This website is a gallery of computer-generated fractal art as well as a text that explains what it is and how it is created. Central to the website are a myriad of fractal images shown in "Stories about Fractal Art" augmented by the text as optional references. In addition, there are two galleries of fractal art: Gallery 2D is a collection of two-dimensional (2D) images made recently, while Gallery 3D comprises such 3D objects as fractal mountains (some are realistic and others imaginative), fractal forests and fractals painted on various nonplanar surfaces. Here are examples:
Digital Artist (Author's Profile): When Junpei Sekino was 10 years old he won first prize for the junior division in a national printmaking contest in Japan.
He now combines art and mathematics to create fractal art.
...from MathThematics, Book 3, Houghton Mifflin, 1998, 2008.
It is given by lighting up the invisible filaments of the boundary of the Mandelbrot set.
The Mandelbrot set conceals infinitely many subsets with exquisite details and captivating compositions often containing irregularities found in nature. Moreover, the detailed patterns are dictated by the boundary of the Mandelbrot set comprising razor-thin "filaments" in the manner illustrated in Figure 0.6(A). Figure 0.4(C) shows a portion of the boundary of the Mandelbrot set, which is a monochromatic line art and is astoundingly complex reflecting the infinitely recursive self-similarity.
Mandelbrot defined fractals using the Hausdorff dimension (aka Hausdorff-Besicovitch dimension), which is a generalization of the familiar dimensions and quantifies the complexities caused by the self-similarity, roughness and irregularities not found in traditional Euclidean geometry. For example, the Hausdorff dimensions of a line and the boundary (perimeter) of the Koch snowflake shown in Figure 0.3 are 1 and approximately 1.26, respectively, showing that the latter is more complex than the line.
Like the Julia sets, the Hausdorff dimension was created in the 1910s and remained rather obscure before 1980 when the Mandelbrot set emerged and fractals became popular. In 1998, Mitsuhiro Shishikura proved that the Hausdorff dimension of the boundary of the Mandelbrot set is the same as the dimension of the plane, which is 2 and maximal for any objects lying in the plane. This effectively proved that the Mandelbrot set containing its boundary is one of the most complex objects ever plotted on a plane.
The last two images are from near the "mini Mandelbrot set" visible in Figure 0.4(A).
Computer users attracted to fractal art soon came to realize that there are infinitely many varieties of Julia sets providing some of the most beautiful fractal art images. Meanwhile, the Mandelbrot set, initially created to show the whereabouts of the Julia sets classified into the dichotomy's two types, grew to be a well designed map on the complex plane to show how the wider varieties of the Julia sets are distributed.
Any image of a subset of the Mandelbrot set featuring a "mini Mandelbrot set" functions as such a map as well. A mini Mandelbrot set is a smaller copy of the Mandelbrot set, and it is known amazingly that any segment of the boundary of the Mandelbrot set tangles with infinitely many mini Mandelbrot sets. Most of them are so small they are invisible in a computer image, but when a computer zooms in on them and they become visible, they may reveal unique, intricate and dazzling patterns surrounding them. The Julia sets "born" from such an object naturally inherit the intricate patterns.
Like "fractal," the word "chaos" was used as a mathematical term for the first time in 1975 when the American Mathematical Monthly published "Period Three Implies Chaos" by T.Y. Li and James Yorke. The paper inspired by May's discovery received a great sensation especially because there appeared very little difference between chaotic and random outcomes even though the former resulted from deterministic processes.
The idea of chaos quickly evolved into comprehensive chaos theory in science and at the same time its affinity with fractal art became clearer. As we'll see, it is the aforementioned sensitivity of chaos that causes drastic changes of colors and patterns in the image, and the boundary of the Mandelbrot set turned out to be chaotic. As we have seen, the boundary is razor-thin and borders between the two regions in the complex plane defined by the dichotomy and comprising totally opposite characters. It is not surprising that such a border becomes "chaotic" like in human interactions.
A Julia set, which is also defined as the boundary of another object, became known to be chaotic as well and its chaotic tendency gets more pronounced if it is born from nearer the boundary of the Mandelbrot set. In fractal art, chaos is generally desirable and the boundary of the Mandelbrot set plays an extremely important role.
Googling we can find a host of websites displaying numerous computer-generated fractal art images that are often stunningly beautiful. It indicates that a large population not only appreciates the digital art forms but also participates in the eye-opening creative activities. Since computer use is essential for such aspirations, a fairly large part of this article is devoted to show how to program a computer and plot popular types of fractals generated by simple dynamical systems. It is not a text on computer programming and instead tells the general principles in everyday language easily translatable to a computer language.
Central to our programming are iterations comprising repetitions of thousands (or occasionally millions) of a simple process defined by a dynamical system. Through iterations our computer zooms in and miraculously reaches a "picoscopic" object like the mini Mandelbrot set of Figure 0.5(A) within a reasonable duration and produces amazing self-similarity like in the fractal output of Figure 0.4(C). Chaos may also ensue after the simple process is iterated so many times. So, it is through iterations that an output image tends to be enormously complex and with unpredictable details. If all goes well, the image may become a breathtaking masterpiece of art.
Particularly exciting in fractal plotting is, therefore, the moment the fractal image generated by our personal program emerges on our computer screen. Even if it turned out not to be an artistic masterpiece, it may still stir our imaginations in the part of mathematics that is in fact quite deep and still filled with unknowns. One might even discover a thing or two in the relatively young field in which computer experiments often lead the way. All in all, 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
p-Canvases and z-Canvases:
Consider a dynamical system, say, the Mandelbrot Equation comprising infinitely many orbits. Each orbit is determined by values of p and z0 and interchangeably called the orbit of p with the initial value z0 or the orbit of z0 with the parameter value p. We have also seen that a canvas is a rectangle in the complex plane consisting of pixels (i, j), each of which has a representative complex number (x, y). In fractal plotting, we view the complex numbers representing the pixels as values of p and call the canvas a p-canvas or view these complex numbers as values of z0 and call the canvas a z-canvas.
