Classic Fractals.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"text-align: right\">Paul Novaes<br>June 2018</div> \n",
    "\n",
    "# Classic Fractals\n",
    "\n",
    "This notebook shows how to render a few classic fractals:\n",
    "1. Koch snowflake\n",
    "2. Sierpinski triangle\n",
    "3. Mandelbrot set\n",
    "4. Hilbert curve\n",
    "\n",
    "## Common code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following function allows to draw the fractals as images instead of plots in a graph:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def init_graphics():\n",
    "    plt.axes().set_aspect('equal')\n",
    "    plt.axis('off')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This utility function will be useful for the Koch snowflake and the Sierpinski triangle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "\n",
    "# Given a segment [P1, P2] returns a point P3 such that [P1, P2, P3] is an\n",
    "# equilateral triangle (counter-clockwise).\n",
    "def complete_triangle(P1, P2):\n",
    "    [x1, y1] = P1\n",
    "    [x2, y2] = P2\n",
    "    return [(x1 + x2) / 2 - (y2 - y1) * math.sqrt(3) / 2,\n",
    "            (y1 + y2) / 2 + (x2 - x1) * math.sqrt(3) / 2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Koch snowflake\n",
    "\n",
    "The Koch snowflake is constructed by starting with an equilateral triangle and replacing each of its sides by 4 connected segments \\_/\\\\\\_ of same length. This process is then iterated on each new segment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Draws the Koch curve that goes from point P1 to point P2.\n",
    "def koch_curve(P1, P2, depth):\n",
    "    [x1, y1] = P1\n",
    "    [x2, y2] = P2\n",
    "    if depth == 0:\n",
    "        plt.plot([x1, x2], [y1, y2], 'black')\n",
    "    else:\n",
    "        # From P1, P2 consider the 4 segments [P1,A], [A,B], [B,C] and\n",
    "        # [C,P2] and draw each corresponding Koch line recursively.\n",
    "        #\n",
    "        #            B\n",
    "        #           / \\\n",
    "        #    P1___A/   \\C__P2\n",
    "        #\n",
    "        A = [x1 + (x2 - x1) / 3, y1 + (y2 - y1) / 3]\n",
