Find Equation of Main Cardioid in Mandelbrot Set & Minibrot

[pierce15]

Assuming that we could interpret the imaginary axis in the complex plane as the output of a relation, how would we find the equation of the curve that bounds the main cardioid of the M-set? Is there a way to find the equation of the main cardioid on a "minibrot" (e.g. if I zoom in on the fractal very deeply and find another quasi-similar M-set)?

 

[kreil]

The Mandelbrot set is obtained from the recursion relation,

[tex]
z_{n+1} = z_n^2 +C
[/tex]

The kidney bean-shaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations¹

[tex]
4x = 2 \cos(t) - \cos(2t)
[/tex]
[tex]
4x = 2 \sin(t) - \sin(2t)
[/tex]


¹ http://mathworld.wolfram.com/MandelbrotSet.html

 

[pierce15]

The Mandelbrot set is obtained from the recursion relation,

[tex]
z_{n+1} = z_n^2 +C
[/tex]

The kidney bean-shaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations¹

[tex]
4x = 2 \cos(t) - \cos(2t)
[/tex]
[tex]
4x = 2 \sin(t) - \sin(2t)
[/tex]


¹ http://mathworld.wolfram.com/MandelbrotSet.html

How would one show/prove that?

 

[kreil]

http://cosinekitty.com/mandel_orbits_analysis.html

 

[alphy]

The main cardioid in the Mandelbrot set is a unique and complex mathematical structure that represents the boundary between the areas of the set that diverge and those that converge. It is characterized by a cardioid shape, with a cusp at the origin, and is surrounded by intricate patterns of smaller cardioids and satellite structures.

To find the equation of the main cardioid in the Mandelbrot set, we need to understand the underlying mathematical principles that govern its formation. The Mandelbrot set is generated by an iterative function called the Mandelbrot function, which takes a complex number, z, and repeatedly applies the function z^2 + c, where c is a constant. The sequence of values generated by this function is then plotted on the complex plane, with the color of each point representing the number of iterations it takes for the sequence to diverge to infinity.

To find the equation of the main cardioid, we need to identify the points on the complex plane that correspond to the boundary between the divergent and convergent regions. This boundary is defined by the points where the sequence remains bounded, meaning that it does not diverge to infinity. This can be expressed mathematically as the condition |z| ≤ 2, where z is a complex number representing a point on the complex plane.

Therefore, the equation of the main cardioid in the Mandelbrot set can be written as z^2 + c ≤ 2, where c is a constant. This equation represents the boundary between the divergent and convergent regions, and it is this boundary that forms the intricate shape of the main cardioid.

Similarly, for a "minibrot" in the Mandelbrot set, we can use the same approach to find the equation of the main cardioid. By zooming in on the fractal and identifying the points where the sequence remains bounded, we can determine the equation of the main cardioid for that specific region.

In conclusion, the Mandelbrot set and its main cardioid are fascinating mathematical structures that can be described and understood through the use of equations and iterative functions. By understanding the underlying principles that govern their formation, we can find the equations that represent these complex and beautiful fractals.