In Introduction we mainly discussed two types of fractals, the Mandelbrot set (and its subsets) and Julia sets (and their subsets), both generated by the Mandelbrot equation. In § 2 and § 4 we plot the former on p-canvases and in § 5 the latter on z-canvases. On a p-canvas, the value of p varies and and the value of z0 is fixed at a constant, and that's where we use orbits of p with fixed z0. On a z-canvas, it's the other way around, and that's where we use orbits of z0 with fixed p.
For example, at each parameter p on the p-canvas (with fixed z0), there corresponds the orbit of p (with fixed z0), and as we'll see, its behavior determines the color of the pixel containing p. As we have seen, the orbits of p belonging to two adjacent pixels on the p-canvas may have totally different behaviors, in which case the colors of the pixels may be drastically different. These ideas will appear over and over with examples and become clear in the upcoming sections.
Because a fractal image plotted on a p-canvas is typically the Mandelbrot set, we often call such a fractal a Mandelbrot fractal. This is especially convenient when we plot a fractal on a p-canvas generated by a dynamical system other than the Mandelbrot equation. Similarly, we often call a fractal plotted on a z-canvas a Julia fractal.
As we'll see, the Mandelbrot set generated by the Mandelbrot equation is bounded so its global figure fits in a circle or a rectangle like in Figure 1.4(B), and from the shape of its black silhouette, it is often nicknamed "Warty Snowman" lying sideways. A part of the global image is called a local image, but a local image in fractal art is usually given by zooming in on a microcroscopic rectangular neighborhood of a point that is very near or on the border of the snowman's silhouette.
Because the Mandelbrot set is also known to be closed, its boundary, which coincides with the snowman's silhouette's border, is a part of the Mandelbrot set. It is also known that the boundary has the "topological dimension" of 1, meaning intuitively that it comprises razor thin "filaments" without thickness, just like the boundary of a circular disk. So, the boundary is mostly invisible in local images unless we "light up" their filaments like in a tungsten light bulb; see Figure 1.4(C) shown below.
As Figure 1.4(B) shows, the filaments are like branched hairs growing outwards from the warty snowman and known to carry infinitely many miniature copies of the snowman, which we call "mini-Mandelbrot sets." One of them is visible in Figure 1.4(C) and, yes, the boundary of the Mandelbrot set is incredibly intricate.
In 2008, PBS broadcast a NOVA program proclaiming that the Mandelbrot set had become "the most famous object in modern mathematics." Naturally then, we begin our stories with the Mandelbrot set and devote a large part of the article to its fascinating attributes. The Mandelbrot set popularized fractal plotting by computers and has been the gold standard for all types of fractals.
In § 2, we state the notion of an orbit that diverges to ∞ and define the Mandelbrot set, denoted by ℳ, using the Mandelbrot equation (2.1) and its critical point. In § 4 we also introduce the notion of an orbit that converges to a cycle of period k for some positive integer k and in conjunction with § 2 explain how to plot ℳ globally and locally on p-canvases using algorithms called the divergence and convergence schemes. Figure 1.4(B) shows global images of ℳ contrasting the two distinct algorithms, while Figure 1.4(C) shows a local image of ℳ together with its complex boundary. As we have noted, ℳ contains its boundary as its subset.
In § 3, we focus on the boundary of ℳ and from an intuitive viewpoint discuss the Hausdorff dimension and Shishikura's theorem. As shown in Introduction, the theorem proves in effect that no figures on the plane are more complex than the boundary of ℳ and hence the Mandelbrot set containing its boundary is one of the most complex figures ever plotted on a plane.
The computer-generated images of the boundary of the Mandelbrot set ℳ in Figures 1.4(B) and 1.4(C) appear to show that ℳ is "connected" in one "piece." We carefully define the important notion of connectedness in fractal geometry and state the Douady-Hubbard Theorem that confirms the connectedness. The idea of connectedness leads to the important idea of connected components or "pieces" at the end of § 3.
In § 4, we turn our attention to the interior of ℳ and use Mandelbrot's idea as a cue to define an atom to be a connected component of the interior. Then atoms are the building blocks of the interior and include all of the disks and cardioids visible in ℳ. Furthermore, 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 below in the periodicity diagram.
In § 5, we define, given a parameter p in the complex plane, the filled-in Julia set of p generated by the Mandelbrot equation and show how to plot it on a z-canvas using the divergence and possibly convergence schemes. If p belongs to an atom of ℳ, then the period of the atom affects the shape of the filled-in Julia set in an amazing and sometimes amusing way, although the exact cause is shrouded in mystery. For example, the image shown below is the filled-in Julia set of a parameter chosen from an atom of period 9 near the "neck" of ℳ, while Figure 1.4(A) shows two filled-in Julia sets born from the same atom of period 17 × 5 near the cusp of ℳ.
The boundary of a filled-in Julia set is called a Julia set. A Julia set is chaotic and mostly self-similar and it plays a leading role in fractal geometry along with the Mandelbrot set. Many of them are also appreciated as dazzling fractal art as described in Introduction. The two images shown above and the images in Figures 5.3 and 5.6 indicate that the Julia sets come with vastly distinct shapes as there are atoms with vastly distinct numerical structures.
If p is near but not in an atom, the Julia set of p still resembles the shape of a Julia set born from the atom as shown in Figure 1.6(C). As p gets further away from the atom, the "gravity" of the atom on p naturally wanes and the Julia set of p gradually loses the resemblance. It makes the periodicity diagram all the more important in plotting many of the Julia sets.
Historically, the Julia sets preceded the Mandelbrot set ℳ by about sixty years. As explained in § 5, Mandelbrot created ℳ to be a graphical interpretation of the Fatou-Julia Theorem applied on the Mandelbrot equation so as to show: If a parameter p belongs to ℳ then the Julia set of p is connected and if not then the Julia set is totally disconnected. In particular, the Julia set shown below is totally disconnected. Today the Mandelbrot set has evolved into a more complete map on the complex plane to show where we can find many of those vastly distinct Julia sets.
