HOME | DD

Description
As a mathematician and a generally artsy sort of person, I cannot regard fractal art and (the people who make it) with a single mind. On the one hand, I am gratified that people are appreciative and often entranced by the beauty of fractals; on the other hand, I am vexed by worries that people who play around with fractals do not fully understand them from a mathematical perspective.

So, put on your learning hats everyone: it's math time!

Although there are a wide variety of fractals, by far the most popular are those generated by iterated function systems. This entails picking some function (z^2, (z^3 -1)/(z + 2), 1 - z + sin(z), etc.) and evaluating it over and over again ("iterating" it) at lots of different points to see what its behavior is like. The variable z is usually a complex number (a complex variable, as we mathematicians call it); this is a number of the form a + bi, where a and b are ordinary "real" numbers, and i is the "imaginary unit", the square root of -1. Unlike real numbers (which can be drawn on a one-dimensional number line), complex numbers are two-dimensional, so, to draw them, you have to graph them as points in a plane (or on a canvas!). The more, the set of all real numbers is contained in the set of all complex numbers; a real number is just a complex number a + bi where the "imaginary part" b is 0. By the Pythagorean Theorem, the magnitude/distance from 0 of the complex number a + bi is equal to the square root of a^2 + b^2.
Since two-dimensional variables are responsible for two-dimensional fractals, it should come as no surprise that fractals of dimension three and hire are generated by variables of dimension three and higher (such as quaternions, though I can't speak much of them, seeing as I have yet to learn about them— it's on my list, though.
There are two keys to understanding fractals and the iteration of functions: fixed points and critical points.
If you have a function f(z) (I use z throughout this discussion to denote a complex variable / complex number), then a fixed point of f is a point that is not changed by f. For instance, the function z^2 has two fixed points, 0 and 1:
(0)^2 = 0 (you plug in z = 0, and get 0 as the result)
(1)^2 = 1 (you plug in z = 1, and get 1 as the result)
The fixed points of a function are important because they remain unchanged no matter how many times you iterate the function at them. There are different kinds of fixed points, classified by how the function behaves when you iterate it at a point near a fixed point.
For instance, 0.1 is pretty close to 0, so when we iterate z^2 at 0.1 several times we get:
(0.1)^2 = 0.01
(0.01)^2 = 0.0001
(0.0001)^2 = 0.00000001
etc.
From the above, it is clear that the more times we iterate our function at 0.1, the closer and closer it gets to the fixed point at 0. This means that 0 is an "attracting fixed point" of z^2; any point that is sufficiently close to zero will be iterated toward 0 by z^2.
On the other hand, if we choose 0.9 (which is pretty close to 1) and iterate z^2 at it, we get:
(0.9)^2 = 0.81
(0.81)^2 = 0.6561
(0.6561)^2 = 0.43046721
etc.
Here too, iterating z^2 at 0.9 will eventually bring up the point close to zero—i.e. AWAY from the fixed point at 1. 1 is therefore what we call a "repelling fixed point"; points close to hey rappelling fixed point of a function are pulled away from that fixed point when iterated by the function. Indeed, if we had picked a point that was close to 1, but slightly larger in magnitude than it, that point would have been repeatedly squared to larger and larger numbers.
In fact, it turns out that on the complex plane, the function z^2 will act in one of three ways when iterated at any point other than its fixed points:
1) The point will be iterated toward 0. This will occur for all points in the complex plane contained inside a disk of radius one drawn at the center of the plane.
2) The point will always have a distance of 1 unit from 0. In the complex plane, the set of all points of distance of 1 unit from 0 is a circle (not the inside of the circle, mind you, just the boundary/circumference).
3) The point will be iterated to larger and larger values, eventually become infinitely large. This only happens if the point is more than 1 unit away from 0; that is to say, such a point is outside of a circle of radius one centered at zero.
Each of these three regions are "disjoint", meaning that they are mutually exclusive; any given point can be in one and only one of those three regions.
Note that both regions 1 and 3 have a non-zero area, while region 2 (the boundary of a circle) is just a curved line, and so, has no area. It turns out that a similar breakdown of the plane into distinct regions (with some of those regions having an area, and others being just lines and/or dots) occurs for almost any function you can think of. The regions that have an area are collectively referred to as the "Fatou set" (Fuh-too); each one of those regions is called a "component" of the Fatou set. On the other hand, the lines and dots that have no area are called the "Julia set". (The person Julia sets were named after was a guy with the last name of Julia. I think he was Italian. Or maybe French.)
Every function has its own unique Julia and Fatou sets. The Julia sets are defined as the boundaries of the Fatou sets: they separate the components of the Fatou sets. That's important, seeing as a function will behave differently on each of the components of its Fatou set.
The function z^2 happens to be one for which the Fatou and Julia sets are extremely simple (both to think about, and to draw). In most cases, the sets are fractals: infinitely complicated regions with repeating details that do not go away, no matter how many times you zoom in to look at them (and thus, rather hard to draw).
The other important aspects of a fractal are the critical points of the function that generates that fractal. Loosely speaking, a critical point is an isolated point in the complex plane where a function does not have an "undo" button; we called this "undo" button the "inverse" of the function.
