The word fractal roughly means “fractional dimensions,” referring to the mathematical tendency of a fractal shape to form somewhere “between” classical geometric dimensions. Fractals are created by iterating a mathmatical formula, geometric progression, or another repeating process to generate an increasingly complex, often beautiful result.
A classic example of a simple, geometric fractal is a Koch Curve. This fractal shape is formed by overlaying an isosceles triangle on top of an inverted isosceles triangle of the same size, creating a six-point star. This forms six more triangles. Repeat for each of these triangles, and again and again… This ultimately creates a sort of fuzzy-star shape, somewhat organic-looking. The figure shows an example of the first two iterations of this process.
Mandelbrot Set
The Mandelbrot Set was discovered by the mathmetician Benoit Mandelbrot at IBM in 1979. He discovered that repeated application of a deceptively simple formula, z=z²+c produced an increasingly complicated result. The value of z starts at zero, and c is a complex (imaginary) number. The “set” is those values of c where the formula result never exceeds 2, (or “escapes”) no matter how many times the formula is iterated.
The colorful graphics that can be generated by a computer are created not by the points that exist in the set, but by those outside that escape the formula. Each escape point took a certain number of iterations of the formula to escape. Different colors can be assigned to the different iteration counts, such as a color gradient or an interference pattern (sine waves). Since complex numbers consist of two axes, real and imaginary, this can be plotted on a two-dimensional graph to create a fractal image.
Multibrot Space
The Mandelbrot set can be extended into three dimensions by adding another variable, the exponent. This exponent is traditionally fixed at the value two (or squared) in the Mandelbrot formula, z=z²+c. Since c is a complex number, the results are mapped onto a two-dimensional plane, with the real component mapped to the horizontal axis and the imaginary component mapped to the vertical axis.
Altering the formula to z=zn+c, we now can vary the exponent, n. A Multibrot simply changes the exponent to another fixed value and draws the resulting set. However, the exponent can be mapped to a third axis, creating a three-dimensional “Multibrot space” for further exploration.
How do we explore such as space? One possibility is to define a plane, or a “slice” through this space and calculate the resulting set on this plane. This is the approach I have taken, allowing a plane of arbitrary angle and position to be requested. See the “Multibrot Slice” formula option in my fractal generator. The image at the left is rendered with the real value on the horizontal axis, the imaginary value fixed at zero, and the exponent on the vertical axis.
Please see the Math and Fractal Everywhere pages for more details.