page_title: Mandelbulb
section_title: Power nine formula
hierarchical_section_title: Mandelbulb / Power nine formula
caption_attribution_description: English: fractal based on z-&gt;z^9 formula
context_page_description: The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.
  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
  ,
  , and
  .
  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 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.
context_section_description: This fractal has cross-sections of the power 9 Mandelbrot fractal. It has 32 small bulbs sprouting from the main sphere. It is defined by, for example:
  These formula can be written in a shorter way:
  and equivalently for the other coordinates.
caption: Fractal with z^9 Mandelbrot cross sections
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