A function like f(x) = x + 1 has no critical points because, the matter what you plug in for x, you can always undo it and recover the previous value. Iterating x + 1 just means adding 1 over and over again; you can undo that simply by integrating the inverse, x - 1, the function that subtracts 1 every time you iterate it.
When you have a region where a function has no critical point, you can apply both that function and its inverse without worry. Because most of the functions in question are mathematically "well-behaved" (The technical term would be "conformal") wherever they have an inverse, if you plug in a set of points to the function, the output will look similar. This is why fractals like the one above have repeating details at smaller and smaller scales: the details arise from repeated applications of a function that is conformal on that part of the plane.
Generally, from both a mathematical and an artistic perspective, the most common fractals are those generated by polynomials (things like 2z^3 - z + 1, z^2 - 3, z^5 + 2iz^2, etc.) of degree two or higher. (The degree of a polynomial is the largest power of z that occurs in it; The polynomials I just listed have degrees of three, two, and five, respectively.) (Also common are fractals made from "rational functions": functions of the form f(z) = P(z) / Q(z) where P and Q are polynomials)
For a polynomial P(z), one of the components of the Fatou set will always be something called "the basin of attraction to infinity". This is the unique region of the plane that always grows larger and larger (without ever stopping) when iterated by P.
A crucial fact about polynomials is that the only ever have a finite number of critical points. Since there's only a finite amount of them, one of those critical points must be further away from 0 then all the other critical points.
Consequently, if you go far enough away from zero, you'll eventually hit a region where, no matter how much further you go out you'll never come across another critical point of P—kind of like passing a "last gas station forever" sign on the freeway. When you've reached that no man's land, you can be assured that you're inside the basin of attraction to infinity of the polynomial P. Furthermore, since there are no critical points in this region, not only can we iterate P, we can also iterate P's inverse! Since iterations of P on the basin make everything larger and larger, iterations of P's inverse will make everything smaller and smaller. However, since there are always going to be components of the Fatou set other than the basin, eventually, you'll reach the boundary of the basin—the place separating the basin from the rest of the plane. That boundary—being part of the boundary of the Fatou set of P—is therefore the Julia set of P.
When your computer is drawing up fractals for you, it's quite likely doing exactly the procedure described above: it's picking points in the plane, and then applying the fractal generating function backwards over and over again. The Julia set is then the boundary that all of those iterated points accumulate at. Colorings of fractals are done to indicate how many times the computer had to iterate backwards to get to that particular region. Different colors correspond to different amounts of iterations required before the point becomes stupidly huge in magnitude. Black usually represents the other side of the Julia set: the component of the Fatou set where the function does not iterate points to infinity.
The fractals depicted in this image and its three companion images were "hand-made" by me (in whatever sense anything made on the computer can be said to be "hand-made" ). They are approximations of the basin of attraction to infinity of the exponential function E(z) = e^z (you can find this function on a you can find this function on any useful scientific/graphing calculator, although it usually has an x in place of the z). I say "approximation" because this and the other four pictures only show how E behaves after five iterations. To get anything close to the "correct" picture, you would have to iterate many, many, many more times than that, far more than my darling MacBook Air can handle. Indeed, the application "Grapher" that I used to make the images crashed several times before I finally got a picture that I liked. Not only that, but, for higher iterations (my laptop can do a bit more than five iterations), The screen becomes filled with increasingly large white patches where the values are so stupidly huge that they cannot be handled by my computer's processor. Then, there's the "alternating" problem.
The regions in red and blue show the places in the plane that the fifth iterate of E sends to extremely large and extremely small (I.e, really close to zero) magnitudes, respectively. (Here, "extremely large" means the biggest number I could plug in my computer without breaking it).
I am posting these fractals in something of an act of celebration. Over the past three or so weeks, I've been performing research on the topics of iterated exponentiation and tetration. Basically, I'm trying to answer the questions "how do you push the e^x button on the scientific calculator a non-integer number of times" (1/2 of a time, -2.37583 times, ∏ times, 1 + 2i times, etc.) (that is fractional exponentiation), and "can you find a nice function that will press that button any specified number of times" (that is tetration).
I am extremely excited to say that, in all likelihood, I may have actually found an answer to both of these questions. Come autumn, I'm going to start graduate school in pursuit of a PhD in pure mathematics, and I am unbelievably eager to explore this idea in greater detail under the guidance of a professorial mentor. (I emailed him a quick go-through of my major results, and I'm still presently awaiting his response.)
My research involves trying to find functions that do what I want them to do, and then to show that they behave nicely on the places in the complex plane where they are defined. The fractals I've posted are relevant to my research is that they represent part of the region of the plane where some of my functions (the specifically, the tetration functions) exist. Those functions are actually defined on the basin of attraction to infinity of the exponential function, thus, the actual region which they are defined is much larger than the region depicted.