    "        C = [x1 + 2 * (x2 - x1) / 3, y1 + 2 * (y2 - y1) / 3]\n",
    "        B = complete_triangle(A, C)\n",
    "        koch_curve(P1, A, depth - 1)\n",
    "        koch_curve(A, B, depth - 1)\n",
    "        koch_curve(B, C, depth - 1)\n",
    "        koch_curve(C, P2, depth - 1)\n",
    "        \n",
    "def koch_snowflake(depth):\n",
    "    init_graphics()\n",
    "    A = [0, 0]\n",
    "    B = [1, 0]\n",
    "    C = complete_triangle(B, A)\n",
    "    koch_curve(A, B, depth)\n",
    "    koch_curve(B, C, depth)\n",
    "    koch_curve(C, A, depth)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Iterating 4 times, we get:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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6JfZ1ymrU+4ZecJvj966JeliGrG+pH//lqnPmx/9LYYY7os53Uw5DndtFufrFdx2kHBJZCh0thUQ+WXXO7HgON1TIUwrXbI7fODKLeitkuhl3dzN7FYWo7QAeRsp7HnrYm6nql7r7QxF62Igq8CX0tp6AblKzu2/volz7zexhylPAXkcvk3chBdwI3F51zm/N7DfI0u9BTpRhqI/twEbvpYtTufvRCG9cgH77MvRyAynzZuBfq85ZFHU8ErVS3kT3r1Tnr7r7si7K1WpmjwKnIwVchVo7Deg+DkCZMivPWRG/ZTR6xlYhJT4dDUO94u47uyJXicIGV8QMnF2oAoe4++GcRUok3oGZrUXN95M8o2VKC9unRU6eIagZ80DOsiQS7yBCIg+h7tydmX1vES1tOCmakXf1AHLzF3oJyETvIhyeG1FTeTt6Ridl0UQupNLCW3Gi4yk7L1YhF32vWaQqUVzCyn4QTfp/Dg0FbW77rNoopCMKwDUYvw71afehYZuByAucSORNaQx+GbDTM7SOhVTaaB6/hCztAeSxu9rdd+QqWCJR5mKUeeRZYJuZDQBOd/ddXf3iQjqi4q3VioZFBgBL3P2X+UqVSJRx93WU48xnoec0kyl+Re7TjqY8AD7Gc0yEnUgci2gRvoBWVjjP3Z9r55SaKKSlDeaiPuxzwAlmNjIyKlqEACYSPYqZnRgZNIeb2QzUEizlpD4rs+sU0dJWDPmMoxxgsQxFn6xGTqkFrtkkiUS3EzN6NlCenVRaznMyag0OJKMhn0Ja2ujTPo7iU9ehQO0z0LjYehQ+dmsodyLRE3wThZ+2IIPShPqyLSjU8hFkYLpMIS1tNZFLaQbKnrfCzB4A3ofyAncqwVciUStmNg1NeFjn7tMqQmz7AyO8E9k926KQlvYYXAt8NhT2u8A1yFP3hXC1JxLdya2olTfVzH4aTtHrgA9mrbDQS5TW3d9w9x9EcS7lfD/nA3+Sm2CJXo+ZXYISyB9EMQMnA7j7o+6eVSrdt1+zNzSPqzGzmcgh9UvgMuByd38qT5kSvY9InfsGcjjNQY7Rdd7Nqz70Ckt7DNaEs6oFBWynIaBEdzAR5aPaicJn13a3wkIvVFozuw/YZWbPo1W/v+fu38lZrEQvxN1fRd2vE1HCgy1mVnNy9M7SK5TWzAZX5ODZi8bJpiHnwF/kJliiL/Bd1BUbhrJEHgKIYJ/R3XHBXtGnNbOnUcTJJHffFulIrgSud/f78pUu0duJvm0zSnXU5O5HzKwFpRAam+UMHyiwpY1QsTlmNhspbCPwuRjimYkiUl7IUcRE32EXckiNBKaY2eUoEmo0cJ2ZjTWzqVldrJCWNiKd1qNm8H5UOetRrOcaFInyPnf/eW5CJvoUkcp3BUo2NwQlZziCrO0OlLliQl/PXPFzFETRjPIIbwbORak9Drn7lTmKl+iDmNmDSDkb0Vzao6gVOAd4yd0zmTRQZKXtj1ztKRtjom5J2RjfzkOoGdKC1lNJJOqKWDj7UBQzWxS8kJY2+rTrKPdpJwAndjVJdSKRFbGcyEZkGLej4aApfb1P24j6D4YcAC3Aqe6+L1fBEgmgYqbZlWgp1YascpgVMrEbgLsfNrNNaHLxAeRib8xXqkTiLU5EHuR1KOAnM+tYSEsbzeOFKED7ABqXvdLdH81TrkSihJlNQsOPu1DQxWDglL6eueIgMAktirQoKWyinnD3DWiJzoMobgD6euYKMxtBeZnLUSkbY6IeMbOFwKnAWe7+UhbfWUhLG5yCok2eAOab2ZgIazQzO/tYJ5jZ2WbWz8xmmtl4M5sfi1OPMLMTe1T6RF1jZhPNbHo8L2eZWYOZnRn/O+dY+cfM7FQzG1TxLM5AC8UBXJSZbEW0tFFhLciNvgsp7wqkyGtQvqjT3P21inPmoQWB16HB7iXx2Yqa2IeAye5+pOd+SaJeMbNFwDz0nDUBz6B1hpeh7Cj3u/sHKo4fjkJpS9kYd6JnalLsG4IWB+/TfdpHKCvhMOA0lMKyGVXWHWbWD95S8ttRWFkzqtz5KMB7bWzjgK/16A9J1CVm9gGUNqYlNgcuQIrYgrzBV5nZyIrTbkTKWnqeJiPjsRYp+kNk1Kft8lLy9bAhK+vAjCg/EOXPRvlDUX40ypOQhW2OcgNKBHcEGJj370lb7s9TSzwfs6L8i3h+Phjlv47yTVGeFuV1Ue6HVrY/iFbLy1a+vCsoo0qeA3w4/r4JWdQdaFB7aHxujxtxSxx3PYoHBfh1VPoa4Dd5/5605fos3RdGoBVYGPuGAp+Pvwcji7kPTX7/LvAUar05cHccdwFwWXfIWMg+bVuY2SMomdsS1MS5F7gBNaXnAU+5+6VV56xGTeW9qFlzirsv6km5E/ljZsNQF6sVNYVHodDD3RXHvBs1dZcAJ6Hm73S0PvJEYKW7L+hWOXub0gKY2RR3b6mYvvcYcAVyNLUc4/j+KCSyFU2p2g6c7O57elDsRI6E3+Nh9Jy8C+V88kqFrTi29Hz9H+B/Iis7G83r3uDdrFSFdES1R4VilprK33fxDoWN448AW1C/9iBS4IE9IWuirpiBXtwbUavrmC/tiufoh8Bh4KfufsDd13e3wkIvtbSVmNlgd29zdfgYAJ+IbtQs4D3u/uuekC9RP5jZFNTM3RnbINTiOu66sjGb50BPKGuJXmlpK2lPYYN9aHpff+D1pLB9k7CgpdDDmcDhthQ2ztnfkwoLfcDS1kIsU7gJTfMb6SkLRp/GzJYAJyCH5Ot5y1NNr7e0NXIKilh5DDjFzEaZ2UkREnl+zrIluhEzOzNCD8eb2VwzmwP8LP59WZ6yHY9kaQEzq1ygeghaT/Q01L+ZBZzt7i/mJmCiWzCzUSiQYjda43g7Gmsdi6Z8DkIZFNtsIvc0ydKKh9E47hqktKWQyBaUzuaOWHM00bu4GQVLlEIPp6A44zXoxf3zelNYSJb2HVT0Z+a6+/JIznUD8EV3vzFf6RJZEUnulwOr3X1mxULQ/YBhXsdTPZOlfSfvRjGmy83se8AHkPv/82Y2JF/REhlyO2pNzTCzu0JJrwSuqGeFhaS078Dd17r7PVGcEZ/NwJnAf89HqkSWmNn7gPPQUN8BNNUOd3/G3Z/OU7ZaSM3jdjCzie6+MZrJHwCu87TcSGGJ5TteR3mb5qHQw009PdbaFZKlbZ+t8bkd1VdTjrIkus5I5Hzai0IQtxZJYSEpbZuY2f3AbjN7EfgD4EZ3/17OYiW6gGsh6C+gGWCvA1vN7Hf5StUxktJWYWbDzGxuFLejsbppaBjgr3ITLJElt6BsEiNi2wlKexqpT+ua1KetwsyeBc4GpsX0q1Km+E+6+235SpfIiujbrkTT6qajoIqtyJCNrucmc7K0QGTPOyVC2M5A9fIFMxuIJjpvBX6Vp4yJzNkAvIwi4aYD1yGn1EjgBjObUtHiqiuSpQXMbD1qJu1F2QpKC1Ovjs9r3P2h/CRMdAdmNo3yQtAj0PBPK4qK242ehXH1FhVV2LV8MuZZlDNqF8pTuxmN4e0AliaF7Z24+9rIbjIOrXNcWnXxXJSJ4pl6U1hIlhZ4K91Myek0zN0P5ixSIkdiAslkFMq6Im95qkl9WvFLlDu5BU0eSPRRzOxBNAkelJmx7kiWFjCzlaj/shfN9DjD3RfmK1Wip4n1odaj/Nc7kVOqqd4S/PVqS2tml5jZdjN7TzuHzkHxp6chxX0olnlI9BEiG+ODyAl1FQq+mNSWwprZVDPbZGY9GpPeK5XWzAbEn3+MLOiXqva/DXdvRd7CAyi0bRjJSdcXGYfGa0tZOY/p26h4jj4OjAf+KBbqOubzlTW9rnkc3sArgDfQmOuzaEbHK2h17ifc/b1V5yxCN2wvSuh1qbs/0YNiJ+oAM5uIhvv2oJGDIcAJVcnKL0LJyt9ES1huR8/OEuS8WuLu53WnnL3C0lYtVbkV5SyeiiJePhSfk5F3eGucMy/SjYBu0Hi0ps8rSWH7Ju6+ES3xsQ9Ny9zn7rsrl7lEGU76oy7VdjREuAQtHTIi9mFmk2McuFsELfyGrOgRlM8H9CZ04KNRvjbKT0R5JGoGL/Hy+ix7UPN4QN6/J225P0/L43kprfV0a5SvjPLXo/yPUZ6CkuKvB/rFvlLye8tavsJaWjObEIv4ngwsQFbyjyJ59HyUJb4UevhbNJF9XjiYPo/elvPM7AwUazwUrbaX0qcmvh2fV8dyllejSKkvhMPqQvSCL8342oTWrx0HzIlJ9mORb+RjERJ5clbCFbJPGxW3AVXKPtQsWYf6o2vi8wp3/03FORcAv0OhiVPj+IkocVsD6ipMcve9PfQzEnVKPF8r0dzp9WiW16uoD1taVPpRd7+y4pxJ6NnbgQzAQdT6K80iGg2M9wwWlS6kh9TdPWbjXIcq5EGUieCcKL9RqbBxztPhpCqlynwZNZMnI4VdkxQ2AW89X4+hIcA9SGFXIn3ZjJTx1apzNkRgRmmlihWoyXw+6v8+m4XCQkEtLbzldt8ONAJDPa0KkKhDYhnVacB8d1+cxXcWtk+L+qtDUPPl0ZxlSSTeQczFbo3ivZl9bxEtbUWfYzhyCEwGTnP313IVLJEIwuG5ATWRt6E+7fQsmsiFVFoAM+uHOvkNyLm0G5jndRYnmuibmNkTwMXABcBCqHkFx3YpbPPYFXq4Hw3RtKKxVstVqESizAg0lrsPOa4OZfXFhbS00TxeguKKD6CO/vnu/myugiUSgZmNQbEBB1DzeASan9vl5nEhLa3rTbMVhR468GJS2EQ94e7bgCdRt20GsDsN+WhdnS2oSTwiDfkk6hEzexMFY2S2QHUhLW3wLtSP/RlwtplNjAWC+5vZVTnLluiDmNmFMXmlKbJ7ngj8MP59XWbXKaKlrQhjLGVQHIIWgD4JeZJnAue6+/M5iZjoY1T0YXejMMbSsiNjUVTVcDShpU/3aR9Ec2ZXIYt7MoonXoe8yj/pqUnJib5NGJEfoimhq9EzOR7N/lmN4pXv6fN92kpiIeh5aFWA5ogxvgb4tLv/KF/pEr2dmDe7GkXnTY3dO1Ba1swXqO4tSjsJZRh43My+D3wGxSWvR0NBaSJAolsIK/sCmjE2BXjA3a+LqXgD3P2VrK9ZyOZxNe6+wd0fj2ITCh1rQfNs/zw3wRJ9gfejBcf3o37sVAB3f6M7FBZ6iaWtxsxGuvtOM7sbuAG43t3vz1uuRO/CzGYgv8pmNKwzJKt+a1v0Ckt7DEohY6Wx20F5CZLo1TSi2PejKMgns1DFtuh1SmtmDwE7zOwV4KPA37n7nTmLleiFuPsy4JNokbZFwEYze6G7r9srlDaWqixly9uIvHZNKM3l13MTLNEX+ClaUX4IGovdBGBmM7trqcxe0ac1s9dRYEVTpP14CHgv8HF3vyNf6RK9HTNrAtaigJ+m2L0NGY/hMSMtMwpraSNU7GwzOxUpbD/gS2Y2DCXgaqZOF1BK9DrWA4+hgIqT0LKpo5D1/aSZTYusn5lQSEtbEcY4HM1XHIredHPicwZwkbs/lZuQiT6FmY1H0Xg70fNYmus9Gq17PIqUjdF+h4ZzdqF8xdvRONluYGFS2ERP4u6bzew+lPqogXI2xouQo+rpPh/GaGalJT76o1CxNDUvUXeY2QqktPPd/Y0svrOwfVrg15SzMaa1dxJ1h5n9AulYK5pCmgmFVNro005BlrYBOD/Ljn4i0VViOZFLUV92DTA29nX9uwvcPDbU4W9ElXIAmJWyMSbqgVgB41y