The current types eFractal has are ones that I have found interesting from what I used in the past or created myself. All of the types have julia fractal associations except the lyapunov and popcorn fractals. Also, all of the types are escape time fractals excluding the lyapunov fractal, which is a 2D population fractal.
For the idea of escape time fractals: consider just multiplying a number by itself and taking the results through the same action. For example we would get the following sequence of numbers . . .
2 -> 4 -> 16 -> 256 -> 65536 -> ... -> infinity 1 -> 1 -> 1 -> ... -> 1 1/2 -> 1/4 -> 1/16 -> 1/256 -> ... -> 0
Going through the process of squaring, 2 escapes to infinity where as 1 and 0 are captured. Likewise for complex numbers we would have something similar for different operations. Some numbers escape to infinity others get captured into orbits. Now guessing where a point will go to is where "Iterations" and "Bailout" from the Fractals/Parameters menu comes in. Iterations is for how many steps you want to go through the process. And Bailout is the distance^2 from the origin that you assume after which a point will go to infinity.
So coloring a fractal is based on the Iterations until escape, Bailout, and starting or ending values of the orbit.
You can get the associated julia fractals to the current types by right clicking on the image when there is no zoombox. It will automatically draw the julia fractal generated from the clicked point. Right click on a julia fractal (with no zoombox) and it will revert back to the generating fractal. Play around with different areas from inside to outside the fractal basin. So we have the following behavior:
With Zoombox on image: Right Click := Zoom in
Without Zoombox on image: Right Click := Generate Julia Fractal
mandel is the traditional fractal that you see all the time. Actually what is weird about this fractal is that you will find its image in many different types. So I stayed with tradition and made this the first fractal that is generated by default.
cube_mandel and forth_mandel are of the same form except they use powers of 3 and 4 respectively for the formula.
z and c are complex numbers. Their real components are represented as the horizontal location and imaginary components are represented as the vertical location.
In the standard form the first z = 0 + 0i and the c is a point on the screen. In the julia form the first z is a point on the screen and the c is the point choosen when you right clicked on the screen (the seed). Then you find the next values in the sequence using:
mandel z = z^2 + c cube_mandel z = z^3 + c forth_mandel z = z^4 + c
Play around with the color functions and see how they bring out different details. If you zoom in you will need to add iterations under the Parameters option.
Also, try generating julia versions where the seed is inside the the set, outside, and very near the boundry.
These ones I liked and borrowed from Fractint, which is a great DOS program. But, I didn't like the X Windows version and I decided to borrow some of the types for eFractal. There some incredible images inside of these sets for you to find.
z and c are complex numbers. Their real components are represented as the horizontal location and imaginary components are represented as the vertical location.
In the standard form the first z = 0 + 0i and the c is a point on the screen. In the julia form the first z is a point on the screen and the c is the point choosen when you right clicked on the screen (the seed). Then you find the next values in the sequence using:
barnsley_1 z = (z - 1)c if real of z >=0 z = (z + 1)c if real of z < 0 barnsley_2 z = (z - 1)c if x_z*y_c + x_c*y_z >=0 z = (z + 1)c if x_z*y_c + x_c*y_z < 0 barnsly_3 z = (x*x - y*y - 1) + i(2*x*y) if x > 0 z = (x*x - y*y -1 + a*x) + i(2*x*y + b*x) if x <= 0
Set the inside color functions to zmag, zmag_diff, or atan. This is an interesting example of not seeing anything at all until you sometimes switch color functions. Then play around with julia variations from near the boundries.
This is an interesting self similar fractal that really shows that sometimes you can zoom in and zoom in and still see basically the same thing.
x is just the horizontal location on the screen and y is the vertical location. Starting with the point on the image (x,y) it the goes through the formula below to form the sequence.
x = x - 0.05*sin(y) + tan(3y) y = y - 0.05 sin(x) + tan(3x)
It is still an escape time fractal like those based in the complex plane.
All the outside color functions are better than the default iteration function.
This is a 2D fractal based upon a changing "vigor" population model. It is an example of a population following two different growth rates at different times. It is probably my all time favorite. The only problem with it is that it can be very unstable (it blows up eFractal every now and then).
The point (x,y) on the screen represents the two growth rates. The formula:
p = 4*r*p*(1 - p)
is varied between the two values according to the Lyapunov Seed from the parameters option. For example if the seed is 1001101010 then for the generations of the population it uses the rates as xyyxxyxyxy and then repeats for a given amount of generations. If the population is stable it gets a color otherwise it is colored as value 1 from the color gradient.
Use Edit Zoombox from the Fractals menu to backout from the default resolution. Something like (-8,-8) to (8,8) are better opposite corners.
Also, leave the color functions as iteration and max. And the boundries of the fractal are the most interesting.
Still need to type this one up. Basically this is a formula based upon a model for magnetism.
Still to come.
Edit the Zoombox to get more of the beginning fractal.
This one is neat for the fragmenting patterns are you move away from the center of the fractal.
Try out different Julia variation from around the center.
Personally I don't think this is very interesting myself. Mainly because the julia variations are so like the original fractal.
Just play around with everything and see what you get.
These are like the mandel forms except for the functions they are based upon.
z and c are complex numbers. Their real components are represented as the horizontal location and imaginary components are represented as the vertical location.
In the standard form the first z = 0 + 0i and the c is a point on the screen. In the julia form the first z is a point on the screen and the c is the point choosen when you right clicked on the screen (the seed). Then you find the next values in the sequence using:
z = sin(z) + c z = cos(z) + c
Move the Bailout value from the Parameter options up to something like 49 or above. This is really needed to see anything on the sine version.
Again, staying with the escape time fractals.
z and c are complex numbers. Their real components are represented as the horizontal location and imaginary components are represented as the vertical location.
In the standard form the first z = 0 + 0i and the c is a point on the screen. In the julia form the first z is a point on the screen and the c is the point choosen when you right clicked on the screen (the seed). Then you find the next values in the sequence using:
z = c*cos(z) z = c*exp(z)
Bump up the Bailout value to larger values. This will bring out more detail from c_exp.
This is my personal mistake from the first time I ever tried to program a Mandelbrot fractal. It is a common mistake when you are using two real numbers to do the complex number calculation.
z and c are complex numbers. Their real components are represented as the horizontal location and imaginary components are represented as the vertical location.
In the standard form the first z = 0 + 0i and the c is a point on the screen. In the julia form the first z is a point on the screen and the c is the point choosen when you right clicked on the screen (the seed). Then you find the next values in the sequence using:
x = x^2 - y^2 + a y = 2xy + b
Play around with different inside color functions and the julia variations.