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<!DOCTYPE html>
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<title>Maxime Munari</title>
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<h1>Mandelbrot Set</h1>
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<h2>Definition</h2>
<p>
The Mandelbrot set, denoted by \(\mathcal{M}\), consists of a set of complex numbers \(c\) for which the
critical point \(z = 0\) of the polynomial \(P(z) = z^2 + c\) does not have an orbit that escapes to infinity.
In other words, the Mandelbrot set \(\mathcal{M}\) includes all points \(c \in \mathbb{C}\) where the following
sequence remains bounded:
</p>
<p>\[ \begin{cases} z_{n+1} = z_n^2 + c \\ z_0 = 0 \end{cases}, \hspace{0.2cm} z_n \in \mathbb{C} \]</p>
<p>
In practice, it can be shown that if \(|z_n| > 2\) for any \(n\), then the sequence \(\{z_n\}\) diverges to
infinity.
</p>
<p>
The Mandelbrot set is a fractal and exhibits an intricate and infinitely complex boundary when visualized. The
set itself is connected and contains numerous interesting features, such as self-similar structures and
minibrots (small copies of the Mandelbrot set).
</p>
<p>To summarize, the Mandelbrot set \(\mathcal{M}\) can be described as follows:</p>
<p>\[ \mathcal{M} := \{ c \in \mathbb{C} \mid \text{the sequence } \{z_n\} \text{ converges} \} \]</p>
<h2>Properties</h2>
<p>
The Mandelbrot set has several fascinating properties. It is a connected set, meaning there are no isolated
points. Each point in the set is part of the same continuous shape. Additionally, the boundary of the Mandelbrot
set is fractal and exhibits infinite complexity. No matter how much you zoom into the boundary, you will always
find more intricate details and self-similar patterns.
</p>
<p>
Another property is that the Mandelbrot set is symmetric with respect to the real axis. This means that if a
complex number \(c\) is in the Mandelbrot set, then its complex conjugate \(\overline{c}\) is also in the set.
</p>
<h2>Visualization</h2>
<p>
Visualizing the Mandelbrot set involves iterating the polynomial \(P(z) = z^2 + c\) for each point \(c\) in the
complex plane and determining whether the sequence remains bounded. Points that belong to the Mandelbrot set are
typically colored black, while points that do not belong to the set are colored based on how quickly the
sequence diverges to infinity.
</p>
<p>
The resulting images reveal the intricate and beautiful structure of the Mandelbrot set, with a large cardioid
shape in the center and various bulbs and spirals emanating from it. These visualizations often use color
gradients to highlight the speed of divergence, creating stunning and colorful fractal patterns.
</p>
<h2>Mathematical Significance</h2>
<p>
The Mandelbrot set is significant in the field of complex dynamics and fractal geometry. It serves as a
graphical representation of the behavior of quadratic polynomials and helps illustrate the concept of chaos in
dynamical systems. The study of the Mandelbrot set has led to many discoveries and advancements in mathematics,
particularly in understanding fractals and complex numbers.
</p>
<p>
The boundary of the Mandelbrot set is also of interest because it contains points that exhibit bifurcation,
where small changes in the parameter \(c\) lead to drastic changes in the behavior of the sequence. This makes
the Mandelbrot set a rich subject for research in mathematical bifurcation theory.
</p>
<h2>References</h2>
<ul>
<li style="list-style-type: none">
[1] "The Fractal Geometry of the Mandelbrot Set." Introduction to the Mandelbrot Set. Accessed June 15, 2024.
<a href="https://math.bu.edu/DYSYS/mandelbrot/MandelbrotSet.html" target="_blank"
>https://math.bu.edu/DYSYS/mandelbrot/MandelbrotSet.html</a
>
</li>
<li style="list-style-type: none">
[2] Wikipedia contributors. "Mandelbrot set." Wikipedia, The Free Encyclopedia. Accessed June 15, 2024.
<a href="https://en.wikipedia.org/wiki/Mandelbrot_set" target="_blank"
>https://en.wikipedia.org/wiki/Mandelbrot_set</a
>
</li>
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