As we have seen, we plot the Mandelbrot set and a Julia set on different canvases, a p-canvas and a z-canvas, respectively, and that's because they belong to two different complex planes, one consisting of the parameters p and the other the initial values zo. Despite the difference, the two sets frequently have local images that look alike as shown in the example below. At the end of § 5, we'll discuss the phenomenon using Tan Lei's Theorem. It confirms that the Mandelbrot set is a "breeding ground" for numerous Julia sets as well as local images of its own, both appreciated as attractive fractal art.
In § 6, we consider more general dynamical systems, and in particular, the dynamical system (6.2) generated by a simple cubic polynomial with a conjugate pair of critical points z0 = ± i / √3 = ± (0, 1 / √3). Thus, (6.2) differs from the Mandelbrot equation (2.1) in the important fact that it comes with two critical points and hence without the dichotomy. It turned out that (6.2) gives rise to an even greater variety of Julia sets than the Mandelbrot equation (2.1). Now we ask: Can we find the "Mandelbrot set" of the dynamical system (6.2) that works as a map guiding us to find these Julia sets?
If we repeat the process of § 2 and § 4 and generate the "Mandelbrot sets" of the two critical points, we get rather comical fractals nicknamed "Speared M Sets" ℳ1 and ℳ2 shown in Figure 1.7(A). To be precise, ℳ1, for instance, corresponds to z0 = i / √3 and is the portion not including the dark green background. It does include, however, its hairy boundary and the mini Mandelbrot set seen near the bottom of Figure 6.1, but it is omitted from Figure 1.7(A) for simplicity. The tip of the "spearhead" turned out to be the origin (0, 0) of the complex plane and ℳ2 is the mirror image of ℳ1 through the real axis of the complex plane.
We again define atoms of say, ℳ1, to be connected components of the interior of ℳ1. Thus, atoms are in a wide variety of shapes that include the interior of a purple disk as well as the interior of the red "spearhead" with jagged edges. Both ℳ1 and ℳ2 contain local images not seen in the Mandelbrot set of (2.1) including the "Burst M Set" of Figure 1.7(B). It contains numerous mini Mandelbrot sets and atoms looking like flying fragments.
Interestingly, all of the fragments and jagged edges of the spearhead atoms, which are the irregularities not seen in the Mandelbrot set, disappear in the third image of Figure 1.7(A), which we call "Atomic Fusion" given by superimposing ℳ1 and ℳ2. The figure shows ℳ1 and ℳ2 fit so perfectly under the superimposition they are made for each other to be "fused together." Figure 1.7(C) shows a part of the fusion in different colors.
As shown in § 6, the "Atomic Fusion" comprising ℳ1 ∪ ℳ2 and ℳ1 ∩ ℳ2 provides a graphical interpretation of the Fatou-Julia theorem with the two critical points and seems to work impeccably as a map guiding us to various Julia sets. For that reason, we call ℳ1 ∪ ℳ2 together with ℳ1 ∩ ℳ2 the Mandelbrot set of the dynamical system (6.2).
In the following, we show a few examples of Julia sets of the dynamical system (6.2) found by using the Mandelbrot set of (6.2) as a guiding map. The first two are Julia sets that are neither connected nor totally disconnected and are generated by parameters chosen from the "symmetric difference" of ℳ1 and ℳ2,
ℳ1 Δ ℳ2 = (ℳ1 ∪ ℳ2) - (ℳ1 ∩ ℳ2).
These are types of Julia sets not involved in the aforementioned dichotomy. The third is generated by a parameter chosen from ℳ1 ∩ ℳ2 and is connected, and the fourth by a parameter chosen from the complement of ℳ1 ∪ ℳ2 and is totally disconnected.
Technical Description: The Julia Set of p = (-1.022, 0.14846) On a z-Canvas Centered at z0 = (0, 0) Generated by the Dynamical System zn+1 = zn(zn2 + p)
The Julia set is connected.
In § 7, we go back to the Mandelbrot set ℳ and discuss possibly the most charming feature it possesses, which is the striking simplicity of the Mandelbrot equation (2.1) generating all the wonders of ℳ we have witnessed. It turned out that the simple form does not diminish the power of the Mandelbrot set in the sense that the Julia sets of any quadratic dynamical systems are geometrically similar to the Julia sets of the Mandelbrot equation.
Rather than showing this fact in full generality (which can be done by high school algebra), we will verify it using a special quadratic dynamical system called the logistic equation. The process involves the important idea of "conjugacy" between two dynamical systems. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.
So, we will show several fractals found by the logistic equation in § 7, even though we may encounter similar fractals using the Mandelbrot equation if we are lucky. In creating fractal art, we can deviate from standard procedures of mathematics such as using critical points and freely conduct computer experiments. Here are examples:
There is a special subset of the Julia fractals consisting of fractals generated by so-called "Newton's rootfinding method." We call them Newton fractals and discuss them in § 8. Here are sample fractals:
People who are familiar with multivariable calculus can venture into plotting fractals in a 3D space. One of the possibilities is to map a fractal from the plane to various surfaces such as a sphere and a torus. We will throw in 3D examples here and there in the upcoming sections.
We say that a sequence 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."
Recall that the Mandelbrot set is denoted by ℳ and is closed so it contains its boundary as its subset. It is also known that the topological dimension of the boundary is 1 like the boundary of a circular disk, so we intuitively picture it as an object made of "razor-thin filaments" without thickness. Does it mean that the area of the boundary is zero? Nobody can find the answer, and we suddenly realize that the boundary is considerably more complex than its appearance in a global image like Figure 2.1.
Although it may not sound obvious unless we know something about the Hausdorff dimension (aka the Hausdorff-Besicovitch dimension), the following celebrated theorem allows us to say 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 Hausdorff dimension of the boundary of the Mandelbrot set is 2 (which is the topological dimension of the plane).
