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:
"Symmetric Rocks in Desert"
"Escher-like Fern Mountains"
"Broken Taiko Drum"
"Mandelbrot Moon Over Fractal Mountains"
1997-2023 Junpei Sekino
The website was last updated on November 20, 2023
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.
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 developed into comprehensive chaos theory in science and at the same time its affinity with fractal art became clearer. It may sound paradoxical, but as we'll learn, the more elaborate and dazzling a fractal image, the more likely its entanglements with chaos.
Through Google, we find numerous websites displaying 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. A fairly large space 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 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 potentially "astounding" 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. 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 x2 + y2 ≤ 1 on R using a computer. We first decompose R into, say, 50 × 32 miniature rectangles of equal size called picture elements or pixels and then represent the pixels by pixel coordinates(i, j) in such a way that the upper left and lower right pixels are (0, 0) and (49, 31), respectively. Thus, the i- and j-axes of the pixel coordinate system are the rays emanating from the upper left corner of R and pointing east and south, respectively; see the diagram in Figure 1.2(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
A fractal plotted this way will be called a Mandelbrot fractal of z0 in § 6 as the Mandelbrot set is a typical of such fractals. The two images shown above are examples.
Plotting a fractal on a z-Canvas: Recall once again that an orbit is uniquely determined by a dynamical system like (1.1) once values of z0 and p are given and that the orbit is conveniently and interchangeably called the orbit of p with the initial value z0 or the orbit of z0 with the parameter value p. We use the latter when we plot a fractal on a z-canvasR. Choose a value of p, say p = (-1.1128, 0.23076). For each pixel (i, j) on the z-canvasR, use its representative initial value z0 and the dynamical system to generate the orbit of z0 with the fixed parameter value p. Then we use its behavior to color the pixel (i, j). The details are shown in § 5.
A fractal plotted this way will be called a Julia fractal of p in § 6. Important fractals called the "Julia sets" are typical examples.
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 that is very near 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 like nits, 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, 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 begin § 2 with 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 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.
Contrary to its unflattering characterization like "warty snowman with nit infested hair," ℳ contains a greater number of varying and dazzling local images than the number of stars in the universe. It is certainly part of the reason why ℳ gained such monumental popularity, but what deserves most of the credit for it is the simplicity of the divergence scheme (aka "escape time algorithm") that allows even amateurs to plot and enjoy these abstract art pieces. If the plotting process was as convoluted as, say, some of the cryptology schemes, ℳ would never have become so popular.
Figure 1.5(B). Another Local Image of the Mandelbrot Set ℳ
In § 3, we discuss important mathematical properties of ℳ, mainly 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 as it gets nearer the mini-Mandelbrot set until it gets an appearance of being solidly painted. It makes the painted area look two-dimensional.
Shishikura's theorem formally explains a phenomenon like that in terms of "fractal dimensions," which measure complexities and space-filling capacities of fractal curves on a plane, 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.
Figure 1.5(C) is another boundary image that shows what the maximum complexity looks like. It also reveals the correlation between the self-similarity of repetitive patterns and the space-filling capacity of the boundary of ℳ.
The boundary of ℳ is highlighted by the goldish color
In § 4, we turn our attention to the interior of ℳ and 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 in the periodicity diagram. Note that the period of every atom visible in the diagram is "recursively" well-defined.
Despite all these astounding complexities of ℳ, Adrien Douady and John H. Hubbard established earlier that ℳ remains, topologically, a pleasantly simple object, which is compact, connected and has no holes. By far the hardest to prove was the connectedness of ℳ, hence the name the Douady-Hubbard Theorem stated in § 3.
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/or 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 its cause is shrouded in mystery. It makes the aforementioned periodicity diagram all the more important in plotting 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, while Figure 1.4(A) shows two filled-in Julia sets born from the same atom of period 17 × 5.
The boundary of a filled-in Julia set is called the Julia set. It determines the basic shape of the filled-in Julia set and it is where chaos and most of the complexities occur, just like in the boundary of ℳ. Thus, Julia sets are fascinating objects to both artists and mathematicians. As in the case of ℳ, an important topological question is whether or not a Julia set is connected as one piece, and it turned out that Pierre Fatou and Gaston Julia already had an answer to it more than 100 years ago. As a corollary to the Fatou-Julia Theorem, we will learn that
(†) the Mandelbrot set is completely characterized as the set comprising parameters whose Julia sets are connected
In the figure shown below, the parameter on the left comes from outside of ℳ while the one on the right belongs to ℳ; hence, the Julia set is totally disconnected on the left and connected on the right. It shows that eyeballing is often helpless in telling whether or not a fractal is connected from its computer output.
