Understanding the mandelbrot set equation(s)

In summary, the conversation discussed the Mandelbrot set and its calculation process. The equation z = z^{2} + c is used to generate new values of z by squaring and adding a chosen constant value c. The resulting set is usually colored based on how quickly the iteration goes to infinity. The Julia fractal is a subset of the Mandelbrot set, where the value of c determines the structure. For some values of c, the Julia fractal is disconnected while for others it is connected. The Mandelbrot set is the set of all values of c for which the Julia fractal is connected. The algorithm for generating the Mandelbrot set is similar to that of the Julia fractal, except
  • #1
ChrisBaker8
24
0
I'm not sure if this is the correct section to be posting in. I'm writing a summary of the mandelbrot set and I'm not sure I understand how the points are calculated.

I've got the equation:

z = z[tex]^{2}[/tex] + c

This means each value is squared, and then a constant value c is added, to get a new value of z, which then goes through the same process. What does this infinite series represent? Is c an actual defined constant, like pi or G, or is it just a value that can be chosen?
 
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  • #2
C is a chosen parameter.

The set is usually colored related to how "quickly" the iteration goes to infinity.
 
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  • #3
If you fix c then you don't get the Madelbrot fractal. What you get is the Julia fractal. This then depends on the value of c you've chosen. Now for some values of c the Julia fractal will be totally disconnected. In tat case, you'll the fractal looks like consisting of islands but if you zoom into these islands, you see that each island consits of smaller islands which in turn consit of even smaller islands and ultimately everything consists of isolated points.

For some other values of c, the Julia fractal is a connected structure. Now, you can prove that a Julia fractal is connected if and if only if the point z=0 belongs to the Julia set. What Madelbrot did was to look at the set of all the values of c for which the Julia fractals are connected.

So, given some C you look if the series:

z_1 = 0^2 + c = c

z_2 = z_1^2 + c

z_3 = z_2^2 + c

etc.

goes to infinity or not. The set of all these values for c is the Mandelbrot set, which is itself a fractal. The algorithm is thus similar to that for the Julia fractal, except that you take c to be z_1.
 
  • #4
By the way, while Benoit Mandelbrot worked for IBM and used the latest computers in his work, Gaston Julia wrote about what are now called "Julia sets" in 1918 and did all computation by hand!
 

Related to Understanding the mandelbrot set equation(s)

Question 1: What is the Mandelbrot set equation?

The Mandelbrot set equation is a mathematical formula that is used to generate the fractal known as the Mandelbrot set. It is written as z = z^2 + c, where z and c are complex numbers. The equation is iterated multiple times for each point in the complex plane to determine whether it is a member of the Mandelbrot set.

Question 2: How is the Mandelbrot set equation related to fractals?

The Mandelbrot set equation is related to fractals because it is a self-similar equation. This means that when the equation is iterated, the resulting patterns are similar to the original pattern. This property is what creates the intricate and detailed structures in the Mandelbrot set, which are characteristic of fractals.

Question 3: What is the significance of the Mandelbrot set equation?

The Mandelbrot set equation has significant implications in mathematics, as it is an example of a complex dynamical system. It also has applications in various fields such as chaos theory, computer graphics, and data compression. The Mandelbrot set has also been used to study the concept of infinity and the boundary between order and chaos.

Question 4: How is the Mandelbrot set equation calculated?

The Mandelbrot set equation is calculated by iterating the equation z = z^2 + c for each point in the complex plane. The value of c is set to the coordinates of the point, and the initial value of z is set to 0. If the resulting value of z remains within a certain threshold after a certain number of iterations, the point is considered to be a member of the Mandelbrot set.

Question 5: Are there variations of the Mandelbrot set equation?

Yes, there are variations of the Mandelbrot set equation that use different values for the exponent and the constant term. These variations can result in different sets, such as the Multibrot set, which has a cubic exponent, or the Burning Ship fractal, which has a different form of the equation. These variations can produce unique and visually striking fractal patterns.

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