Let's pause for a moment and look at its local image in, say, Figure 2.3, in which a part of ℳ is visible. The intricate image surely looks impressive, but exactly where is the boundary of ℳ and what does it have to do with the colorful patterns? It turned out that the boundary of ℳ is all over the image as we can see in Figure 3.1 given by darkening the entire Figure 2.3 and lighting up its razor-thin filaments:
The image shows that the boundary of ℳ in the rectangular area is vividly self-similar, making it a fractal as per our informal definition. Shishikura's theorem also makes it a fractal according to Mandelbrot's definition: A fractal means a set for which the Hausdorff 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 Hausdorff dimension of the boundary of ℳ is the same as the topological dimension of the plane (which is 2).
In fact, the Hausdorff dimension of any object on a plane is a nonnegative real number (rather than an integer) not exceeding the topological dimension of the plane and is designed to quantify the complexities often caused by the self-similarity, roughness and/or irregularities not found in traditional Euclidean geometry.
For this reason, we may define the complexity of the object to be its Hausdorff dimension. Because the boundary of ℳ has topological dimension 1 and Hausdorff dimension 2, it is a line art with a maximum complexity, part of which is revealed in Figure 3.1. The relation between Figure 3.1 and Figure 2.3 in full color is illustrated in Figure 0.6(A).
The Hausdorff dimension of the boundary (perimeter) of the Koch Snowflake in Figure 0.3 is known to be approximately 1.26, showing that it is far less complex than the boundary of the Mandelbrot set but more complex than a circle whose Housdorff dimension is 1. After seeing Figure 3.1, it is not surprising to learn that the boundary of ℳ has an infinite length while bounding a finite area. It is also the case with the boundary of the Koch Snowflake and a good calculus student can actually prove it using a geometric series.
One of the most important topological properties in fractal geometry is "connectedness" of a set and Figure 3.1 appears to show that the Mandelbrot set ℳ with its complex boundary is "connected" as "one piece." To give precision to the intuitive concept involving "one piece," R. C. Buck adopts the following formal definition in his classical textbook for Advanced Calculus: Suppose S is a nonempty set of points in the xy-plane. S is said to be connected if it is impossible to split S into two disjoint sets, neither one empty, without having one of the sets contain a boundary point of the other.
For example, it is known that the "neck" of the "snowman" in Figure 2.1 is the point (-3/4, 0), and if we cut the head off the body of the snowman with the vertical line x = -3/4, then either the head or the body contains the boundary point of the other, namely (-3/4, 0). Thus, the particular attempt fails to show that ℳ is disconnected. Because of the complexity of its boundary, proving whether or not ℳ is connected is by no means a simple task, as evidenced by the fact that Mandelbrot initially conjectured ℳ to be disconnected and reversed it later without substantiation―before Adrien Douady and John H. Hubbard settled it:
The Douady-Hubbard Theorem (1982): The Mandelbrot set is connected. They also proved that ℳ is "simply connected," which means ℳ has no holes. Topologically speaking therefore, ℳ is well-behaving as a compact set in one piece without a hole. As described by Wikipedia, Douady and Hubbard established many of the fundamental properties of ℳ at an early stage and created the name "Mandelbrot set" in honor of Mandelbrot. They were the pioneers of the mathematical study of ℳ.
"Who Discovered the Mandelbrot Set?" is the title of an interesting read that appeared in Scientific American in 2009. It writes: Douady now says, however, that he and other mathematicians began to think that Mandelbrot took too much credit for work done by others on the set and in related areas of chaos. "He loves to quote himself," Douady says, "and he is very reluctant to quote others who aren't dead."
Figure 3.3. A mini Mandelbrot Set under the Microscope
M = 1,500,000
M = 500,000
For the above image on the left, we used whopping 1,500,000 iterations of the Mandelbrot equation for each black pixel. If we use M = 500,000 (still a large number) instead, the outline of the mini-Mandelbrot set becomes blurry as shown in the above picture on the right. Fortunately, computers (especially used ones) are inexpensive nowadays and we can easily afford a second or third computer to do tedious jobs. Programming carefully so as to minimize computing time is not as important as it used to be.
Shown below is a nighttime view of the fractal on the left that reveals the boundary of the mini Mandelbrot set. In Figure 3.1, the boundary of ℳ appears to be made of razor-thin filaments and ℳ certainly looks simply connected (with no holes), but it is not the case in Figure 3.4. That is because the maximum number of iterations, M, is not large enough and still made the boundary near the mini Mandelbrot set a little blurry.
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; see Figure 1.4(B).
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 have seen, the topological dimension of a curve is 1, but the topological dimension of a totally disconnected set is 0. In § 5 and onward, we'll see fractals composed of the latter as well as those composed of curves. Just like in art, therefore, mathematics has its own stippling art as well as line art.
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. We now turn our attention to its interior, namely, ℳ minus its boundary; see Figure 1.4(B).
The Mandelbrot set has become so illustrious, everybody interested in fractals knows its "warty snowman" shape by heart. To its main cardioid-shaped body, a bunch of (circular) disks are tangentially attached, and to each of these disks another bunch of disks are tangentially attached. The pattern repeats as if the cardioid has children, grandchildren, great grandchildren and so on and so forth. Here, a "cardioid" means, instead of the familiar curve, the curve together with all the points inside the curve.
As Figure 3.1 shows, ℳ also contains infinitely many mini-Mandelbrot sets, each of which is a smaller copy of ℳ, again comprising a cardioid (which may be distorted) with infinite generations of disks (which may be distorted) and even smaller mini-Mandelbrot sets. If we remove the boundary of ℳ from ℳ, we are left with the interior of ℳ comprising the interiors of these disks and cardioids, etc., which are the connected components of the interior of ℳ; again see Figure 1.4(B).
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 nobody have encountered yet. An atom and the interior of 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; again see Figure 1.4(B). 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). It will in turn lead us into a complex world portrayed by one of the most important open questions in fractal geometry called the "density of hyperbolicity" conjecture.
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 critical orbits do not diverge to ∞. Then apply various convergence schemes with ε = 10-8 on ℳ. Figure 4.2 shows the (resized) output images of three molecules.
The first image is generated by the convergence scheme with period index k = 1 and shows that the interior of the cardioid is an atom of period 1. Painting in subtle shades of red is done by a basic technique included in the Fractal Coloring site.