As mentioned, all these facts connecting the Julia sets and ℳ stem from the Fatou-Julia Theorem, in which the critical points of the dynamical system play crucial roles. This explains why the critical point of the Mandelbrot equation was used to define the Mandelbrot set ℳ in the first place in § 2.
To further consolidate the close-knit relations between ℳ and the Julia sets, we will also discuss Tan Lei's Theorem in § 5, which explains why we frequently observe striking similarities in appearance between local images of ℳ and Julia sets. The phenomenon is so prevalent that the Mandelbrot set ℳ is often called an "index" to all Julia sets.
As we have just explained, the Mandelbrot set is defined using the unique critical pointz0 = 0 of the Mandelbrot equation. So, what happens if we deal with a dynamical system with multiple critical points?
Atoms of ℳ1, for instance, are again defined to be "connected components" of the interior of ℳ1 and are in a wide variety of shapes that include the interior of a blue disk as well as the interior of the red spearhead.
In addition to numerous local images such as Figure 1.2(C) that are similar to those from the Mandelbrot set, ℳ1 and ℳ2 have numerous others including Figure 1.7(B) and the first image of Figure 1.7(C) shown below where some of the atoms appear to break into fragments. Interestingly, all of the fragments disappear in the third image of Figure 1.7(A), which we call "Atomic Fusion," given by superimposing ℳ1 and ℳ2.
As we will see in § 6, "Atomic Fusion" has a handy application in the pictorial interpretation of the Fatou-Julia Theorem and the classification of the Julia sets given by the dynamical system (1.5). There we learn that the Julia set of a parameter p is connected if p belongs to ℳ1 ∩ ℳ2 and it is totally disconnected if p belongs to the complement of ℳ1 ∪ ℳ2. Thus, if the Julia set of a parameter p defies the dichotomy, then p belongs to the "symmetric difference"
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 (1.1) generating all the wonders of ℳ we have witnessed. Does it come at a cost? It turned out that the answer is "no" and any quadratic dynamical system we can think of in § 6 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. By virtue of (†), it means that the Mandelbrot set is loaded with all fractal information from 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.
§ 7 also shows many "Mandelbrot fractals" and "Julia fractals" generated by the logistic equation. Although they may be found through the Mandelbrot equation in light of the conjugacy, it is unlikely to happen because getting a certain fractal depends on a rare chance encounter involving various parameters used in fractal plotting. To make the matchup even more unlikely in plotting the Mandelbrot fractals, we use noncritical points of the functionfp for the initial values to deviate from the standard procedure used in earlier sections. Here's an example:
As a sidenote, we encourage people learning fractal plotting to be free-spirited and carry out frequent computer experiments, because fractal art is wide open to new ideas and discoveries. The next two images are examples that popped up from rather aberrant experiments given by twisting the standard algorithms. The cause for the latter became known quickly and has been documented as in the "eyeball effect" but the former is still under investigation.
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 zn of complex numbers diverges to ∞ if the real sequence |zn| diverges to ∞, i.e., if zn gets further away from the origin of the complex plane without bound as n gets larger. 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 |zm|2 > θ2 instead of |zm| > θ to avoid using the hidden square root in |zm| 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, 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 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, 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.
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 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 Japanese manga, anime and 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. Quite similarly, the boundary image controls the shape and a large part of the quality of a multicolored fractal image like Figure 2.3.
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.
"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 z0 is fixed at the critical point z0 = 0 of the function fp. 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 z0 in the complex plane whose orbits zn (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 z0 = 0 of the function fp, 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.
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.
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 z0 = 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 z0 = 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 generated by the simple process 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 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 z0, 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 z0. For example, all of our local images of the Mandelbrot set are Mandelbrot fractals of the critical point z0 = 0 given by the dynamical system (2.1). Thus, z0 is often a critical point of fp 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 z0, on R, (6.1) defines the orbit of z0, 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 dark 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.
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 z0 = i / √3, which is a critical point of fp in (6.2), and it turned out that the Mandelbrot fractal of the conjugate critical point z0 = - 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 z0 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.
