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. For a family of polynomial maps, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected.
"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. Starting instead with 
gives the same boundedness criterion, since 
is the first iterate of 0. 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 exceed 
. The kidney bean-shaped portion of the Mandelbrot
set turns out to be bordered by a cardioid with equations
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(2)
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(3)
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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, the conjecture that the Mandelbrot set is locally connected remains open (Dudko 2025). If it is locally connected, 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
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(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,
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(5)
|
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(6)
|
(OEIS A054670 and A054671; Ewing and Schober 1992). The recursion for 
is given by
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(7)
|
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(8)
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(9)
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where 
.
These coefficients can be computed recursively, but a closed form is not known. Furthermore,
the sum converges very slowly. An empirical extrapolation by Munafo estimates approximately
terms for two decimal places
and 
terms for three decimal places. These estimates are not rigorous convergence bounds.
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 value
lies between 
and 
.
The area of the set obtained by pixel counting is 
(OEIS A098403;
Munafo; Lesmoir-Gordon 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 computer-generated plots can be then be created by
coloring nonmember points depending on how quickly they diverge to 
. For an escaping point, an integer
called the count can be defined as the first iteration
such that 
. With 
, exceeding this threshold certifies escape for the
critical orbit. Points with different counts are colored
differently. 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 equation,
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(10)
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The first few lemniscates are therefore given by
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(11)
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(12)
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(13)
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(14)
|
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(15)
|
 |  | ![C+[C+(C+C^2)^2]^2](/images/equations/MandelbrotSet/Inline71.svg) |
(16)
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(17)
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(OEIS A114448).
When writing 
and taking the absolute square of each side, the
lemniscates can be plotted in the complex plane,
and the first few are given by
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(18)
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 |  | ![(x^2+y^2)[(x+1)^2+y^2]](/images/equations/MandelbrotSet/Inline81.svg) |
(19)
|
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(20)
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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
![[(x^'-1/2)^2+y^2][(x^'+1/2)^2+y^2]=r^2,](/images/equations/MandelbrotSet/NumberedEquation4.svg) |
(21)
|
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.
A connection with pi occurs near the cusp 
of the main cardioid.
For a real number 
, let 
be the first positive
integer 
for which 
when 
and 
.
Then
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(22)
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For example, 
gives 
(Klebanoff 2001, Numberphile 2015). This is a limit statement.
It does not identify the finite iteration counts with decimal prefixes of pi
at every scale.
For 
,
where 
,
2, ..., the iteration counts begin 2, 3, 4, 5, 6, 6, ... (OEIS A332061).
This integer sequence uses the threshold 
, which agrees with 
for these parameters because equality
cannot occur. For 
,
where 
,
1, ..., the iteration counts begin 2, 3, 5, 7, 11, ... (OEIS A332062).
Both integer sequences start with 
and count applications of the quadratic
recurrence equation.
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 different-looking 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."
Taking the absolute values of the real and imaginary parts of 
before each squaring instead gives the nonanalytic burning
ship fractal.
in the Mandelbrot Set." Fractals 9, 393-402,
2001.