The grainy-looking parts are where the detail of the fractal became too intricate for my laptop to render properly. They are actually supposed to be clusters and trails of star-like hyperbolae, as can be seen in the zoomed in shot at: Exponential (5th Iterate) Real Part Escape Set - 4

Also worth mentioning is the fact that the exponential function is a "transcendental": it cannot be written as a polynomial, or a rational function, or involving square roots or anything like that. Consequently—like all other transcendental functions—the fractals generated by e^z are markedly different from those generated by polynomials or rational functions. The study of fractals of rational functions or polynomials (known as "complex dynamics") started in the early 20th century (Fatou basically invented the subject back in 1919); since the 1950s or so, these fractals have been studied to death (such as those fractals generated by quadratic polynomials of the form z^2 + c, where c is a complex number). On the other hand, the theory of fractals of transcendental functions is still very new (even though Fatou himself also began it in 1919), and there are still many unanswered questions. Hopefully, it'll give me something to do while in graduate school.

Anywho, if you've made it this far:
Congratulations, you've learned something!

*Flies off to work on his next project*

Description
As a mathematician and a generally artsy sort of person, I cannot regard fractal art and (the people who make it) with a single mind. On the one hand, I am gratified that people are appreciative and often entranced by the beauty of fractals; on the other hand, I am vexed by worries that people who play around with fractals do not fully understand them from a mathematical perspective.

So, put on your learning hats everyone: it's math time!

