Fractals are geometric shapes that exhibit self-similarity. That is, they have the same pattern at different scales. In fact, fractals continue to show intricate detail at arbitrarily small scales.
One variety of fractal is the escape-time fractal. These fractals are generated by iterating a formula on each point in a given space. If a point diverges as the formula is iterated, it escapes; otherwise, it remains bounded. Three of the more well known escape-time fractals are the Mandebrot set, Julia set, and Burning Ship fractal.
The Mandelbrot set uses complex values, and is generated by iterating the following formula on each point c:
The Julia set use the same formula , but the iteration is different. For a fixed parameter , the Julia set is found by iterating the formula on each point z.
The Burning Ship fractal is again similar to the Mandelbrot set. It uses the same formula, but before squaring z, the absolute values of its real and imaginary components are taken and used in place of the original z point:
The Newton fractal uses the formula
This formula is somewhat different from the three others listed above, as it takes a function p as a parameter. The Newton fractal is also different in that instead of checking whether a point diverges, it checks whether a point converges to a root.
Despite their similar underlying mathematics, the stunning images of these fractals all have their own unique features.