From be06edbcfd1213ea1658d36f8934493a92205aac Mon Sep 17 00:00:00 2001 From: pommicket Date: Fri, 1 Jul 2016 17:43:58 -0400 Subject: Created pommicket.github.io --- mandelbrot_explanation.html | 72 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 72 insertions(+) create mode 100644 mandelbrot_explanation.html (limited to 'mandelbrot_explanation.html') 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 @@ + + + + + + +Mandelbrot Set Explanation + + + + +

Explanation of the Mandelbrot Set

+ + +
+ +Consider the function +
+f_c(z) = z^2+c\\ +

+Where z and c are complex numbers. Complex numbers are numbers in the form of +
+\\ +ai+b\\ +$Where $i=\sqrt{-1} +
+
+Now let's check if 0.5 is in the Mandelbrot Set. To do so, start at 0 +
+\\ +f_{0.5}(0) = 0^2 + 0.5 = 0.5\\ +f_{0.5}(0.5) = 0.5^2 + 0.5 = 0.75\\ +f_{0.5}(0.75) = 1.0625\\ +f_{0.5}(1.0625) = 1.62890625\\ +f_{0.5}(1.62890625) = 3.15333557\\ +
+
+It can be proven that if the function passes 2, it will go to infinity if you continually apply the function. +Since this function has passed 2, 0.5 is not in the Mandelbrot Set. Compare this to 0.25. + +
+\\ +f_{0.25}(0) = 0^2+0.25 = 0.25\\ +f_{0.25}(0.25) = 0.3125\\ +f_{0.25}(0.3125) = 0.34765625\\ +f_{0.25}(0.34765625) = 0.370864868\\ +f_{0.25}(0.370864868) = 0.38754075\\ +f_{0.25}(0.38754075) = 0.400187833\\ +f_{0.25}(0.400187833) = 0.410150302\\ +f_{0.25}(0.410150) = 0.418223\\ +f_{0.25}(0.418223) = 0.424911\\ +f_{0.25}(0.424911) = 0.430549\\ +f_{0.25}(0.430549) = 0.435373\\ +f_{0.25}(0.435373) = 0.439549\\ +f_{0.25}(0.439549) = 0.443204\\ +f_{0.25}(0.443204) = 0.446429\\ +f_{0.25}(0.446429) = 0.449299\\ +
+
+This will never pass 2, so 0.25 is in the Mandelbrot Set. +
+This process can also be done to complex numbers.
+
+
+M(x) =$ the number of iterations required for $f_x$ to pass 2.$ +
+ +The website is just a 2d plot of M(x). + + + + -- cgit v1.2.3