The Mandelbrot set is a particular mathematical set of points

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

, whose boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

 generates a distinctive and easily recognisable two-dimensional fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

 shape. The set is closely related to the Julia set

Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

 (which generates similarly complex shapes), and is named after the mathematician Benoît Mandelbrot

Benoît Mandelbrot

Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

, who studied and popularized it.

More technically, the Mandelbrot set is the set of values of c in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 for which the orbit

Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...

 of 0 under iteration

Iterated function

In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

 of the complex quadratic polynomial

Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...

 z_n+1 = z_n² + c remains bounded.