) can be written in the form

(7.5) a ζ_{n+1 } + b = (a ζ_{n} + b)^{2} + p,

which is the result of applying (7.4) on (7.3). The process involved is precisely the same as finding the vertex of the parabola given by a quadratic function in high school algebra. Rewrite (7.2) as

-q ζ_{n+1 } = q^{2} ζ_{n}^{2} - q^{2} ζ_{n} ,

i.e., a ζ_{n+1 } = (a ζ_{n})^{2} + 2b(a ζ_{n}) ,

where a = -q and b = q/2. Completing the square with respect to a ζ_{n} , we get

Example: The first image of Figure 7.11. shows the filled-in Julia set of the parameter q = (3.02382, 0.1) generated by the logistic equation (7.2) and the second image the filled-in Julia set of p = q(2 - q)/4≈ (-0.77146, -0.10119) generated by the Mandelbrot equation (7.3). By the aforementioned theorem, they are geometrically similar. Although the two images are painted by exactly the same coloring routine, the artist's renderings of the filled-in Julia sets turned out to be a little different. It shows that the conjugacy relation preserves the geometric shape of the filled-in Julia set but not necessarily its coloring.
Finally, the dynamical system

Figure 7.12 is a global Mandelbrot fractal of the critical point z_{0} = 1/√3 of the function f_{p}. Figure 7.13 shows two local Mandelbrot fractals of noncritical points z_{0} = 0.1 and z_{0} = 0.5. The circular atoms of the global image are cracked and deformed by the use of the noncritical points and give birth to interesting figures like the ones shown in Figure 7.13. These figures often have strong resemblance to Julia fractal born from the atoms. Figure 7.14 shows a closeup of a crack painted on a plane and on an egg.

Figure 7.15. "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 equationz_{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 7.19. "Dancing Bouquet" by the Fifth Degree Logistic Equation

The idea of the Julia set of a polynomial we saw in § 5 naturally extends to a rational function with minor modifications. So, suppose g is a polynomial of degree greater than one and consider a dynamical system of the form

(8.1)
z_{n+1 }= z_{n} - g(z_{n})/g'(z_{n}),

which generates, like in § 5, a (filled-in) Julia set of (8.1). The reader may have noted already that (8.1) is nothing but the Newton-Raphson Root-Finding Algorithm, aka
Newton's Method for finding roots of g. For this reason, we call a Julia Fractal featuring the Julia set of (8.1) a Newton fractal of g.

As a benefit of using Newton's method, each orbit of (8.1) converges to a root of g quickly more often than not, 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. Also, because g is a polynomial, we may take advantage of the time-saving scheme called
Horner's Method to efficiently evaluate bothg and g' that appear in (8.1). 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.

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, "orbit" and "canvas" always mean an orbit of z_{0} and z-canvas, respectively, in this section. It is important to remember that z_{0} is an initial value for computing a root by Newton's Method (8.1).

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

g(z) = z^{ n} - 1 ,

as its roots r_{0}, r_{1}, r_{2}, ... , r_{n-1}, called the nth roots of unity, are given in a trigonometric expression by

r_{k} = cos(2kπ/n) + i sin(2kπ/n) with r_{0} = 1.

The fact that each r_{k} is indeed a root of the polynomial g(z) follows immediately from De Moivre's formula.

The second image of Figure 8.1 is a variation of the first. The image shown below, called "Crab Queue," is given by zooming in on one of the "bands" in the second image. It is accompanied by a fractal showing the Julia set in "Crab Queue."

Example 2 (Cyclotomic Polynomials): Another interesting example with known roots is a
Cyclotomic Polynomial. The picture on the left in Figure 8.2 is a Newton fractal of the "30th cyclotomic polynomial"

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"

Once our computer program starts running smoothly, plotting Newton fractals provides us with great entertainment. It is easy to pick an input polynomial from infinitely many choices with anticipation from not knowing what to expect in the output. Furthermore, a high-res output image generally emerges within minutes rather than hours and days of runtime. Figures 8.3through 8.8 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.