The Mandelbrot set is the set of complex numbers
c
{\displaystyle c}
for which the function
f
c
(
z
)
=
z
2
+
c
{\displaystyle f_{c}(z)=z^{2}+c}
does not diverge when iterated from
z
=
0
{\displaystyle z=0}
, i.e., for which the sequence
f
c
(
0
)
{\displaystyle f_{c}(0)}
,
f
c
(
f
c
(
0
)
)
{\displaystyle f_{c}(f_{c}(0))}
, etc., remains bounded in absolute value.
Its definition and name are due to Adrien Douady, in tribute to the mathematician Benoit Mandelbrot. The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.
Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point
c
{\displaystyle c}
, whether the sequence
f
c
(
0
)
,
f
c
(
f
c
(
0
)
)
,
…
{\displaystyle f_{c}(0),f_{c}(f_{c}(0)),\dotsc }
goes to infinity (in practice -- whether it leaves some predetermined bounded neighbourhood of 0 after a predetermined number of iterations). Treating the real and imaginary parts of
c
{\displaystyle c}
as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence
|
f
c
(
0
)
|
,
|
f
c
(
f
c
(
0
)
)
|
,
…
{\displaystyle |f_{c}(0)|,|f_{c}(f_{c}(0))|,\dotsc }
crosses an arbitrarily chosen threshold, with a special color (usually black ) used for the values of
c
{\displaystyle c}
for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement). If
c
{\displaystyle c}
is held constant and the initial value of
z
{\displaystyle z}
—denoted by
z
0
{\displaystyle z_{0}}
—is variable instead, one obtains the corresponding Julia set for each point
c
{\displaystyle c}
in the parameter space of the simple function.
Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self -similarity applies to the entire set, and not just to its parts.
The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules.
