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 is the article entitled "Stories about Fractal Art." It shows a host of fractal images intended to be the main attraction and offers the text as optional references. The text occasionally touches upon certain areas of advanced mathematics, but it is written in a way that is accessible to people who have not experienced beyond introductory calculus.

The website has additional galleries of fractal art: Gallery 2D mainly shows two-dimensional (2D) images given by the basic programming algorithms described in "Stories about Fractal Art," 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.

It was in 1980 when Mandelbrot showed the famous fractal now called the Mandelbrot set generated by a simple dynamical system and a computer. Almost immediately after that, the novelty and complexity of the Mandelbrot set reinvigorated the interests in fractals and stimulated mathematicians to develop further theories in fractal geometry.
On the other hand, chaos often associated with fractals was basically born as a brand new subject in 1974 from biologist Robert May's computer simulations of population dynamics through the dynamical system called the logistic equation. Like "fractal," the word "chaos" was used as a mathematical term for the first time in 1975 when the American Mathematical Monthly published "Period Three Implies Chaos" by T.Y. Li and James Yorke. The paper received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes. It soon became known that fractals and chaos are closely related and together they provide applications in science as well as in art.

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 activities of fractal plotting. A fairly large part of what follows is dedicated 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 or coding but instead tells the general principles for fractal plotting in everyday language assuming the readers' basic programming experience.

Particularly exciting is the moment the fractal image generated by our personal program emerges in our computer screen because of its possibly "amazing" artistry and built-in chaos-related unpredictability. 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.

People who wish to understand fractals in some depth need to know (a) elementary algebra and geometry of complex numbers and (b) beginning calculus. In addition, people interested in learning fractal plotting should have some basic computer programming experience.

(a) includes the practice of writing a complex number z as a point (x, y) in the xy-plane as well as the standard algebraic expression z = x + yi and ability to do basic arithmetic of complex numbers such as addition and multiplication. The complex plane means the set of all complex numbers z = (x, y) which coincides with the Cartesian xy-plane. For each complex number z = (x, y), the absolute value of z means |z| = √(x^{2} + y^{2}) and it represents the distance of z from the origin O of the complex plane. More generally, if u and v are complex numbers, |u - v| represents the distance between u and v, which satisfies the triangle inequality

|u - v| ≤ |u| + |v|.

Setting w = u - v, it follows that |w| ≤ |w + v| + |v|; hence, another way of writing the triangle inequality is

|w + v| ≥ |w| - |v|.

(b) includes the basic ideas about the
derivative and a critical point of a function where the derivative vanishes.

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

Note that a z-canvas is associated with a fixed parameter value p, hence the orbits used to paint a fractal on the z-canvas have variable initial values z_{0} while the orbits used to paint a fractal on a p-canvas have variable parameter p and fixed initial value z_{0}. Here's an example:

In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set, which is a fractal generated by the Mandelbrot equation, had become "the most famous object in modern mathematics." Naturally then, a large part of the article, § 2 - § 5 and § 7, is devoted to the Mandelbrot set and a myriad of its attributes that fascinate mathematicians and artists alike. We'll define the Mandelbrot set carefully in § 2 and denote it by ℳ.

In § 2 and § 4, we 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 difference between the two algorithms. Because fractal art is wide open to new ideas and discoveries, people learning fractal plotting should be free-spirited and carry out frequent computer experiments. The next two images are examples that popped up from rather aberrant experiments by a computer and the algorithms. The cause for the latter has been documented as in the eyeball effect but the former is still under investigation.

The Mandelbrot set ℳ turned out to be the most complex object ever plotted on a plane and we show why from an intuitive viewpoint in § 3. We also show the meticulous numerical structure of the interior of ℳ interweaving its atoms with periods in § 4; see the periodicity diagram. Despite all these astounding intricacies, ℳ remains, topologically, a relatively simple object, which is compact and connected and without any holes.

In § 5, we define the Julia set of a parameter p generated by the Mandelbrot equation and show how to plot it on a z-canvas using the divergence and/or convergence schemes. Frequently, we observe striking similarities in appearance between local images of ℳ and Julia sets and learn that it is no accident. To further consolidate the close-knit relations between ℳ and Julia sets, we also learn that ℳ is completely characterized as the set comprising parameters whose Julia sets are connected.

If a parameter p belongs to an atom of ℳ, then the period of the atom affects the shape of the Julia set of p in an amazing and sometimes amusing way, although its cause is shrouded in mystery. It makes the aforementioned periodicity diagram all the more important in plotting Julia sets. For example, the image shown below is the Julia set of a parameter chosen from an atom of period 9, while Figure 1.4(A) shows two Julia sets born from the same atom of period 85 = 17 × 5.