The second image is given by the convergence scheme with period indices k = 1, 2, 3, 4, which is basically defined as the natural sequence of the four convergence schemes, the one with period index k = 1 followed by the one with period index k = 2, etc. It shows that the interior of the largest disk is an atom of period 2 and painted in subtle shades of orange. Similarly, the green and purple atoms are of periods 3 and 4, respectively.
The third image is given by a straightforward extension of the scheme described in the preceding paragraph. Because there aren't enough colors that are easily distinguishable, the correspondence between the periods and colors of the atoms is not one-to-one. For example, the atoms of periods 2 and 5 are painted orange in the third image.
where k is the period of the corresponding atom of ℳ. We call λ the base period of ℳ '.
For example, the most visible mini-Mandelbrot set of Figure 4.3 happens to have the base period λ = 4 and is shown in Figure 4.4 and the inset of the periodicity diagram. There, it is painted by the convergence scheme with period indices λ k = 4, 8, 12, ..., 100 and with the colors of the Mandelbrot set of Figure 4.2 so as to emphasize the one-to-one correspondence between the atoms of ℳ and the atoms of the mini-Mandelbrot set.
Here's an additional technical detail: For the convergence scheme with period indices λ k = 4, 8, 12, ..., 100, we used the variable maximum number of iterations
(4.2) M = 20000 - 150k,
rather than a constant like M = 20000 to speed up the computation without notably sacrificing the appearance of the output image. The reason for such manipulations is that the convergence scheme gets markedly slower than the divergence scheme if its period index gets greater. For example, most of the mini Mandelbrot sets we encounter have base periods λ greater than 100 and it is not practical to decorate them using the convergence scheme and computers currently available in a store. That's why the mini Mandelbrot sets of Figures 1.5(A) and 3.3 are left in a single color black.
Figure 4.4. The the mini-Mandelbrot Set of Base Period λ = 4 and Its Boundary
The picture shown below on the left is essentially the same as Figure 3.3 (but in different colors) and is given by the divergence scheme alone, while the one on the right is painted by the divergence scheme followed by the convergence scheme with period index k = 1 (using different colors) on the complement of ℳ. The "eyeballs" painted by the convergence scheme are caused by its "mistake" of confusing some of the slowly divergent orbits as convergent. The images show which parameters are affected. Figures 1.8(D) and 5.8 illustrate the "eyeballs" more vividly. Note: The "eyeball effect" shows that the converse of the proposition is false.
Figure 4.5. The Eyeball Effect (Right) Given by the Convergence Scheme
Periodicity Diagram: If we label the atoms of the Mandelbrot set in Figures 4.2 and 4.4 by their periods instead of colors, we get the following periodicity diagram. The periods in the diagram show meticulously aligned numerical patterns that are easy to recognize and will play an important role in plotting many of the "Julia sets" in the next section. The numerical patterns are yet another amazing property of the Mandelbrot set ℳ.
As mentioned in Introduction, the Julia sets preceded the Mandelbrot set by about sixty years and were created by Pierre Fatou and Gaston Julia before the computer age. To show the main ingredients of their work, we begin with a polynomial function of a complex variable z,
(5.1)
f(z) = cm zm + cm-1 zm-1 + · · · + c2 z2 + c1 z + c0,
where m ≥ 2, and cm, cm-1, · · ·, c1, c0 are complex constants with cm ≠ 0. Then consider the dynamical system
consisting of all orbits of z0 in the complex plane. For a technicality, we may view any of the coefficients cj to be the constant parameter p but p plays no roles at this stage.
Define the filled-in Julia set of f denoted by 𝒦(f) to be the set of all initial values z0 in the complex plane whose orbits do not diverge to ∞ and define the Julia set of f denoted by 𝒥(f) to be the boundary of 𝒦(f). Then it can be shown that 𝒦(f) and 𝒥(f) are both compact and contain infinitely many points. In particular, 𝒦(f) is bounded and contains 𝒥(f) as its subset. The Julia set 𝒥(f) turned out to be chaotic just like the boundary of ℳ as well as self-similar "almost" always, and because of the fractal complexity, it plays a leading role in fractal geometry. It can be connected or disconnected, and if it is totally disconnected, we call it a Cantor set or Cantor dust.
Julia and Fatou independently proved the following powerful theorem in 1918-1919:
The Fatou-Julia Theorem: (1) If all of the critical points of f belong to 𝒦(f), then 𝒥(f) is connected as one piece, and (2) if none of the critical points of f belong to 𝒦(f), then 𝒥(f) is totally disconnected and forms a Cantor set.
As in the case of ℳ, therefore, the image shows the complement of 𝒦(p), and hence, the critical point of fp at its center does not belong to 𝒦(p). Because it is the only critical point of fp, it follows from the Fatou-Julia theorem that the Julia set 𝒥(p) is a Cantor set.
Using the fact that the boundary of 𝒦(p) coincides with the boundary of the complement of 𝒦(p), we can also paint the Julia set 𝒥(p) by the divergence scheme. Figure 5.0(B) shows the Julia set, which is a fractal as it is vividly self-similar.
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 critical orbit converges to a cycle of period 11 at the center of the canvas. Therefore, the convergence scheme with period index k = 11 is a natural choice in decorating the filled-in Julia set 𝒥p. Note that the critical point belongs to 𝒦(p): hence, 𝒥(p) is connected by the Fatou-Julia theorem.
It is another fascinating fact about the Mandelbrot set that the period of the parameter p is always reflected in the shape of the filled-in Julia set of p, as in the number of "Hydra's heads," although why it is so is not completely understood.
Example 3: The filled-in Julia set of p = (-0.6891, 0.27896) called "Hydra's Ash" in Figure 1.6(C) is given by the parameter near the aforementioned atom of period 11 but lies outside of ℳ. We'll see a quick way to tell why 𝒥(p) is a Cantor set. Interestingly, 𝒦(p) still retains a hydra's shape even though the parameter is detached from the atom.
