1 files changed, 72 insertions, 0 deletions
diff --git a/mandelbrot_explanation.html b/mandelbrot_explanation.html
new file mode 100644
index 0000000..3a59639
--- /dev/null
+++ b/mandelbrot_explanation.html
@@ -0,0 +1,72 @@
+<html>
+<head>
+<script src="js/latexit.js"></script>
+<link rel="stylesheet" href="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 $i=\sqrt{-1}
+</div>
+<br>
+Now let's check if 0.5 is in the Mandelbrot Set. To do so, start at 0
+<div lang="latex">
+\\
+f_{0.5}(0) = 0^2 + 0.5 = 0.5\\
+f_{0.5}(0.5) = 0.5^2 + 0.5 = 0.75\\
+f_{0.5}(0.75) = 1.0625\\
+f_{0.5}(1.0625) = 1.62890625\\
+f_{0.5}(1.62890625) = 3.15333557\\
+</div>
+<br>
+It can be proven that if the function passes 2, it will go to infinity if you continually apply the function.
+Since this function has passed 2, 0.5 is not in the Mandelbrot Set. Compare this to 0.25.
+
+<div lang="latex">
+\\
+f_{0.25}(0) = 0^2+0.25 = 0.25\\
+f_{0.25}(0.25) = 0.3125\\
+f_{0.25}(0.3125) = 0.34765625\\
+f_{0.25}(0.34765625) = 0.370864868\\
+f_{0.25}(0.370864868) = 0.38754075\\
+f_{0.25}(0.38754075) = 0.400187833\\
+f_{0.25}(0.400187833) = 0.410150302\\
+f_{0.25}(0.410150) = 0.418223\\
+f_{0.25}(0.418223) = 0.424911\\
+f_{0.25}(0.424911) = 0.430549\\
+f_{0.25}(0.430549) = 0.435373\\
+f_{0.25}(0.435373) = 0.439549\\
+f_{0.25}(0.439549) = 0.443204\\
+f_{0.25}(0.443204) = 0.446429\\
+f_{0.25}(0.446429) = 0.449299\\
+</div>
+<br>
+This will never pass 2, so 0.25 is in the Mandelbrot Set.
+<br>
+This process can also be done to complex numbers.<br>
+<br>
+<div lang="latex">
+M(x) =$ the number of iterations required for $f_x$ to pass 2.$ 
+</div>
+
+The website is just a 2d plot of M(x).
+
+</body>
+
+</html>