In § 6, we extend the ideas of the Mandelbrot and Julia sets to Mandelbrot fractals and Julia fractals generated by general dynamical systems. We take quite a bit of time on a comical fractal we call the "Speared Mandelbrot Set" generated by a cubic dynamical system with two critical points. By comparison, the Mandelbrot equation has a single critical point that provides us with a simple and elegant classification of its Julia sets called the Dichotomy Theorem.

As we'll see, the two critical points make the matter more challenging as well as interesting and help us forecast highly convoluted circumstances if the dynamical system has three or more critical points. The next image shows a disconnected non-Cantor Julia set defying the Dichotomy Theorem.

In § 7, we turn our attention to the logistic equation, which became famous with the advent of chaos in the 1970s, and show (using high school algebra) that the Mandelbrot equation and the logistic equation are "conjugate" to each other, which implies that they share the same Julia sets. The process effectively indicates that all quadratic dynamical systems are in fact conjugate to the Mandelbrot equation and explains part of the reasons why the Mandelbrot equation in such a simple form generates such a complex and fascinating object as the Mandelbrot set.

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."

Benoit Mandelbrot rocked the mathematics world in 1980, when he introduced his computer-generated images of the fractal now called the Mandelbrot set. Its novelty and intricacy and the fact that it can be generated by such a simple process invigorated a great many mathematicians and scientists to engage in their research and numerous articles on its extraordinary properties appeared in mathematics books and journals, popular magazines and major newspapers. In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set had become "the most famous object in modern mathematics." In addition to the fact that it has limitless varieties of astounding local images as indicated in § 1 and § 2 (rather modestly), we will see many more reasons for the fame in § 3, § 4, § 5 and § 7.

Recall that the Mandelbrot set is closed so it contains its boundary as its subset. It is 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 the most complex object ever plotted on a plane.

Shishikura's Theorem (1998): 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, recall that the Mandelbrot set is denoted by ℳ 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.

One of the most important topological properties in fractal geometry is "connectedness" of a set and Figures 3.1 and 3.1(B) appear 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 Figures 3.1 and 3.1(B) show, 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. Atoms and the interiors of mini-Mandelbrot sets and ℳ 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.

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 have interesting 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.

We have so far viewed the complex plane as the set of parameters p and the Mandelbrot equation (2.1) as the collection of all orbits of p varying through the complex plane while the initial value z_{0} is fixed at the critical point z_{0} = 0 of the function f_{p}. As shown in § 2, we use these critical orbits to define the Mandelbrot setℳ, which possesses the dazzling features, mathematical and artistic, we have witnessed in § 2 through § 4. In this section, we will show, as yet another fascinating attribute of the Mandelbrot set ℳ, that almost every parameter p on or near ℳ gives rise to an intricate fractal called the "Julia set" of p.

We now define the filled-in Julia set of p to be the set of all initial values z_{0} in the complex plane whose orbits z_{n} (with the fixed value of p) do not diverge to ∞. Because the definition of the filled-in Julia set is almost identical to the definition of the Mandelbrot set, we expect that the divergence and convergence schemes are again effective in plotting the filled-in Julia sets, this time on a z-canvas instead of a p-canvas. The use of these plotting schemes is explained in detail in Fractal Coloring Algorithms.

If p is a parameter in the interior of ℳ then p belongs to a unique atom of ℳ by what we have seen in the preceding section. So, we say (rather fancifully) that the filled-in Julia set of such a parameter p is "born from the atom of ℳ." Here are examples:

where |p| ≤ 2; see proposition A. Like the Mandelbrot set of Figure 2.1, the white "Hydra" is an approximation of the filled-in Julia set of p.

The second image of Figure 5.1(B) is given by applying the convergence scheme with period index k = 11 on the filled-in Julia set. It is an artist's rendering of the filled-in Julia set. 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. From now on, we will normally omit mentioning "approximation" and "artist's rendering."

We also 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 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 of Example 1.
The filled-in Julia set of Figure 5.1(A) is painted by the same divergence and convergence schemes with p = (-0.692712, 0.273012) belonging to the same atom of period 11. People with sharp eyes may have noted, however, that the "Hydra" coils differently from the "Hydra" of Figure 5.1(B). This is because the coiling direction depends on the position of p in the atom. "Medusa Lions" of Figure 5.0 are generated by parameters chosen from the atoms of periods 10 and 21 near the cusp of the cardioid of the Mandelbrot set.

By a Cantor set or Cantor dust is meant a totally disconnected set with infinitely many components and a fractal structure. It was named after Georg Cantor, the pioneer of set theory, who discovered the early form of the fractal in 1883.

