From 7c8ffd449ab7365cbe6e135f1a0a0de122d00e9e Mon Sep 17 00:00:00 2001 From: pommicket Date: Mon, 29 Aug 2016 23:07:21 -0400 Subject: Changed navbar --- mandelbrot_explanation.html | 123 ++++++++++++++++++++++---------------------- 1 file changed, 61 insertions(+), 62 deletions(-) (limited to 'mandelbrot_explanation.html') diff --git a/mandelbrot_explanation.html b/mandelbrot_explanation.html index 961076c..36f7032 100644 --- a/mandelbrot_explanation.html +++ b/mandelbrot_explanation.html @@ -1,72 +1,71 @@ - - - - + + + + + + Mandelbrot Set Explanation + -Mandelbrot Set Explanation - + - + + -

Explanation of the Mandelbrot Set

- - -
+

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