diff options
Diffstat (limited to 'mandelbrot_explanation.html')
| -rw-r--r-- | mandelbrot_explanation.html | 123 |
1 files changed, 61 insertions, 62 deletions
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 @@ <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"> + <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"> + <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script> + <title>Mandelbrot Set Explanation</title> + </head> -<title>Mandelbrot Set Explanation</title> -</head> + <body> -<body> + <div id="navbar"></div> + <script src="navbar.js"></script> -<h2>Explanation of the Mandelbrot Set</h2> -<div id="header_links_div"></div> -<script src="js/header_links.js"></script> -<hr> + <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> -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 + <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> + <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). + The website is just a 2d plot of M(x). -</body> + </body> </html> |