Example 2: The parameter p = (-0.6891, 0.27896) used for "Hydra's Ash" shown above is near the atom of the period 11 but lies outside of the Mandelbrot set; hence, the Julia set of p is a Cantor set.

The two images are painted by the convergence scheme with period index 85 and its 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.

Figure 5.4. "Twin Lions" born from the same atom of period 17 × 5

Example 5: "Esmeralda Lion" with a technical description 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

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 5.10 and 5.11 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 5.12 shows we can zoom out from Figures 5.10 and 5.11 while retaining some degree of similarity. The cuttlefish on the left has the mini-Mandelbrot set of Figure 5.11 at the midpoint between its "eyes" and the cuttlefish on the right the "black hole" of the Julia set of Figure 5.10 instead. Each figure contains, as we can see, infinitely many cuttlefish pointing exactly where we can find other mini-Mandelbrot sets and "black holes" through the self-similarity. The "black holes" that are not part of the Julia set indicate an enormous topological complexity of "Cuttlefish Lion" of Figure 5.9.

Figure 5.12. "Cuttlefish" Swimming in the Mandelbrot Set (Left) and in the Julia Set (Right)

Soon after Mandelbrot published its computer plot 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 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 local images of the Mandelbrot set or Julia sets that look drastically diffferent 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.

If R is a p-canvas associated with a constant initial value z_{0}, then for each pixel (i, j), er, parameter p, on R, (6.1) defines the orbit of p, which allows us to use the divergence and/or convergence schemes to plot a fractal on R called a Mandelbrot fractalof z_{0}. For example, all of our images involving the Mandelbrot set are Mandelbrot fractals of the critical point z_{0} = 0 given by the dynamical system (2.1). Thus, z_{0} is often a critical point of f_{p} but it is not a requirement. In § 7, we will see quite a few Mandelbrot fractals of noncritical points.

Similarly, if R is a z-canvas associated with a constant parameter p, then for each pixel (i, j), er, initial value z_{0}, on R, (6.1) defines the orbit of z_{0}, which allows us to plot a fractal on R. The fractal is called a Julia fractalof p plotted on the z-canvas. For example, all images of the preceding section involving the (filled-in) Julia sets are Julia fractals generated by the dynamical system (5.1).

Like in Figure 4.1, the central object in Figure 6.1 is the portion comprising the parameters whose orbits do not diverge to ∞ called the "Speared Mandelbrot Set." It is the complement of the green background comprising the parameters whose orbits diverge to ∞.

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 Mandelbrot 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"

A closeup of the Toddler Mandelbrot Set is shown below. Compared to the Mandelbrot set, the Toddler Mandelbrot Set has a proportionately larger head (like a toddler) and its boundary is disconnected from the boundary of the Giant Mandelbrot Set. 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.

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 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 6.1 also contains two "Squished Mandelbrot 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."

Recall that Figure 6.1 is a Mandelbrot fractal of z_{0} = i / √3, which is a critical point of f_{p} in (6.2), 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 6.1 through the real axis. If we superimpose the two mirror images, we get a surprising results as shown in Figure 6.6: The big Spearhead in one image fits perfectly in the cardioid body of the "the Giant Mandelbrot Set" in the mirror image and the lips of the "Broken Balloons" in Figure 6.3 are beautifully repaired by the "Squished Mandelbrot Sets" of Figure 6.4.

Figure 6.6. The "Giant Mandelbrot Set" with the "Mandelbrot Balloons"

The concept of Julia set naturally extends from the Mandelbrot equation to a more general dynamical system (6.1). Thus, the filled-in Julia set of a parameter p in (6.1) is the set of all possible initial values z_{0} of (6.1) in the complex plane whose orbits with the fixed value of p do not diverge to ∞ and the Julia set of p is the boundary of the filled-in Julia set. A lot of things about the general Julia sets are still in mystery, however, and belong to experimental mathematics by the use of computers.

Here is a fascinating and useful fact, however. Gaston Julia and Pierre Fatou independently proved it in 1918-1919, way before the computer era. It is a version of what is now known as the Fatou-Julia Theorem.
Theorem: Consider a dynamical system of the form

with two critical points i/√3 and -i/√3 of f_{p}. Let A be the set of parameters p 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 precisely the "Speared Mandelbrot Set" depicted by Figure 6.1 and B the mirror image of A through the real axis. As we saw in Figure 6.6, A and B have interesting relations when they are superimposed.

Figure 6.7 shows a pseudo Venn Diagram of A∪B and A∩B which we call the "Venn Diagram." Here, the union A∪B is painted by colors other than black and the intersection A∩B by colors other than black and green. Thus, the black zone is the complement of A∪B denoted by [A∪B]^{c} and the green zone is the symmetric difference

AΔB = A∪B - A∩B.

