Mandelbulb
The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers. White and Nylander's formula for the "nth power" of the vector in ℝ3 is where The Mandelbulb is then defined as the set of those in ℝ3 for which the orbit of under the iteration is bounded. For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" depending on n. Many of their graphic renderings use n = 8. However, the equations can be simplified into rational polynomials when n is odd. For example, in the case n = 3, the third power can be simplified into the more elegant form: The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (p, q) given by Since p and q do not necessarily have to equal n for the identity |vn| = |v|n to hold, more general fractals can be found by setting for functions f and g. (Wikipedia).