Here are additional examples of filled-in Julia sets 𝒦(p) born from circular atoms attached to the red cardioid near its cusp:
The Fundamental Dichotomy: For any parameter p in the complex plane, the Julia set of p is either connected or totally disconnected. For example, the Julia set of Figure 1.1(A) is neither connected nor totally disconnected and the dichotomy assures us that a Julia set like this never arises from the dynamical system (5.3). Mandelbrot, who once studied under Gaston Julia and later became an "IBM fellow," used a computer to visualize the fundamental dichotomy that divides up the complex plane into two parts. He initially defined the Mandelbrot set to be
(†) the set ℳ comprising all parameters p in the complex plane whose Julia sets are connected.
He knew how to compute ℳ as the Fatou-Julia theorem clearly implies that the Julia set of p is connected if and only if the critical orbit of p does not diverge to ∞; see the computer-friendly definition of § 2. Thus, the famed Mandelbrot set ℳ was born from the dichotomy of the Julia sets. It also explains why the critical point of (5.3) is indispensable in computing ℳ.
Henceforth, we call (†) the alternative definition of ℳ. It shows that the Julia sets 𝒥(p) of Figures 5.1(B)(C) (and hence 𝒦(p) as well) are connected, while "Hydra's Ash" mentioned in Example 3 is a Cantor set.
The Julia Set of p = (0.25000316374967, -0.00000000895972)
Example 4: The parameter p used for Figure 5.2(A)(B) shown above belongs to the interior of the cardioid atom of the mini-Mandelbrot set of Figure 0.5(A) whose period is large and unknown. So, Figure 5.2(A) is painted by the divergence scheme alone and shows the complement of 𝒦(p) with Figure 5.2(B) showing its boundary 𝒥(p). Because p belongs to ℳ, the Julia set 𝒥(p) is connected by the alternative definition of ℳ.
The two "lions" are painted by the convergence scheme with period index 85 and the background by the divergence scheme with the threshold θ = 2. The curling directions of the mane of the "Twin Lions" are opposite to each other and depend on the locations of the parameters in the atom.
Figures 5.3 & 5.4. "Twin Lions" born from the same atom of period 17 × 5
"Esmeralda Lion" shown in Gallery 2D is an enlarged version of the filled-in Julia set shown above on the left. "Ruby Lion" shown below is an enlarged version of the filled-in Julia set shown above on the right.
The Jordan curve theorem states that a Jordan curve divides the plane into two parts, a bounded region called "inside" and an unbounded region called "outside." The theorem seems utterly obvious from a typical image like the one shown above, but the Julia set as a Jordan curve can get extremely convoluted geometrically if the parameter gets arbitrarily close to the boundary of the cardioid. In fact, the proof of the Jordan curve theorem is far from obvious involving algebra, analysis and topology and provides one of the fascinating topics in mathematics.
Example 7: The image shown below is the filled-in Julia set of the parameter p = (-1.0073, 0.2552) chosen from a circular atom of period 2 × 4. The atom is the leftmost blue disk shown in the first image of Figure 4.3 and is attached to the orange atom of period 2. Both factors 2 and 4 are visible in the Julia set.
Figure 5.7. "Run for the Sun"
A Filled-in Julia Set Born from an Atom of Period 2 × 4
It is a local image of the Julia fractal shown in Figure 5.2(A) but is painted by using different colors and the eyeball effect. The eyeball effect makes it easier to identify the numerous "cuttlefish" swimming in the global image, in which their eyes are closed. Note that one of the cuttlefish is at the center of the image.
Figure 5.2(A) is a global Julia set of the parameter p = (0.25000316374967, -0.00000000895972) and if we zoom in on the center of the figure between the eyes of the central cuttlefish, we get another local image revealing the cuttlefish's mouth painted black:
Recall that the Mandelbrot set and a (filled-in) Julia set belong to two different complex planes, one comprising parameters p and the other initial values z0 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
z0 = 0, z1 = -2 ,z2 = 2 ,z3 = 2 ,z4 = 2 ,· · · .
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 5.9 and 5.10 are strikingly similar even though the parameter p belonging to the interior of the mini-Mandelbrot set is not a Misiurewicz point. The sidenote to Figure 3.3 shows that the distance between p and a nearby Misiurewicz point is much less than 10-13. Figure 5.11 shows we can zoom out from Figures 5.9 and 5.10 while retaining some degree of similarity.
Figure 5.11. "Cuttlefish" Swimming in the Mandelbrot Set (Left) and in the Julia Set (Right)
Soon after Mandelbrot published its computer plot generated by the simple process of § 2 in 1980, the Mandelbrot set became so popular that a great many computer hobbyists, digital artists, mathematicians and scientists have explored around it and shown their fractal art on a variety of objects including posters, book covers, T-shirts, coffee mugs and webpages. Although the hidden beauty of the Mandelbrot set is inexhaustible, it has become quite a challenge to unearth local images of the Mandelbrot set or Julia sets that look drastically different from what have been published by using available computers and software. An easy way to find a new pattern such as the one shown below is to use a dynamical system other than the Mandelbrot equation and there are infinitely many of them.
The origin (0. 0) of the complex plane is at the tip of the spearhead, which coincides with the upper left corner of the closeup image shown below. We call the area "Spearhead Bay." Like the Mandelbrot set, the Giant M Set contains infinitely many circular atoms that satisfy the numerical pattern of the periodicity diagram. These circular atoms include the largest and the second largest blue atoms shown in Spearhead Bay, whose periods happened to be 7 and 8, respectively. Note that the "seaweed" growing out of the blue atom of period 7 contains seven-way junctions and likewise for the "seaweed" around the atom of period 8.
We also note that in Spearhead Bay, the seaweed grows only on the side of the Giant Mandelbrot Set and tangles with infinitely many extra atoms that look like tropical fish.
Interestingly, the fish-like atoms begin to disintegrate near the circular atom of period 6, which is painted purple at the mouth of Spearhead Bay, and they become extinct near the circular atom of period 5, which is located just outside of the bay.
