What is the Mandlebrot Set?

Question

I've heard a little about this but I was wondering if anybody could explain it simply to me (maybe in a sentence or less). Thanks.

Answer 1

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Answer 2

The Mandelbrot set is an example of what is called an orbit diagram based on an equation like y = x^2 + c. Except the function is iterated. So it's more like z(n) = (z(n-1))^2 +c. The n and n-1 are supposed to be subscripts.

So, for example, look at a c value of 1 and start with z-sub-0 = 0.

z1 = z0^2 + 1 = 0^2 + 1 = 1
z2 = z1^2 + 1 = 1^2 + 1 = 2
z3 = 2^2 +1 = 5
z4 = 5^2 + 1 = 26
z5 = 26^2 + 1 . . .

Notice that the values keep increasing. The limit of this sequence is called the ORBIT of the iterated function. In this case it is infinity.

If c = 0, and z0 is 0,
z1 = z0^2 + c = 0^2 + 0 = 0
z2 = z1^2 + 0 = (0)^2 + 0 = 0
z3 = (0)^2 + 0 = 0
z4 = (0)^2 + 0 = 0

The orbit here is a fixed value of 0.

Now select c = -1. z0 is still 0.

z1 = z0^2 + -1 = 0^2 + -1 = -1
z2 = z1^2 + 1 = (-1)^2 + 1 = 0
z3 = 0^2 + -1 = -1
z4 = (-1)^2 + 1 = 0
z5 = 26^2 + 1 . . .

This function has an orbit that is a cycle with period 2.

This can also be done with complex values of c. c might be 1 + i. Or 3i. or -0.5 + 0.2i. For each number in the complex plane you calculate the orbit. Then you color the point according to the nature of its orbit. If the point has a finite orbit (including a cyclic orbit) you color it black. If the orbit is not finite you color it something else. In most pictures the color is determined by how fast the orbit gets large. The set of black points is the Mandelbrot set.

Answer 3

The Mandelbrot set is a fractal that has become popular far outside of mathematics both for its aesthetic appeal and its complicated structure, arising from a simple definition. This is largely due to the efforts of Benoît Mandelbrot and others, who worked hard to communicate this area of mathematics to the general public.

Mathematically, the Mandelbrot set M is defined as the connectedness locus of the family

f_c:\mathbb{C}\to\mathbb{C}; z\mapsto z^2 +c.

of complex quadratic polynomials. That is, the Mandelbrot set is the subset of the complex plane consisting of those parameters c for which the Julia set of fc is connected.

The Mandelbrot set can also be defined as the set of parameters c for which the set {0, | fc(0) | , | fc(fc(0)) | , | fc(fc(fc(0))) | ,...} has a finite upper bound. This definition lends itself immediately to the production of computer generated renderings

Answer 4

The MANDELBROT SET is the most complex object in mathematics, its admirers like to say. The Mandelbrot set is a fractal that has become popular far outside of mathematics both for its aesthetic appeal and its complicated structure, arising from a simple definition. This is largely due to the efforts of Benoît Mandelbrot and others, who worked hard to communicate this area of mathematics to the general public.

Mathematically, the Mandelbrot set M is defined as the connectedness locus of the family of complex quadratic polynomials. That is, the Mandelbrot set is the subset of the complex plane consisting of those parameters c for which the Julia set of fc is connected.

The Mandelbrot set can also be defined as the set of parameters c for which the set {0, | fc(0) | , | fc(fc(0)) | , | fc(fc(fc(0))) | ,...} has a finite upper bound. This definition lends itself immediately to the production of computer generated renderings, (see below).

The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The first pictures of it were drawn in 1978 by Brooks and Matelski as part of a study of Kleinian Groups.[1]

Mandelbrot studied the parameter space of quadratic polynomials in an article which appeared in 1980.[2] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,[3] who established many fundamental properties of M, and named the set in honor of Mandelbrot.

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. It would be futile to attempt to make a list of all the mathematicians who have contributed to our understanding of this set since then, but such a list would certainly include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz.

Visit http://en.wikipedia.org/wiki/Mandelbrot_set for a more detailed explanation with pictures and symbols