Welcome to my nice little project about complex fractals. First of all some simple definitions- a fractal is something that infinitely repeats it self, or is infinitely self-similar. A complex fractal is a fractal that is generated on an inductive (or recursive) formula on Complex numbers, or numbers that have a real part and an imaginary part. The classic example is the Mandelbrot Set, which is the fractal that the applet starts out with if you don't mess around with it. This quadratic mandelbrot set is a good example of a complex fractal. Here's the inductive formula associated with this relatively well-known fractal. First of all, you might have read the word quadratic and said that doesn't look like a parabola, and you'd be right, but graphing a quadratic function is different from an inductive quadratic formula. Here's basically how it works. Take some arbitrary complex number (I'll call it c). c represents a point on the graph on the right, where the x axis is the real 'number line' and the y axis is the imaginary numbers, so while I say x and y in those those settings at the top, don't be fooled, those are really real and imaginary bounds. Now for the inductive definition. If you create a number z

z

So you might be wondering what the heck that simple-looking function has to do with the crazy fractal picture. Well, if you do this over and over again, as n=>infinity, one of two things happens- it either diverges to infinity or it doesn't. The values of c for which z

Another thing you can do to play with it is change the color scheme on it- My applet gives you full control over the color scheme, and while I like the defaults, they're not the only way to do it. Basically, you pick (in "red start", "blue start", and "green start") what color 0 is, and then pick (in "inc red by", "inc green by", and "inc blue by", inc means increment) how much the color changes for each count. Each of the three numbers (red, green and blue values) are numbers from 0-255 defined as black*(count*inc+start)%256, where black is 0 if count is greater than the number defined in "# of iterations", otherwise it's 1. If you put all the start values the same and all the inc values the same, the picture will be greyscale. If the inc and start for one color are both 0, that color won't be in there (if "blue start" and "inc blue by" are 0, for instance, only green red and yellow (the mix of green and red) shades will be present).

A third way to play with it is to find a cool part, zoom in on it, and then use the "Draw 3D" checkbox and hit the Redraw button. You'll see an angled view of the same section of the fractal where the 'height' of the pixel is also based on the count (as well as the color). The topography of the Mandelbrot set can be almost as interesting as the coloring.

Yet another way to play with it is to acknowledge that the mandelbrot set doesn't HAVE to be quadratic. It can be raised to any power! Change the number at "power-Real" to 3 and hit Redraw (if you're zoomed in, it's better to zoom back out first by hitting the "Show full set" button before you hit "Redraw"). A completely different fractal! Then try changing it to 4 and 5 and so on. After awhile, they get to be similar images with more points coming out of them. If you think of the Mandelbrot set as an infinite set of cardiods and ellipses (I know of a graduate student who attempted to prove that somehow), the quadratic has one ellipse on each segment recursively (I don't know how to really describe that better), and the cubic has 2 and the quartic has 3 and the 5th power has 4, etc. So the question is, if you raise it to the 2.5 power, do you have 1.5 ellipses on every segment? Put 2.5 into "power-Real" and change the box in the top right (the one that should say "Integer Power") so that it says "Real Power", and hit "Redraw". This one will take a little longer to calculate (what can I say, the math is a little more complicated). This one looks like you twisted the quadratic mandelbrot set around about 135 degrees counterclockwise and then tried to tear it appart with your hands. Now try every power from 2 to 3 in smaller divisions, every .1 or so if you have somewhere to go in the next 20 minutes, or every .05 if you have a little time. It rotates and changes somewhat drastically within .1, really. You'll notice that some of these have some smooth, cut-off looking edges, which I said usually means your breakout isn't right, but in this case, it probably is right. You'll notice that using 2.1 as the power makes it look alot more similar to the quadratic mandelbrot set, where 2.9 looks like an angled and slightly twisted cubic mandelbrot set, and in between, it twists and turns every which-way.

Of course, Mandelbrot sets don't HAVE to use real number, the exponent can also be imaginary or complex. As it turns out, imaginary exponents on the mandelbrot set are incredibly un-exciting, and complex exponents (they obviously take a little more time to calculate than real ones) are only worth looking at if the imaginary component of the power is significantly smaller than the real component of the power (like 1/10 maybe). Just a few tips. The more 'imaginary' the exponent is, the more likely the graph is to look like a big black circle. Remember, if it's taking longer than you think it should, you can hit the cancel button, and it will still display as much of the set as it's calculated so far.

