In this studio, you will find explorers for the Mandelbrot set with all his variants. If you have made some fractal explorers yourself, please write in the comments. If you don't know what a Mandelbrot set is, please read this text:
All numbers on the number line, when squared, become positive. So numbers that their squares were negative had to be invented.
There are numbers which can not be found on the number line, for example sqrt(-4). This number can be written as 2*sqrt(-1). Because sqrt(-1) isn't on the number line, it's called an "imaginary number". so sqrt(-4) or 2*sqrt(-1) is written as 2i, or for another example 4+sqrt(-9) is 4+3i. In fact, i means sqrt(-1). Because you never will find one of this terms on the number line, John Wallis has invented the "complex plane". A "Mandelbrot set" (invented by Benoît Mandelbrot) is the graphical visualization of Z´=Z^2+C on the complex plane. Every point Z on the complex plan is written as x+iy, so 4+3i is on the complex plane on the point x=4 and y=3. Benoît's formula Z´=Z^2+C means therefore Z´=(x+iy)^2+C (points on the complex plane are always named Z). Z´=(x+iy)^2+C=x^2+xiy+iyx+(iy)^2 =x^2+2ixy+i^2*y^2. We don't now the equivalent to i, but i^2 must be -1 because i=sqrt(-1) and (sqrt(x))^2is always x, so Z^2=x^2+2ixy+(iy)^2=x^2+2ixy+(-1)*y2=x^2-y^2+2ixy. Now, because Z=x+iy, every addend which is multiplied by i is a part of the imaginary part of formula. That means, Z´=z^2+C can be written as x´=x^2-y^2 and y´=2xy. So, we dissected the formula in 2 parts and then we factorized i in the imaginary part. The other part of formula will be named "real part". That's the theoretical how-to. Now let's practice that; we have a point C or Z0 on the complex plane. For first, we need the coordinates of the point; in this example I will reach with the point X0=-1 and Y0=1. Then we have second point Z1, his coordinates are X1=X0*X0- Y0*Y0+X0=(-1)^2-1^2+(-1)=-1 and Y1=2*X0*Y0+Y0=2*(-1)*1+1=-1. Now, we do the second iteration with point Z2. X2=X1*X1-Y1*Y1+X0=(-1)^2-(-1)^2+(-1)=-1 and Y2=2*X1*Y1+Y0=2*(-1)*(-1)+1=3. Then we do the third iteration with point Z3 and then with Z4 and then with Z5 and... So we have a infinity list of points on the complex plane. If this points converge to one ore more points, the point Z0 will be a part of the Mandelbrot set. If the points diverge to infinity, it isn't. The points in the Mandelbrot set are always colored black, all the over points are colorful.