2.5.11.14 Fractal Patterns

Fractal patterns supported in POV-Ray:

• The Mandelbrot set with exponents up to 33.(The formula for these is: z(n+1) = z(n)^p + c, where p is the correspondent exponent.)
• The equivalent Julia sets.
• The magnet1 and magnet2 fractals (which are derived from some magnetic renormalization transformations; see the fractint help for more details).
  Both 'Mandelbrot' and 'Julia' versions of them are supported.

For the Mandelbrot and Julia sets, higher exponents will be slower for two reasons:

1. For the exponents 2,3 and 4 an optimized algorithm is used. Higher exponents use a generic algorithm for raising a complex number to an integer exponent, and this is a bit slower than an optimized version for a certain exponent.
2. The higher the exponent, the slower it will be. This is because the amount of operations needed to raise a complex number to an integer exponent is directly proportional to the exponent. This means that exponent 10 will be (very) roughly twice as slow as exponent 5.

Syntax:

MANDELBROT:
  mandel ITERATIONS [, BUMP_SIZE]
  [exponent EXPONENT]
  [exterior EXTERIOR_TYPE, FACTOR]
  [interior INTERIOR_TYPE, FACTOR]
  
JULIA:
  julia COMPLEX, ITERATIONS [, BUMP_SIZE]
  [exponent EXPONENT]
  [exterior EXTERIOR_TYPE, FACTOR]
  [interior INTERIOR_TYPE, FACTOR]

MAGNET MANDEL:
  magnet MAGNET_TYPE mandel ITERATIONS [, BUMP_SIZE]
  [exterior EXTERIOR_TYPE, FACTOR]
  [interior INTERIOR_TYPE, FACTOR]

MAGNET JULIA:
  magnet MAGNET_TYPE julia COMPLEX, ITERATIONS [, BUMP_SIZE]
  [exterior EXTERIOR_TYPE, FACTOR]
  [interior INTERIOR_TYPE, FACTOR]

Where:

ITERATIONS is the number of times to iterate the algorithm.

COMPLEX is a 2D vector denoting a complex number.

MAGNET_TYPE is either 1 or 2.

exponent is an integer between 2 and 33. If not given, the default is 2.

interior and exterior specify special coloring algorithms. You can specify one of them or both at the same time. They only work with the fractal patterns.
EXTERIOR_TYPE and INTERIOR_TYPE are integer values between 0 and 6 (inclusive). When not specified, the default value of INTERIOR_TYPE is 0 and for EXTERIOR_TYPE 1.
FACTOR is a float. The return value of the pattern is multiplied by FACTOR before returning it. This can be used to scale the value range of the pattern when using interior and exterior coloring (this is often needed to get the desired effect). The default value of FACTOR is 1.

The different values of EXTERIOR_TYPE and INTERIOR_TYPE have the following meaning:

• 0 : Returns just 1
• 1 : For exterior: The number of iterations until bailout divided by ITERATIONS.
      Note: this is not scaled by FACTOR (since it is internally scaled by 1/ITERATIONS instead).
      For interior: The absolute value of the smallest point in the orbit of the calculated point
• 2 : Real part of the last point in the orbit
• 3 : Imaginary part of the last point in the orbit
• 4 : Squared real part of the last point in the orbit
• 5 : Squared imaginary part of the last point in the orbit
• 6 : Absolute value of the last point in the orbit

Example:

  box {
    <-2, -2, 0>, <2, 2, 0.1>
    pigment {
      julia <0.353, 0.288>, 30
      interior 1, 1
      color_map { 
	    [0 rgb 0]
        [0.2 rgb x]
        [0.4 rgb x+y]
        [1 rgb 1]
        [1 rgb 0]
      }
    }
  }