Although there are a wide variety of fractals, by far the most popular are those generated by iterated function systems. This entails picking some function (z^2, (z^3 -1)/(z + 2), 1 - z + sin(z), etc.) and evaluating it over and over again ("iterating" it) at lots of different points to see what its behavior is like. The variable z is usually a complex number (a complex variable, as we mathematicians call it); this is a number of the form a + bi, where a and b are ordinary "real" numbers, and i is the "imaginary unit", the square root of -1. Unlike real numbers (which can be drawn on a one-dimensional number line), complex numbers are two-dimensional, so, to draw them, you have to graph them as points in a plane (or on a canvas!). The more, the set of all real numbers is contained in the set of all complex numbers; a real number is just a complex number a + bi where the "imaginary part" b is 0. By the Pythagorean Theorem, the magnitude/distance from 0 of the complex number a + bi is equal to the square root of a^2 + b^2.
Since two-dimensional variables are responsible for two-dimensional fractals, it should come as no surprise that fractals of dimension three and hire are generated by variables of dimension three and higher (such as quaternions, though I can't speak much of them, seeing as I have yet to learn about them— it's on my list, though.
There are two keys to understanding fractals and the iteration of functions: fixed points and critical points.
If you have a function f(z) (I use z throughout this discussion to denote a complex variable / complex number), then a fixed point of f is a point that is not changed by f. For instance, the function z^2 has two fixed points, 0 and 1:
(0)^2 = 0 (you plug in z = 0, and get 0 as the result)
(1)^2 = 1 (you plug in z = 1, and get 1 as the result)
The fixed points of a function are important because they remain unchanged no matter how many times you iterate the function at them. There are different kinds of fixed points, classified by how the function behaves when you iterate it at a point near a fixed point.
For instance, 0.1 is pretty close to 0, so when we iterate z^2 at 0.1 several times we get:
(0.1)^2 = 0.01
(0.01)^2 = 0.0001
(0.0001)^2 = 0.00000001
etc.
From the above, it is clear that the more times we iterate our function at 0.1, the closer and closer it gets to the fixed point at 0. This means that 0 is an "attracting fixed point" of z^2; any point that is sufficiently close to zero will be iterated toward 0 by z^2.
On the other hand, if we choose 0.9 (which is pretty close to 1) and iterate z^2 at it, we get:
(0.9)^2 = 0.81
(0.81)^2 = 0.6561
(0.6561)^2 = 0.43046721
etc.
Here too, iterating z^2 at 0.9 will eventually bring up the point close to zero—i.e. AWAY from the fixed point at 1. 1 is therefore what we call a "repelling fixed point"; points close to hey rappelling fixed point of a function are pulled away from that fixed point when iterated by the function. Indeed, if we had picked a point that was close to 1, but slightly larger in magnitude than it, that point would have been repeatedly squared to larger and larger numbers.
In fact, it turns out that on the complex plane, the function z^2 will act in one of three ways when iterated at any point other than its fixed points:
1) The point will be iterated toward 0. This will occur for all points in the complex plane contained inside a disk of radius one drawn at the center of the plane.
2) The point will always have a distance of 1 unit from 0. In the complex plane, the set of all points of distance of 1 unit from 0 is a circle (not the inside of the circle, mind you, just the boundary/circumference).
3) The point will be iterated to larger and larger values, eventually become infinitely large. This only happens if the point is more than 1 unit away from 0; that is to say, such a point is outside of a circle of radius one centered at zero.
Each of these three regions are "disjoint", meaning that they are mutually exclusive; any given point can be in one and only one of those three regions.
Note that both regions 1 and 3 have a non-zero area, while region 2 (the boundary of a circle) is just a curved line, and so, has no area. It turns out that a similar breakdown of the plane into distinct regions (with some of those regions having an area, and others being just lines and/or dots) occurs for almost any function you can think of. The regions that have an area are collectively referred to as the "Fatou set" (Fuh-too); each one of those regions is called a "component" of the Fatou set. On the other hand, the lines and dots that have no area are called the "Julia set". (The person Julia sets were named after was a guy with the last name of Julia. I think he was Italian. Or maybe French.)
Every function has its own unique Julia and Fatou sets. The Julia sets are defined as the boundaries of the Fatou sets: they separate the components of the Fatou sets. That's important, seeing as a function will behave differently on each of the components of its Fatou set.
The function z^2 happens to be one for which the Fatou and Julia sets are extremely simple (both to think about, and to draw). In most cases, the sets are fractals: infinitely complicated regions with repeating details that do not go away, no matter how many times you zoom in to look at them (and thus, rather hard to draw).
The other important aspects of a fractal are the critical points of the function that generates that fractal. Loosely speaking, a critical point is an isolated point in the complex plane where a function does not have an "undo" button; we called this "undo" button the "inverse" of the function.
A function like f(x) = x + 1 has no critical points because, the matter what you plug in for x, you can always undo it and recover the previous value. Iterating x + 1 just means adding 1 over and over again; you can undo that simply by integrating the inverse, x - 1, the function that subtracts 1 every time you iterate it.
When you have a region where a function has no critical point, you can apply both that function and its inverse without worry. Because most of the functions in question are mathematically "well-behaved" (The technical term would be "conformal") wherever they have an inverse, if you plug in a set of points to the function, the output will look similar. This is why fractals like the one above have repeating details at smaller and smaller scales: the details arise from repeated applications of a function that is conformal on that part of the plane.
