Julia Sets

Julia Sets

The above is a fractal called the Julia Set that was drawn with the function . For those who know it and its properties, you can set the magnification factor, as well as the value of c, in the code(JS tab).

This fractal normally exists in the complex plane, which consists of y-values that are imaginary and x-values that are real. The y-values are imaginary because they contain the square root of -1, which is denoted by i and can be used to find the square roots of other negative numbers.

Basically, to check what coordinate points are in the Julia Set, you take a function of the form:

f(z) = z² + c

Both c and z are complex(they contain a real number and imaginary number in the form a + bi). What this does is square z, and move it by c. z happens to be the coordinates of the current point on the complex plane(in form x + yi). In the code, I use this formula to "convert" the coordinates to new ones:

(x, y) -> (x² - y² + a, 2xy + b)

I iterate this a 100 times, and if the values of x and y never go beyond a distance of 5 from the center, it makes up the set of numbers that form the Julia Set. These numbers are colored in white.

However, if the numbers happen to exceed a distance of 5, then it is colored according to it's "escape time", or the time it takes to become extremely large(A.K.A head on to infinity). This explains the shades of red and black that surround the white points.

For those that are interested in the coloring recipe, it's here:

hsl(0, 100%, rate%)

Pretty simple.

For those who want to learn more, I suggest taking a look at the book "Fractals: A Short Introduction" by Kenneth Falconer, and the immense amount of Numberphile videos there are on fractals.