About Fractals

In a fundamental way, fractals are merely shapes, just like circles, triangles, and squares are shapes. Yet, fractals are so different from the more common shapes that we know. You can tell immediately that fractals are very different. Circles, for example, are smooth, and we can see that even more precisely if we look at all the time smaller image of pieces of a circle:

When the magnification of the piece of the circle increases, the circle’s edges will flatten out to eventually get indistinguishable from straight lines. For a smooth shape, this is characteristic: if only you’ll zoom in enough, the edges will look just like straight lines.

Fractals are different: no matter how often or how much you’ll zoom in, their shape will never flatten out. With fractals, regardless how much you are zooming in, you’ll never be without detail. In every fractal, there is an infinite amount of detail, literally. Usually, the detail is made up of tiny copies of the fractal’s overall shape. If we are zooming in on the right spots of the Julia or Mandelbrot set, for example, we can see tiny copies of the sets that are buried inside. 

The differences between fractals and smooth shapes are arising from the way the shapes are formed mathematically. A smooth shape may usually be described with a relatively simple equation, for example, Ax + By = C for a triangle’s side, or x2 + y2 = r2 for a circle. A fractal is generated in a sort of iterative process, at which one step’s result is the input for the next step.

To generate an escape-time fractal, each pixel within an image is representing a complex number (i.e. number that has two parts), just like a set of map coordinates. Each fractal comes with its own characteristic formula, and within this formula, ‘z0′ is the number used to generate number sequences. With each iteration, the previous outcome is the formula’s new input.

Different things will be happening to the sequence, all depending on the numbers chosen and the formula. With the set of Mandelbrot, we can see that some numbers, or pixels, are leading to sequences that get bigger and bigger, to never turn back. These pixels or numbers are named ‘Outside’ points. If you use other numbers, sequences will come out that to produce repeating patterns, or meander to never get too big. These pixels or numbers are called ‘Inside’ points. The boundary between ‘Outside’ and ‘Inside’ points now is the fractal, and this is exactly where the most interesting details are found.

In a mathematical sense, the ‘outside’ points include the pixels of which their sequence’s numbers are infinitely large, while the ‘inside points’ include everything else. To make sure the formula will not iterate endlessly, there is a ‘bailout’ condition, and if the sequence is crossing that ‘bailout’ threshold, it won’t return and we deal with an ‘outside’ point. If the sequence is never growing bigger than the bailout threshold (of course within a certain number of iterations), we consider that point to be an ‘inside’ point. Generally, the image is more details in line with the iterations that are calculated.

A fractal’s basic shape is determined by the calculation formula, though the gradient and the coloring formulas also affect its final appearance. The coloring formulas are communicating with the calculation formulas to transform the numbers sequence into a specific value that’s used by the gradient for the assignment of the numbers to various areas of the palette, just like ‘painting by number’.