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<html>
<head>
<script src="js/latexit.js"></script>
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.6/css/bootstrap.min.css">
<link rel="stylesheet" href="css/style.css">
<title>Mandelbrot Set Explanation</title>
</head>
<body>
<h2>Explanation of the Mandelbrot Set</h2>
<div id="header_links_div"></div>
<script src="js/header_links.js"></script>
<hr>
Consider the function
<div lang="latex">
f_c(z) = z^2+c\\
</div><br>
Where z and c are complex numbers. Complex numbers are numbers in the form of
<div lang="latex">
\\
ai+b\\
$Where $i=\sqrt{-1}
</div>
<br>
Now let's check if 0.5 is in the Mandelbrot Set. To do so, start at 0
<div lang="latex">
\\
f_{0.5}(0) = 0^2 + 0.5 = 0.5\\
f_{0.5}(0.5) = 0.5^2 + 0.5 = 0.75\\
f_{0.5}(0.75) = 1.0625\\
f_{0.5}(1.0625) = 1.62890625\\
f_{0.5}(1.62890625) = 3.15333557\\
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<br>
It can be proven that if the function passes 2, it will go to infinity if you continually apply the function.
Since this function has passed 2, 0.5 is not in the Mandelbrot Set. Compare this to 0.25.
<div lang="latex">
\\
f_{0.25}(0) = 0^2+0.25 = 0.25\\
f_{0.25}(0.25) = 0.3125\\
f_{0.25}(0.3125) = 0.34765625\\
f_{0.25}(0.34765625) = 0.370864868\\
f_{0.25}(0.370864868) = 0.38754075\\
f_{0.25}(0.38754075) = 0.400187833\\
f_{0.25}(0.400187833) = 0.410150302\\
f_{0.25}(0.410150) = 0.418223\\
f_{0.25}(0.418223) = 0.424911\\
f_{0.25}(0.424911) = 0.430549\\
f_{0.25}(0.430549) = 0.435373\\
f_{0.25}(0.435373) = 0.439549\\
f_{0.25}(0.439549) = 0.443204\\
f_{0.25}(0.443204) = 0.446429\\
f_{0.25}(0.446429) = 0.449299\\
</div>
<br>
This will never pass 2, so 0.25 is in the Mandelbrot Set.
<br>
This process can also be done to complex numbers.<br>
<br>
<div lang="latex">
M(x) =$ the number of iterations required for $f_x$ to pass 2.$
</div>
The website is just a 2d plot of M(x).
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