Note that the "Venn diagram" quickly gets a lot more complex if f_{p} has three or more critical points.

In terms of the "Venn Diagram," the Fatou-Julia theorem states:

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

Thus, the symmetric difference in (2) plays the role not seen in the dichotomy theorem. Although the "Venn Diagram" works well for our purpose of forming examples that illustrate the Fatou-Julia Theorem, we need to be a little careful as the diagram does not include the hairy boundaries of A and B seen in Figure 6.1. Also omitted in the diagram are the Toddler Mandelbrot Set in A and its mirror image in B. Both of them belong to the green zone AΔB.

Example 1: The Julia set of Figure 0.2 called "Twin Dragons" and shown at the outset of this website is given by the parameter p = (0.185, 0.00007666) belonging to A∩B; 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 the "Venn diagram" and it intersects both A∩B and the symmetric difference AΔB which is the green zone.

The connected "Roses" of Figure 6.9(A) is a Julia fractal of the parameter p = (0.02912, -1.093853) belonging to an atom of period 3 × 7 in "Broken Balloons." The parameter p also belongs to A∩B, 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) is a Julia fractal of the parameter p = (0.07761, -1.12427) belonging to an atom of period 3 × 4 in "Broken Balloons." The parameter p also belongs to the symmetric difference 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 ?

Example 3: The "Toddler Mandelbrot Set" seen near the bottom edge of Figure 5.1 belongs to AΔB but it is omitted from the "Venn diagram." 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—except that they are all disconnected as "the Toddler Mandelbrot Set" belongs to AΔB. For example, the image which is shown below and resembles the "Hydra" of Figure 5.1 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 is a baby hydra.

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

Example 4: While the "Toddler Mandelbrot Set" generate Julia sets that resemble Julia sets of the Mandelbrot set seen in § 5, the "Giant Mandelbrot 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 from near the real axis through the Giant Mandelbrot set. Here is another, this time from near the neck of the giant.

There is one remaining and possibly the biggest selling point of the Mandelbrot set we would like to discuss and it is the striking simplicity of the Mandelbrot equation (1.1) from which the Mandelbrot set is defined and all the wonders we have witnessed are generated. It turned out that the simplicity is a disguise and any quadratic dynamical system is "conjugate" to (1.1) for some p in the sense that any Julia set of the former is geometrically similar to the Julia set of the latter (and vice versa). We recall that the Mandelbrot set is completely characterized as the set comprising parameters p whose Julia sets are connected per its alternative definition.

Mathematically speaking, therefore, the Mandelbrot set is loaded with information on all quadratic dynamical systems. Rather than showing the "conjugacy" in full generality which merely involves completing the square used in high school algebra, we will verify it using an example of quadratic dynamical systems called the logistic equation. The logistic equation became famous in the 1970s with the advent of chaos and it is interesting in its own right.

Artistically speaking, the "conjugacy" may be a letdown but we should still retain our interests in plotting fractals using the logistic equation. Figure 7.11 shows that the "conjugacy" that preserves the geometric shape of a Julia set need not preserve the colors and textures of the fractal. Besides, getting just the right shapes and colors in fractal plotting is very often a chance encounter, and it is unlikely that what we get from the logistic equation someday emerges from the Mandelbrot equation.

Figure 7.1 depicting a mini-Mandelbrot set given by the logistic equation may well be a fractal that cannot be found by the Mandelbrot equation. It also shows that a coloring change alone may drastically alter the appearance of a fractal, indicating that art is more sensitive to a variety of factors than mathematics.

So, it is natural that we plot Mandelbrot fractals of the dynamical system (7.1) by zooming in on the interval [α, 4]. Figure 7.3 is one of them and uses a noncritical point z_{0} = 0.1 (10% of the sustainable population) of f_{p} as the initial value for the orbits of various species p. Note the bifurcation pattern on the leaves.

Figure 7.4 is a Mandelbrot Fractal of z_{0} = 0.2 generated by the Logistic equation. The use of the noncritical point as the initial value makes the circular atoms of the logistic set crack like eggs and give birth to elephants. The next three images are Mandelbrot fractals of the noncritical point z_{0} = 0.1 of f_{p} in (7.1) from Elephant Bay. For example, the third image comes from a microscopic rectangular neighborhood of the parameter p = (2.999997892, 0.0079284853).

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

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

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

where the parameter p = 0 is invisible and g is a holomorphic function with its derivative g'. In this section, g(z) is a polynomial in complex variable z which allows us to take advantage of the time-saving scheme called
Horner's Method to efficiently evaluate bothg 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 (8.1) is nothing but the Newton-Raphson Root-Finding Algorithm, aka
Newton's Method. Hence, each orbit of (8.1) 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, "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} = con(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.

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.