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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Mandelbrot Sets"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "----------------"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Imports"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pyplot as plt\n",
      "import numpy as np\n",
      "from pylab import cm\n",
      "\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Settings"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Establish the area of the plot to look at. The \n",
      "# Mandelbrot set is a fractal defined to exist in\n",
      "# the set of imaginary numbers.\n",
      "#\n",
      "# You can see the whole set when you \"zoom out\" so that:\n",
      "#\n",
      "#     real axes go from  -2 to 1\n",
      "#     imaginary axes from -1 to 1\n",
      "#\n",
      "# For this notebook the view is defined by two points. The \n",
      "# \"upper left\" and the \"lower right\". To see the whole set \n",
      "# you would use an upper of -2 + 1j and a lower of -1 + 1j.\n",
      "\n",
      "# The view is defined by two complex numbers.\n",
      "g_view = {\"upper\": -2 + 1j, \"lower\": 1 - 1j}\n",
      "#g_view = {\"upper\": 0+0j, \"lower\": 1 - 1j}\n",
      "\n",
      "# The resolution of the image in pixels\n",
      "# The larger the image, the more time it will take to calculate\n",
      "g_resolution = 320\n",
      "\n",
      "# The maximum iterations. The more iterations, the longer it will\n",
      "# take to calculate the mandelbrot set.\n",
      "g_max_iterations = 128"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Functions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def complex_range(upper, lower, resolution):\n",
      "    \"\"\"Yields an iterable range of pixel points and complex numbers\"\"\"\n",
      "    step_y = (upper.imag - lower.imag) / (resolution - 1)\n",
      "    step_x = (lower.real - upper.real) / (resolution - 1)\n",
      "    for x in range(resolution):\n",
      "        real = upper.real + (x * step_x)\n",
      "        for y in range(resolution):\n",
      "            yield x, y, complex(real, lower.imag + (y * step_y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def test_point(point, max_iterations):\n",
      "    \"\"\"Test a point for inclusion in Mandelbrot set.\n",
      "    1.0 means it is in the set. Anything else is out.\"\"\"\n",
      "    zn = complex(0,0)\n",
      "    c = point\n",
      "    for n in range(1, max_iterations + 1):\n",
      "        zn = zn * zn + c\n",
      "        if abs(zn) >= 2:\n",
      "            return 1 - (float(n) / max_iterations)\n",
      "    return 1.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def aspect_ratio(upper, lower):\n",
      "    return (upper.imag - lower.imag) / (lower.real - upper.real)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Generate an Image"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Calculate the mandelbrot\n",
      "i = np.array([[0]*g_resolution]*g_resolution, dtype=float)\n",
      "for x, y, c in complex_range(g_view[\"upper\"], g_view[\"lower\"], g_resolution):\n",
      "    i[y][x] = test_point(c, g_max_iterations)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Show the mandelbrot\n",
      "plt.figure(figsize=(10,10))\n",
      "#plt.axis([g_view[\"upper\"].real, g_view[\"lower\"].real, g_view[\"lower\"].imag, g_view[\"upper\"].imag])\n",
      "img = plt.imshow(i, aspect=aspect_ratio(g_view[\"upper\"], g_view[\"lower\"])) #, cmap=cm.gray)\n",
      "\n",
      "# Try different colormaps\n",
      "#img.set_cmap('Greys')\n",
      "img.set_cmap('RdBu')\n",
      "#img.set_cmap('gray')\n",
      "#img.set_cmap('binary')\n",
      "#img.set_cmap('jet')\n",
      "#img.set_cmap('gist_rainbow')\n",
      "#img.set_cmap('hot')\n",
      "#img.set_cmap('afmhot')\n",
      "#img.set_cmap('RdGy')\n",
      "#img.set_cmap('RdYlGn')\n",
      "#img.set_cmap('RdYlBu')\n",
      "#img.set_cmap('Dark2')\n",
      "#img.set_cmap('hsv')\n",
      "\n",
      "#plt.colorbar()\n",
      "#img.write_png(\"mandelbrot.png\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "-------------------------"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Resources"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* [Benoit Mandelbrot](http://en.wikipedia.org/wiki/Benoit_Mandelbrot)\n",
      "* [Mandelbrot Set](http://en.wikipedia.org/wiki/Mandelbrot_set)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "--------------------"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Copyright Notice"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<html>\n",
      "<p xmlns:dct=\"http://purl.org/dc/terms/\" xmlns:vcard=\"http://www.w3.org/2001/vcard-rdf/3.0#\">\n",
      "  <a rel=\"license\"\n",
      "     href=\"http://creativecommons.org/publicdomain/zero/1.0/\">\n",
      "    <img src=\"CC0-88x31.png\" style=\"border-style: none;\" alt=\"CC0\" />\n",
      "  </a>\n",
      "  <br />\n",
      "  To the extent possible under law,\n",
      "  <a rel=\"dct:publisher\"\n",
      "     href=\"http://charles.stanho.pe\">\n",
      "    <span property=\"dct:title\">Charles Stanhope</span></a>\n",
      "  has waived all copyright and related or neighboring rights to\n",
      "  <i><span property=\"dct:title\">Mandelbrot Sets</span></i>.\n",
      "This work is published from:\n",
      "<span property=\"vcard:Country\" datatype=\"dct:ISO3166\"\n",
      "      content=\"US\" about=\"http://charles.stanho.pe\">\n",
      "  United States</span>.\n",
      "</p>\n",
      "</html>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
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