"Spearhead Bay"
The boundary of the Giant M Set near the circular atom of period 5 is depicted in the image shown below. It shows no signs of fish but, like in the Mandelbrot set, it contains five-way junctions and encloses numerous mini-Mandelbrot sets. Unlike the Mandelbrot set however, the boundary now appears to be disconnected.
"Seaweed with Five-Way Junctions"
Another area in Figure 6.1 that provides a rich fishing ground for attractive fractals is in and around the blue molecule located between the spearhead and the Toddler M Set that looks like a pair of balloons. We call it "Broken Balloons" because of its "bursted lips" with jagged edges and small fragments; see the image shown below. It is generated by the convergence scheme with period indices k = 3, 6, 9, ..., 60. Like the cardioid body of the Mandelbrot set, we again painted the atoms of the smallest period 3 red.
Figure 6.1 also contains two "Squished M Sets," each of which has a "bursted" cardioid. The molecule can be seen near the top of Figure 6.1, but its magnified image shown below uses different colors. It is generated by the convergence scheme with period indices k = 3, 6, 9, ..., 60 with k = 3 corresponding to the red atoms, just like in the "Broken Balloons."
Let ℳ1 = ℳ (i / √3) and ℳ2 = ℳ (-i / √3) so ℳ1 and ℳ2 are the Mandelbrot sets of the conjugate critical points i / √3 and -i / √3 of fp in (6.2), respectively. Thus, ℳ1 comprises the parameters p whose orbits with the initial value z0 = i / √3 do not diverge to ∞ and ℳ2 likewise. As mentioned earlier, ℳ1 is given by Figure 6.1, and as shown in Figure 1.7(A), ℳ2 turned out to be symmetric to ℳ1 through the real axis of the complex plane.
As shown in Figure 1.7(A), if we superimpose ℳ1 and ℳ2, we get a surprising result illustrated by what we call "Atomic Fusion" in fractal geometry. For example, "Big Spearhead" atom in ℳ2 fits perfectly in the cardioid atom of ℳ1 and the lips of the "Broken Balloons" in ℳ2 are beautifully repaired by the "Squished M Sets" of ℳ1; see Figure 6.6 shown above.
Figure 6.7 shows a simplified version of "Atomic Fusion," where most of the boundary curves in ℳ1 and ℳ2 are omitted for the ease of its interpretation. 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
ℳ1Δℳ2 = ℳ1∪ℳ2 - ℳ1∩ℳ2.
The concept of Julia set naturally extends from (2.1) to our dynamical system (6.2). Thus, the filled-in Julia set of a parameter p, denoted by 𝒦 (p), is the set of all possible initial values z0 of (6.2) in the complex plane whose orbits with the fixed value of p do not diverge to ∞ and the Julia set of p, denoted by 𝒥 (p), is the boundary of the filled-in Julia set.
Therefore, if z0 is a critical point of (6.2), then z0 belongs to 𝒦 (p) if and only if the orbit of p with the initial value z0 (which is the same as the orbit of z0 with the fixed parameter p) does not diverge to ∞, i.e., p belongs to ℳ1 if z0 = i / √3 and p belongs to ℳ2 if z0 = -i / √3.
It is now straightforward to prove that the Fatou-Julia Theorem applied on (6.2) can be written as follows:
(1) If p belongs to 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.
All of our relevant computer output seems to show that the converse of (3) is true, but the Fatou-Julia Theorem does not confirm it.
Recall that the Mandelbrot set of (2.1) is initially defined so as to visualize the fundamental dichotomy, which is the Fatou-Julia theorem applied on (2.1) with exactly one critical point. Note also that we have just shown that the "Atomic Fusion" comprising ℳ1∪ℳ2 and ℳ1∩ℳ2 illustrates the Fatou-Julia theorem applied on (6.2) with exactly two critical points. It is natural, therefore, that we call ℳ1∪ℳ2 together with ℳ1∩ℳ2 the Mandelbrot set of the dynamical system (6.2).
In general, the Mandelbrot set of the dynamical system (6.1) comprises the union and the intersection of all of the Mandelbrot sets of the critical points of (6.1) satisfying the properties comparable to (1), (2) and (3). In particular, if (6.1) has exactly one critical point, then the Mandelbrot set of (6.1) coincides with the Mandelbrot set of the critical point as is the case in (2.1).
Just like the Mandelbrot set of (2.1), the Mandelbrot set of (6.2), er the "Atomic Fusion" diagram, is indispensable in finding the Julia sets of (6.2) with infinitely many varieties, although we have to be a little careful with its hairy boundary not shown in the diagram.
Example 1: The Julia set of Figure 0.1(B) called "Twin Dragons" and shown at the outset of this website is given by the parameter p = (0.185, 0.00007666) belonging to 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 Figure 6.7 that it intersects with both ℳ1∩ℳ2 and the symmetric difference ℳ1Δℳ2.
The connected "Roses" of Figure 6.9(A) is the Julia set of the parameter p = (0.02912, -1.093853) belonging to an atom of period 3 × 7 in "Broken Balloons." The parameter p belongs to 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) shows the Julia set of the parameter p = (0.07761, -1.12427) belonging to an atom of period 3 × 4 in "Broken Balloons." The parameter p is chosen from the symmetric difference 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.
Example 3: The "Toddler M Set" seen near the bottom edge of Figure 6.1 belongs to ℳ1Δℳ2 but it is omitted from Figure 6.7. Recall that it comprises atoms of periods k = 2 × 1, 2 × 2, 2 × 3, · · · . It produces a great many attractive fractals but they are naturally similar to the fractals coming out from the Mandelbrot set for the quadratic system—except that they are all disconnected. For example, the image which is shown below and resembles the "Hydra" of Figure 5.1(A) is the Julia set of p = (0.00399109,-1.98545775) belonging to an atom of period 2 × 13. It contains numerous dots in its background each of which is a baby hydra.
Figure 6.11. "Lernaean Hydra with Thirteen Heads and Offsprings"
Example 4: While the "Toddler M Set" generate Julia sets that resemble Julia sets of the Mandelbrot set seen in § 5, the "Giant M Set" produces Julia sets that do not resemble anything from the Mandelbrot set, apparently affected by the "Spearhead." "Twin Dragons" of Figure 6.8 are such examples. Here is another, this time from near the neck of the giant.