Recall that a critical orbit of (6.1) means an orbit whose initial value z0 is a critical point of the function fp. Here is a hugely useful theorem Gaston Julia and Pierre Fatou independently proved in 1918-1919 when computer-generated images of Julia sets were not available!
The Fatou-Julia Theorem: Consider a polynomial dynamical system
where m ≥ 2, cm, cm-1, · · ·, c1, c0 are complex constants with cm ≠ 0, and one of the coefficients, cj , assumes the role of the parameter p. Then we have:
(1) If all critical orbits of p stay within a finite bound, the Julia set of p is connected;
(2) if no critical orbits of p stay within a finite bound, the Julia set of p is a Cantor set.
In case of the Mandelbrot equation (5.1), fp has just one critical point and hence just one critical orbit of the parameter p. If p belongs to the Mandelbrot set ℳ, then the critical orbit does not diverge to ∞, i.e., it stays within a finite bound as per the divergence criterion and hence the Julia set of p is connected. If p does not belong to ℳ then the critical orbit of p diverges to ∞ and hence the Julia set of p is a Cantor set.
with two critical points i/√3 and -i/√3 of fp. Let ℳ1 be the set of parameters p whose critical orbits with the initial value i/√3 stay within a finite bound and ℳ2 the same with the initial value -i/√3. ℳ1 is precisely the "Speared Mandelbrot Set" depicted by Figure 6.1 and ℳ2 the mirror image of ℳ1 through the real axis. Figure 1.7(A) shows interesting relations between ℳ1 and ℳ2 when they are superimposed.
Figure 6.7 shows a pseudo Venn Diagram of ℳ1∪ℳ2 and ℳ1∩ℳ2 which we call the "Venn Diagram." Here, the union ℳ1∪ℳ2 is painted by colors other than black and the intersection ℳ1∩ℳ2 by colors other than black and green. Thus, the black zone is the complement of ℳ1∪ℳ2 denoted by [ℳ1∪ℳ2]c and the green zone is the symmetric difference
Note that the "Venn diagram" quickly gets a lot more complex if fp has three or more critical points.
In terms of the "Venn Diagram," the Fatou-Julia theorem states:
(1) If p belongs to ℳ1∩ℳ2 then the Julia set of p is connected;
(2) if p belongs to [ℳ1∪ℳ2]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 ℳ1Δℳ2.
Our computer experiments show that the converse of (3) seems true, but the Fatou-Julia Theorem does not confirm it. Although the "Venn Diagram" works well for our purpose of finding examples that illustrate the Fatou-Julia Theorem, we need to be a little careful as the diagram does not include the hairy boundaries of ℳ1 and ℳ2 seen in Figure 6.1. Also omitted in the diagram are the Toddler Mandelbrot Set in ℳ1 and its mirror image in ℳ2. Both of them belong to the green zone ℳ1Δℳ2.
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 ℳ1∩ℳ2; hence, it is connected. It is actually given by rotating the output image 90o 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" that it intersects both ℳ1∩ℳ2 and the symmetric difference ℳ1Δℳ2 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 ℳ1∩ℳ2, 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 ℳ1Δℳ2, 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 ℳ1Δℳ2 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 ℳ1Δℳ2. 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 z0 = 0.1 (10% of the sustainable population) of fp 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 z0 = 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 z0 = 0.1 of fp 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."
where p and q ≠ 0 are constant parameters, while the initial values ζ0 and z0 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 z0 in the complex plane whose orbits zn 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 ζn diverges to ∞ if and only if zn 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 ζ0 to the complex plane as the set of z0 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 ζn+1 + b = (a ζn + 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 ζn+1 = q2 ζn2 - q2 ζn ,
i.e., a ζn+1 = (a ζn)2 + 2b(a ζn) ,
where a = -q and b = q/2. Completing the square with respect to a ζn , 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 z0 = 1/√3 of the function fp. Figure 7.13 shows two local Mandelbrot fractals of noncritical points z0 = 0.1 and z0 = 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.
A Julia Fractal is called a Newton fractal if it is given by a dynamical system of the form
zn+1 = zn - g(zn)/g'(zn)
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 z0 and z-canvas, respectively, in this section. It is important to remember that z0 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 r0, r1, r2, ... , rn-1, called the nth roots of unity, are given in a trigonometric expression by
rk = con(2kπ/n) + i sin(2kπ/n) with r0 = 1.
The fact that each rk 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.