MandelbrotSetIterationCount

MandelbrotSetIterationCount[c]

returns the number of iterations of the function , beginning with , that are needed to determine whether c is in the Mandelbrot set.

Details and Options

  • The Mandelbrot set is the set of all complex numbers c for which the sequence does not diverge to infinity when starting with .
  • With the option MaxIterations->n, the sequence will be iterated at most n times to determine if the sequence diverges.
  • The default setting is MaxIterations->1000.
  • If the maximum number of iterations is reached, z is assumed to be in the Mandelbrot set.
  • For some points that the algorithm knows in advance to be inside the Mandelbrot set, MandelbrotSetIterationCount will return one greater than the value of MaxIterations.

Examples

open allclose all

Basic Examples  (4)

Find the number of iterations needed to determine that is not in the Mandelbrot set:

In[1]:=
Click for copyable input
Out[1]=

Zero is known to be inside the Mandelbrot set and therefore returns :

In[1]:=
Click for copyable input
Out[1]=

It takes a few additional iterations to determine that 0.250001 is not in the Mandelbrot set:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=

MandelbrotSetIterationCount works on all kinds of numbers:

In[1]:=
Click for copyable input
Out[1]=

Options  (2)

Properties & Relations  (1)

Possible Issues  (1)

Neat Examples  (3)

See Also

MandelbrotSetPlot  MandelbrotSetMemberQ  MandelbrotSetDistance  JuliaSetIterationCount

Introduced in 2014
(10.0)