Definition:Mandelbrot Set

Definition

Definition 1

The Mandelbrot set $M$ is the subset of the complex plane defined as follows:

Let $c \in \C$ be a complex number.

Let $T_c: \C \to \C$ be the complex function defined as:

$\forall z \in \C: \map {T_c} z = z^2 + c$

Then $c \in M$ if and only if the sequence:

$\tuple {0, \map {T_c} 0, \map { {T_c}^2} 0, \ldots}$

is bounded.

Definition 2

The Mandelbrot set $M$ is the subset of the complex plane defined as follows:

Let $c \in \C$ be a complex number.

Let $T_c: \C \to \C$ be the complex function defined as:

$\forall z \in \C: \map {T_c} z = z^2 + c$

Then $M$ is the set of points $c \in \C$ for which the Julia set of $T_c$ is connected in the extended complex plane $\overline \C$.

Graphics

Complete Set

The following is a depiction of the complete Mandelbrot set.

The Mandelbrot set itself is white.

The region of the complex plane depicted is the rectangle $\closedint {-2.2} {2.2} \times \closedint {-1.65 i} {1.65 i}$.

Also see

• Definition:Multibrot Set

• Results about the Mandelbrot set can be found here.

Source of Name

This entry was named for Benoît B. Mandelbrot.

Historical Note

The Mandelbrot set was discovered by Benoît B. Mandelbrot in $1980$.

Online Fractal Generator

Fractals presented on this page were generated using this online fractal generator.