As we have seen, the Mandelbrot equation (1.1) generates infinitely many Julia sets through the iterations of (1.1), which in turn give rise to the enormously complex Mandelbrot set through the fundamental dichotomy of the Julia sets. In the whole process, the first thing we note is probably the striking simplicity of the formula (1.1).
When we paint a fractal, we use a canvas comprising millions of pixels. Since coloring each pixel easily requires thousands of orbit evaluations, one extra addition or multiplication of complex numbers in a formula like (1.1) can make a significant difference in a computer's runtime. So, the simplicity is not only aesthetically pleasing but also important from a practical viewpoint.
Fortunately, it turned out that (1.1) with its simple form is not as constraining as it first appears. The reason is that if (6.3) is any quadratic dynamical system, then we use high school algebra of "completing the square" to show that it is "conjugate" to (1.1), guaranteeing that any Julia set generated by (6.3) is geometrically similar to a Julia set generated by (1.1) and vice versa. Thus, by knowing the Julia sets of (1.1), we effectively know the Julia sets of (6.3).
Rather than showing the "conjugacy" in full generality, we will verify it using a special quadratic dynamical system called the logistic equation. As mentioned in Introduction, the logistic equation became famous with the advent of chaos and is interesting in its own right.
What is the logistic equation? In 1838 Pierre Verhulst introduced a differential equation called the "logistic equation," which became widely used to describe the population dynamics with self-limiting growth. If we replace the derivative in the differential equation by its approximating difference quotient and do some algebra, we get the following "difference equation," which is more suitable for computer applications and again called the logistic equation:
(7.1)
zn+1 = fp(zn) = p(1 - zn) zn .
Expand its variables and parameters to complex numbers and let ℳ ' be the aforementioned "Mandelbrot set" defined by (7.1). Then by the reason shown in the remark, we get ℳ ' shown below by applying the divergence and convergence schemes on the critical orbits of p with the fixed initial value z0 = 0.5. For convenience, we call the entire moleculeℳ ' comprising the atoms together with its hairy boundary the logistic set, short for the "logistic equation's Mandelbrot set."
As we'll see later, there is a two-to-one function F mapping the logistic set onto the Mandelbrot set such that for any parameter p, the Julia sets of p and F(p) are geometrically similar; see Figure 7.4. The function F becomes a one-to-one correspondence if it is restricted to the left half or the right half of the logistic set. Thus, F actually fails to be two-to-one at the intersection point (1, 0), which works as a "double cusp" of the Mandelbrot set.
Julia Sets by the Logistic Equation: We now show a few (filled-in) Julia sets generated by the logistic equation. Figure 7.8 shows the filled-in Julia set of the parameter p = (2.994915, 0.1) belonging to the red atom of period 1 in the logistic set at Elephant Bay. Hence, it is decorated by the convergence scheme with period index 1 and its complement by the divergence scheme. The second image shown below is the boundary of the filled-in Julia set, namely, the Julia set of p = (2.994915, 0.1). It is a Jordan curve which is homeomorphic to a circle.
Julia Fractal of q = (3.0014564, 0.08) by the Logistic Equation (7.2)
The parameter p = (3.0237615, 0.1) that generates "Dancing Seahorses" shown below belongs to the orange atom of period 2 in the logistic set at Elephant Bay, hence the filled-in Julia set is painted by the convergence scheme with period index 2. Elephant bay is sandwiched by a red atom of period 1 and an orange atom of period 2, and interestingly, a parameter from the orange shore generates "seahorses" instead of "elephants."
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.
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 equationzn+1 = fp(zn) = p(1 - zn4) zn. The Julia set is emphasized in the nighttime fractal on the right.
Figure 7.19. "Dancing Bouquet" by the Fifth Degree Logistic Equation
The idea of the Julia set of a polynomial we saw in § 5 naturally extends to a rational function with minor modifications. So, suppose g is a polynomial of degree greater than one and consider a dynamical system of the form
(8.1)
zn+1 = zn - g(zn)/g'(zn),
which generates, like in § 5, a (filled-in) Julia set of (8.1). The reader may have noted already that (8.1) is nothing but the Newton-Raphson Root-Finding Algorithm, aka
Newton's Method for finding roots of g. For this reason, we call a Julia Fractal featuring the Julia set of (8.1) a Newton fractal of g.
As a benefit of using Newton's method, each orbit of (8.1) converges to a root of g quickly more often than not, and it allows us to plot most of the Newton fractals by the convergence scheme (with period index k = 1) alone with a relatively small maximum number of iterations like M ≤ 500. Also, because g is a polynomial, we may take advantage of the time-saving scheme called
Horner's Method to efficiently evaluate bothg and g' that appear in (8.1). Horner's method is nothing but "synthetic division" taught in high school algebra, and it should be interesting for the reader to see how (differently) it is applied in computer programming.
Furthermore, if we know all the roots of g prior to the fractal plotting, we can modify the convergence scheme fairly easily so as to add more colors to Newton fractals of g; see Example 1 below. Because a Newton fractal is a Julia fractal, "orbit" and "canvas" always mean an orbit of 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 = cos(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.
The second image of Figure 8.1 is a variation of the first. The image shown below, called "Crab Queue," is given by zooming in on one of the "bands" in the second image. It is accompanied by a fractal showing the Julia set in "Crab Queue."
Example 2 (Cyclotomic Polynomials): Another interesting example with known roots is a
Cyclotomic Polynomial. The picture on the left in Figure 8.2 is a Newton fractal of the "30th cyclotomic polynomial"
g(z) = z 8 + z 7 - z 5 - z 4 - z 3 + z + 1
with the unit disk highlighted. Since g happens to be a factor of z 30 - 1, its roots are among the 30th roots of unity that lie on the unit circle. In the picture, the thirty dots on the unit circle show where the roots of unity are located and eight of them colored yellow show the whereabouts of the roots of g. The picture on the right is a Newton fractal of the "20th cyclotomic polynomial"
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.
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