Like fractals?

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I created these about 5 years ago with a program called UltraFractal. The patterns are all based on iterative mathematical formulas. The best designs come from experimenting with the mathematical code to create interesting patterns and effects. You create an artistic effect through layering and the choice of aesthetically pleasing colors.

wow these are great ! !! !
i want to learn it too !
*jumps up and down enthousiastily*


yours are really great!

(did you also make the purple one i saw here not long ago? (i wanted to watch it again but forgot where i saw it)

I am very interested in fractals. Those are very, very cool. Where did you learn how to do them?

Spectacular! And all this beauty out of one mathematical formula? Awesome possum! How do you make the mathematical formulae?

I think fractals are amazingly beautiful. I did download a few programmes to create them myself, but like many things, I have not got round to testing them yet.

wow. awesome!

Breathtaking... this is the sort of thing I keep trying to draw by hand. Figures it would be mathematical, the kind of art we like.

I'll try and explain a little about one of the most famous fractal sets; the Mandlebrot set.

The basic starting point is the family of quadratic polynomials:

f_c: C --> C
z --> z^2 + c

with parameter c where c is in C.

The Mandlebrot set is then the subset of the complex plane: {c \in C: lim sup_{n in N} |f^n_c (0)| \in C}

that is the set of values of the parameter c such that the sequence a_n = f^n_c(0) is convergent.

So that gives you the set but it doesn't tell you how to make the pictures.

The simplest way is to make a plot in the argand diagram (complex plane) colouring a point if it is in the set and leaving it blank otherwise. Of course, it isn't always practical on a computer to check for definite whether a point is in the set or not so various algorithms are used. One simple way is just to keep iterating the function until its modulus exceeds 2. If it doesn't exceed two you let it run for a chosen finite number of iterations and then you can say that if the finite number of iterations is high enough; it is likely that the point is in the set. If you want to make a fancy colourful version of the set, you choose a different colour to correspond to a certain number of iterations and colour points according to how many iterations they take to get over modulus 2.


Of course, there are much simpler fractals that are easier to explain but I know when people use the term in general usage they are really referring to the pretty pictures!

Many works by the artist Jackson Pollock are pseudo-fractal in nature in that his paintings feature fractal like self similarity. Of course not all fractals have to neccesarily be self similar.

Thanks JakeG...I'll check out Jackson Pollack as well...
I did pretty well in math until highschool...but did terrible in calculus...so actually, I think I'll have to read your explanation several times and put some thought into it in order to understand it...but thanks for letting us know!

Most people won't understand the mathematical notation in defining the Mandelbrot set if they haven't had college level math.

Basically, complex numbers are a two dimensional set of numbers wgich have the same type of algebraic rules as the real numbers (for example a*b = b*a, a*(b+c) = a*b+a*c, etc). The full set of algebraic rules define what is called a "field". Both the real numbers and the complex numbers are fields. While the real numbers (which most people are familiar with) form a line going from negative infinity to positive infinity, the complex numbers form an infinite plane. The complex numbers are a generalization of the real numbers. The real numbers are just complex numbers with no imaginary part. The real numbers lie on the line y=0 in the complex plane.

All the fractal images you see lie somewhere on the complex plane. Each pixel corresponds to a complex number. Plug that complex number into an iterative formula and two things can happen. In one case the number will diverge with time, meaning the point will become farther and farther from the origin each time it is plugged into the formula. In other cases the number will either converge to a fixed point, alternate between a number of fixed points (called a limit cycle), or just bounce around randomly within a fixed area. The points that diverge lie outside the set and the points that converge lie inside. The outside points can be colored based on the number of iterations it takes for the point to exit a particular region. The inside points can be colored based on how quickly the formula converges to a fixed point or limit cycle. There are also more complicated coloring criteria you can use to get interesting patterns. Getting good patterns is mostly a matter of experimenting with different formulas and zooming into an interesting region of the complex plane.

You can explore the Mandelbrot set by going to Mandelbrot/Julia Set Explorer

Though the default is 50, you will need to increase the number of iterations to get more detail as you zoom in.

JakeG:

Three of my images are based on divergent fractals like the Mandelbrot set, but using more interesting function maps such as

z --> sin(z) + c.

The complex sine gives interesting "lobed" patterns when you use the modulus as the escape criteria. A couple of my images are actually convergent fractals based on variations of Newton’s method. These are not like the typical Mandelbrot set, as you tend to color the convergent pixels which occupy the majority of the plane. What you get is a webbed appearance with narrow strands of slow convergence surrounding regions that quickly converge to fixed points. Sometimes there are small divergent regions in the web intersections that look like mini-Mandelbrot or mini-Julia sets. These areas are really interesting to zoom in on.

I also like to use the "trap" methods for coloring. These methods are based on the location of the point after a fixed number of iterations. This tends to creates repeated shapes around the perimeter of the set. That's how you get the ribbon and star effects. You can also experiment with averaging the values over many iterations in different ways to get a variety of "smooth" coloring effects. Also, in a few of my fractals I made my output correspond to an elevation rather than a color. Then you can add a 3D light/shadow effect based on the terrain slope.

The hardest part is getting the colors to look good. I do this mostly through layering and playing with the different transparency effects. Of course the more layers you add the more processing time it takes to generate the image.

I might have some talent for naming... in order of appearance, I would call your fractals thus:

Through the Portal of the Springtime Goddess

Do Good Flies go to Heaven?

When Autumn Comes

Bubbles of a Drowning Man

Flight of the Phoenix Army

I already had names for them, but I like your names better than most of mine, especially the last one. Deciding on names is so hard for me.

I think the second one looks like the head of a mosquito with its sucker pointed down. Interesting that you also see a bug. Everyone else I've shown it to sees an alien face. The third one I thought looked like what you would see staring up at snow falling into a forest. That's what I went for in the coloring. The fact that the trees are twisted in a spiral gives it a nice surreal quality.

Is that the Newton-Raphson method? I think I remember playing around with things like that in maple one time.

Yea. The Newton-Raphson method also works for finding roots on the complex plane. The second fractal is the Newton-Ralphson method for a cubic polynomial. The, red, green, and blue regions are convergence basins for each of the three complex roots. For the fourth one I started with the Newton method but added some extra "fudging" in the formula to slow the convergence. The thing that looks like a Mandelbrot set is actually a divergent region. I have no idea why the divergent regions always look like Mandelbrot sets whan you do this.