Finally, another quite interesting way to play with mandelbrot sets is to use them to draw julia sets. What's a julia set, you might wonder? Well, some would say that the mandelbrot set is the set of all julia sets, which is just a bunch of technical mathematical nonsense. I don't think that the mandelbrot set should be defined by julia sets, but that it should be the other way around- I consider julia sets to be

How similar complex fractals are to Mandelbrot/Julia sets really depends on the fractal. Two things can pretty much be laid down- first of all, they all use at least one complex number, and second of all, they're all defined inductively- the value of z is based on the value of the previous z (and it's also conventional to use the letter 'z' to denote this inductively defined complex number). It's also a convention to define any other number involved as c, but not all complex fractals have a c value, while some have multiple 'c' values. Almost all (probably about 95%, and all but one of the types that I demonstrate) complex fractals are colored according to how long it takes them to diverge, or go to infinity. Depending on the base function, the definition of divergence could be 2 away from the origin, e

The "Phoenix curve" was discovered by Shigehiro Ushiki in 1988. To see the exact curve he discovered, put 0.56667 and -0.5 in as C-real and C-imaginary and change the second choice box to "Phoenix". Kind of cool, eh? It's a little harder to find interesting-looking Phoenix sets than it is to find good julia sets, because the mandelbrot version of phoenix sets isn't as helpful. Still, I put it in, that's the one I called "Ph-index". The center pixel of phoenix sets is still the same color as the pixel it's drawn from on the index set, but the difference is that even when the origin is black, the set might be disconnected, which is different from Julia sets. Several look like wings, (like the one I showed you above), and several look sort of like swords. The formula sort of stretches the rules a bit, because it seperates the real and imaginary parts of c, and uses them both as real scalars, which is probably the reason that the index is asymmetrical and the phoenix is symmetrical on the real axis rather than being radially symmetrical. In other words, c isn't really a complex number at all in this case, which may be why it doesn't have the symmetry that most complex fractals do. I guess you could say that The quadratic formula looks like this:

z

It also requires me to store not just the previous value of z, but also the value of z from 2 iterations ago.

There were several things that I spent long periods of time getting right in this applet, and even getting something that resembled what looked like the Newton-Raphson sets I saw pictures of on the internet was among those things. Just for kicks, keep the function type on "Integer Power" and the "power-Real" on 2 and choose "Newton-Raph." for the fractal type. I say just for kicks because you'll look at a straight line with some circles getting bigger and bigger and you'll be like "I'm not amused". Apparently (I don't remember learning this in any math class, but it works) the Newton-Raphson method is a fairly effective way of finding the roots to an equation. It works to find the zeros in polynomials with complex numbers, too, and alot of polynomials with real coefficients have complex or imaginary roots anyways. When using complex numbers in the Newton-Raphson method, there's two things that are fractal about it (and my applet only shows one of them). Here I show the amount of time it takes to find a root of the polynomial, which forms an interesting repeating pattern on higher powers (ok, change "power-Real" to 3 and hit "Redraw" again and see what I mean). The other thing that's fractal about Newton-Raphson sets (I might make a seperate applet for Newton-Raphson sets for just this reason) is which root you'll converge to. You'll notice that it goes out infinitely in all directions. The number of 'chains' going out depends on the power. The angles are always equidistant (assuming integer powers), but are rotated depending on what you use for c. The reason the Newton Raphson sets go out infinitely is because the color here is based on how long it takes to

z

Yes, that f'(z

z

Division is an interesting operation when you're using complex numbers. (Skip this part if you don't care about the arithmetic involved here). You can't divide a number by a complex number par se, but what you can do is you can multiply the top and the bottom by the

The main reason I added non-integer and complex exponents in this applet is because they make really cool Newton-Raphson sets. In fact, with the amount of research, math, and tweaking that it took to do complex powers, I would have never done it and also never known whether I had it right if it wasn't for the fact that I knew that complex powers made for really cool Newton-Raphson sets. When taking the derivative of a function with a complex exponent, you still just subtract one, you don't do anything to the imaginary component of the exponent, but of course, you still multiply the derivative by the whole complex number. First of all, though, let's change the function type from "Integer power" to "Real power" and type in a number for "power-Real" that's not an integer, say 3.5. Remember that there's 3 chains coming out if the power is 3, and there's 4 chains if the power is 4. Strangely enough, while there's really no such thing as a 'half' chain, this Newton-Raphson set is about how I'd imagine 3.5 fractal chains. Notice, however, that the chains still go in straigt directions outward. Now change the function type from "Real power" to "Complex power" and pick an imaginary number to go with it. You'll notice now that the chains twist around, and your eyes aren't deceiving you, there's sort of a funny cut-off thing going directly down the negative imaginary axis. I thought it was cool, if not worth all the effort associated with getting complex powers to work, let alone getting Newton-Raphson sets to work at all.

The Quartet set is a mandelbrot-type set with a much simpler formula-

z

z

You probably noticed in between all of them, there is 4 fractals called "My Fractal 1", "Index 1", "My Fractal 2", and "Index 2". These are all complex fractals I came up with in an effort to figure out the Phoenix set fractal, so you'll notice the formula is similar (for both of them) to that of the Phoenix set:

z

The difference between this and the Phoenix set is that this time, c is still divided, but the imaginary component of c is actually treated as imaginary when I multiply it with z

One thing you get alot of with the "My Fractal" sets is spiral patterns, especially with higher powers of "My Fractals". Let's try a 4th-power one. Change "power-Real" to 4 and set c to .65+.32i and draw a "My Fractal 1". Just 4 little things going around in a circle. Now make a "My Fractal 2" from the same numbers. Now the top-right thingy is really big, the top-left one is a little smaller, and the two on the bottom are just tiny.

By now you've certainly noticed that "Integer Power", "Real Power", "Imaginary Power", and "Complex Power" aren't the only function types I have listed. I also included sine, cosine, tangent, natural logarithms, and e