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<html>
<head>
<script src="js/latexit.js"></script>
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<title>Mandelbrot Set Explanation</title>
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<body>
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<h2>Explanation of the Mandelbrot Set</h2>
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 $i=\sqrt{-1}
</div>
<br>
Now let's check if 0.5 is in the Mandelbrot Set. To do so, start at 0
<div lang="latex">
\\
f_{0.5}(0) = 0^2 + 0.5 = 0.5\\
f_{0.5}(0.5) = 0.5^2 + 0.5 = 0.75\\
f_{0.5}(0.75) = 1.0625\\
f_{0.5}(1.0625) = 1.62890625\\
f_{0.5}(1.62890625) = 3.15333557\\
</div>
<br>
Since this function has passed 2, 0.5 is not in the Mandelbrot Set. Compare this to 0.25.
<div lang="latex">
\\
f_{0.25}(0) = 0^2+0.25 = 0.25\\
f_{0.25}(0.25) = 0.3125\\
f_{0.25}(0.3125) = 0.34765625\\
f_{0.25}(0.34765625) = 0.370864868\\
f_{0.25}(0.370864868) = 0.38754075\\
f_{0.25}(0.38754075) = 0.400187833\\
f_{0.25}(0.400187833) = 0.410150302\\
f_{0.25}(0.410150) = 0.418223\\
f_{0.25}(0.418223) = 0.424911\\
f_{0.25}(0.424911) = 0.430549\\
f_{0.25}(0.430549) = 0.435373\\
f_{0.25}(0.435373) = 0.439549\\
f_{0.25}(0.439549) = 0.443204\\
f_{0.25}(0.443204) = 0.446429\\
f_{0.25}(0.446429) = 0.449299\\
</div>
<br>
This will never pass 2, so 0.25 is in the Mandelbrot Set.
<br>
This process can also be done to complex numbers.<br>
<br>
<div lang="latex">
M(x) =$ 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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