Generally, from both a mathematical and an artistic perspective, the most common fractals are those generated by polynomials (things like 2z^3 - z + 1, z^2 - 3, z^5 + 2iz^2, etc.) of degree two or higher. (The degree of a polynomial is the largest power of z that occurs in it; The polynomials I just listed have degrees of three, two, and five, respectively.) (Also common are fractals made from "rational functions": functions of the form f(z) = P(z) / Q(z) where P and Q are polynomials)
For a polynomial P(z), one of the components of the Fatou set will always be something called "the basin of attraction to infinity". This is the unique region of the plane that always grows larger and larger (without ever stopping) when iterated by P.
A crucial fact about polynomials is that the only ever have a finite number of critical points. Since there's only a finite amount of them, one of those critical points must be further away from 0 then all the other critical points.
Consequently, if you go far enough away from zero, you'll eventually hit a region where, no matter how much further you go out you'll never come across another critical point of P—kind of like passing a "last gas station forever" sign on the freeway. When you've reached that no man's land, you can be assured that you're inside the basin of attraction to infinity of the polynomial P. Furthermore, since there are no critical points in this region, not only can we iterate P, we can also iterate P's inverse! Since iterations of P on the basin make everything larger and larger, iterations of P's inverse will make everything smaller and smaller. However, since there are always going to be components of the Fatou set other than the basin, eventually, you'll reach the boundary of the basin—the place separating the basin from the rest of the plane. That boundary—being part of the boundary of the Fatou set of P—is therefore the Julia set of P.
When your computer is drawing up fractals for you, it's quite likely doing exactly the procedure described above: it's picking points in the plane, and then applying the fractal generating function backwards over and over again. The Julia set is then the boundary that all of those iterated points accumulate at. Colorings of fractals are done to indicate how many times the computer had to iterate backwards to get to that particular region. Different colors correspond to different amounts of iterations required before the point becomes stupidly huge in magnitude. Black usually represents the other side of the Julia set: the component of the Fatou set where the function does not iterate points to infinity.
The fractals depicted in this image and its three companion images were "hand-made" by me (in whatever sense anything made on the computer can be said to be "hand-made" ). They are approximations of the basin of attraction to infinity of the exponential function E(z) = e^z (you can find this function on a you can find this function on any useful scientific/graphing calculator, although it usually has an x in place of the z). I say "approximation" because this and the other four pictures only show how E behaves after five iterations. To get anything close to the "correct" picture, you would have to iterate many, many, many more times than that, far more than my darling MacBook Air can handle. Indeed, the application "Grapher" that I used to make the images crashed several times before I finally got a picture that I liked. Not only that, but, for higher iterations (my laptop can do a bit more than five iterations), The screen becomes filled with increasingly large white patches where the values are so stupidly huge that they cannot be handled by my computer's processor. Then, there's the "alternating" problem.
The regions in red and blue show the places in the plane that the fifth iterate of E sends to extremely large and extremely small (I.e, really close to zero) magnitudes, respectively. (Here, "extremely large" means the biggest number I could plug in my computer without breaking it).
I am posting these fractals in something of an act of celebration. Over the past three or so weeks, I've been performing research on the topics of iterated exponentiation and tetration. Basically, I'm trying to answer the questions "how do you push the e^x button on the scientific calculator a non-integer number of times" (1/2 of a time, -2.37583 times, ∏ times, 1 + 2i times, etc.) (that is fractional exponentiation), and "can you find a nice function that will press that button any specified number of times" (that is tetration).
I am extremely excited to say that, in all likelihood, I may have actually found an answer to both of these questions. Come autumn, I'm going to start graduate school in pursuit of a PhD in pure mathematics, and I am unbelievably eager to explore this idea in greater detail under the guidance of a professorial mentor. (I emailed him a quick go-through of my major results, and I'm still presently awaiting his response.)
My research involves trying to find functions that do what I want them to do, and then to show that they behave nicely on the places in the complex plane where they are defined. The fractals I've posted are relevant to my research is that they represent part of the region of the plane where some of my functions (the specifically, the tetration functions) exist. Those functions are actually defined on the basin of attraction to infinity of the exponential function, thus, the actual region which they are defined is much larger than the region depicted.

The grainy-looking parts are where the detail of the fractal became too intricate for my laptop to render properly. They are actually supposed to be clusters and trails of star-like hyperbolae, as can be seen in the zoomed in shot at: Exponential (5th Iterate) Real Part Escape Set - 4

Also worth mentioning is the fact that the exponential function is a "transcendental": it cannot be written as a polynomial, or a rational function, or involving square roots or anything like that. Consequently—like all other transcendental functions—the fractals generated by e^z are markedly different from those generated by polynomials or rational functions. The study of fractals of rational functions or polynomials (known as "complex dynamics") started in the early 20th century (Fatou basically invented the subject back in 1919); since the 1950s or so, these fractals have been studied to death (such as those fractals generated by quadratic polynomials of the form z^2 + c, where c is a complex number). On the other hand, the theory of fractals of transcendental functions is still very new (even though Fatou himself also began it in 1919), and there are still many unanswered questions. Hopefully, it'll give me something to do while in graduate school.

Anywho, if you've made it this far:
Congratulations, you've learned something!

*Flies off to work on his next project*