The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable.
"The" Mandelbrot set is the set obtained from the quadratic recurrence equation
(1)

with , where points in the complex plane for which the orbit of does not tend to infinity are in the set. Setting equal to any point in the set that is not a periodic point gives the same result. The Mandelbrot set was originally called a molecule by Mandelbrot. J. Hubbard and A. Douady proved that the Mandelbrot set is connected.
A plot of the Mandelbrot set is shown above in which values of in the complex plane are colored according to the number of steps required to reach . The kidney beanshaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations
(2)
 
(3)

The adjoining portion is a circle with center at and radius .
The region of the Mandelbrot set centered around is sometimes known as the sea horse valley because the spiral shapes appearing in it resemble sea horse tails (Giffin, Munafo).
Similarly, the portion of the Mandelbrot set centered around with size approximately is known as elephant valley.
Shishikura (1994) proved that the boundary of the Mandelbrot set is a fractal with Hausdorff dimension 2, refuting the conclusion of Elenbogen and Kaeding (1989) that it is not. However, it is not yet known if the Mandelbrot set is pathwiseconnected. If it is pathwiseconnected, then Hubbard and Douady's proof implies that the Mandelbrot set is the image of a circle and can be constructed from a disk by collapsing certain arcs in the interior (Douady 1986).
The area of the Mandelbrot set can be written exactly as
(4)

where are the coefficients of the Laurent series about infinity of the conformal map of the exterior of the unit disk onto the exterior of the Mandelbrot set,
(5)
 
(6)

(OEIS A054670 and A054671; Ewing and Schober 1992). The recursion for is given by
(7)
 
(8)
 
(9)
 
(10)

These coefficients can be computed recursively, but a closed form is not known. Furthermore, the sum converges very slowly, so terms are needed to get the first two digits, and terms are needed to get three digits. Ewing and Schober (1992) computed the first values of , found that in this range, and conjectured that this inequality always holds. This calculation also provided the limit and led the authors to believe that the true values lies between and .
The area of the set obtained by pixel counting is (OEIS A098403; Munafo; LesmoirGordon et al. 2000, p. 97) and by statistical sampling is with 95% confidence (Mitchell 2001), both of which are significantly smaller than the estimate of Ewing and Schober (1992).
To visualize the Mandelbrot set, the limit at which points are assumed to have escaped can be approximated by instead of infinity. Beautiful computergenerated plots can be then be created by coloring nonmember points depending on how quickly they diverge to . A common choice is to define an integer called the count to be the largest such that , where can be conveniently taken as , and to color points of different count different colors. The boundary between successive counts defines a series of "Mandelbrot set lemniscates" (or "equipotential curves"; Peitgen and Saupe 1988) defined by iterating the quadratic recurrence,
(11)

The first few lemniscates are therefore given by
(12)
 
(13)
 
(14)
 
(15)
 
(16)
 
(17)
 
(18)

(OEIS A114448).
When writing and taking the absolute square of each side, the lemniscates can plotted in the complex plane, and the first few are given by
(19)
 
(20)
 
(21)

These are a circle (black), an oval (red), and a pear curve (yellow). In fact, the second Mandelbrot set lemniscate can be written in terms of a new coordinate system with as
(22)

which is just a Cassini oval with and . The Mandelbrot set lemniscates grow increasingly convoluted with higher count, illustrated above, and approach the Mandelbrot set as the count tends to infinity.
The term Mandelbrot set can also be applied to generalizations of "the" Mandelbrot set in which the function is replaced by some other function. In the above plot, , , and is allowed to vary in the complex plane. Note that completely different sets (that are not Mandelbrot sets) can be obtained for choices of that do not lie in the fractal attractor. So, for example, in the above set, picking inside the unit disk but outside the red basins gives a set of completely differentlooking images.
Generalizations of the Mandelbrot set can be constructed by replacing with or , where is a positive integer and denotes the complex conjugate of . The above figures show the fractals obtained for , 3, and 4 (Dickau). The plots on the bottom have replaced with and are sometimes called "mandelbar sets."