0TtTryBWTScRUIS1tBV6xpUyMiXqipFutlJ/RTCikpQ0ruwxlrjiAZlac7e4v5ipYIhHE2sctKB55G2omz+rrmSs2oKUFW9HiRklhE3WDu+9ADtIdKG5gWxry0ZDPNvTiGZ6GfBL1iJktB2aT4ZBPIYMrgttRbqiNwG1mtgN1+ncDO9z9huoTzOweYAxyYL2MIqqakOKvdvff76pQZnYrmlt5GLUGtgLnoSCQne5+/THOuRslABuCllAcDEyPf6919491Va56JTKNLEDdnC0o/PRS9ELejeZCe9U5P0WZIhqBpej+zUGZ/De4+0cykOtmtNTlvpBlLXA5up/jgIfd/ctV59wJTEJ6tRy1Aku+lptQBtGuk/XqvBN2AAAOa0lEQVTS8j2xRUVsjArdgkLGFqMIlJWo039h1Tlnxv7V6OYuRxFU21Do2U5gUBflakAToveFfAfRlK1WFCHjwFVV55wccq9Fir6KUHAU5bUHBY/kXu/ddB/XoH7f2qiHl6OelsXn71ed0xTHb4j7vuEYdT4hA9mWx3OyJu7fiyHPm/H5q6rjx8e1N8VztQXFEByIfYeAkZnUW943rguVOg251E+jPAn5b9Dq8C3xEAyJYweG0mxGDoG/pOzROw+4Kv7+d6LL0EmZ/iG+5z8BJ1BeZ+if0ET8lfGQjYjjG1Hmg51x0/9bhVyXARfG33d2Ra563YD/Eb/vS8hC7Yl7eVu8AF9EfcLJcXwD8FQowGzgUxX19REUrH8YLX3arwtyfTG+889Ry2xbKPA98aL5LbFyQBzfD3g4jpkPfLBCrj9AraazM6u3vG9cRjd/UdzsSVH+RVTYJ6L8gSg/FuXRUcHLozwwHphDwMBOymBxIw9VKOULobSzovwfIccXo3xplJ+P8pA4vznK/ZG1PYL67bnXdcb3rbQU5NSq+/auKH8jyl+P8glRXhH13S/u284oG3pZt5YUqpNyrY7vmBflO+O6V0f5a1H+lyhPjeObK+QoxcJ3+uVxXPnyvnEZ3fyxwDnx9w+iAreiRbomxOfmqOhb47gziGYU8vKVms7PdVKGJ+N8L30HGpI6P/6+OeTahppx30FN+o1xzh1x3KnAlPj7kfjfGrReUe51neE9e7RCOV6PfQOBy+LvG5HV3Bl1dBMKUmiJOrk3jpsLzIm/H4j/NSPfQIdbJ2ix8tXxMlkS+/oDV8TfI1FTfA966XwfLWXZHNd+MI6bhTKEZl93ed+8bngYHkLW6pWoxMfjcyHqczx8jHOWoj5ISenO7+A1T6ScAXJ9KFm/qmMeiIew1Gcr3eTXUL/nsWN87+vojV3qp1+ed/1mdI8mxz3aEnW1kSp/AnBXRX0djXr1eNHtBZ45xvc+H8q0JI79aAflGkt5lbuVId+wqmMujWNeIRxfca2lqKXV7S/X3G9gNz0UQ+Oz1Bx9oHL/MY43yulCdoSijKrxWgNDWfcjj+aQ473hK+S6hbc3148pV+l/yJu8OR6K8XnXbxfvTb9QqiPISg4CGtqpr69HfS2K89urrwbkSDqAlpesRS4LRWwFTon72r8duf4i5FqJrPFx5cpyK2RwRXt4OTn5d1BT5ttV+49FA+X+yFF082rlSMW5/ThOSGXF9b+LrMWNbckVkV+l7zPKIXFFp1RfDdRWXz9EL6zvuXtrO/exsr5KzqCOyAXlZ+GY+lFx/dtQ8/0Wdz/SjlyZUdjgiloxM/N2fqSZLUPOqb3IK32Ou9ecVS8SeC1Brn1DN3+6t5EbqEa53kBNyVJUzaXuXvhpiGY2ETWL96CX6iC0pMuBNs6ppb5eRM6q9ciKf8jd7+6AXKNRq+kAUsYRyKG1uytyZU2vtLSV1Fihzci1D+ordSgNpiuV5lOozzwRaGlLYTsgVwtyfDQgD3PhFRbA3TeiYZNdaNx1Y1sKG+fUeh+Hoi7KK2iIpiNybUc+kB3o5b2pLYXtgFyZ0ustbS3E6nqV2fMOduI7SkM+A4Cx7d3sGr+zATlDhgOjs/jOesLMNqLoounu3pzB95WCNZrQMNvqTn7PGjSMc0K8kOuKIocxZskd6A29CYVElkIPdwK73P0D1SeY2b2oST0UeaaHojd0E/AT4NoM5LodBYNsAf49HvIL4zq73T2zhYq7mwjVHI8cPIvRs7cftfZuQ8EkXeUHSNl2AN80s7XxvVtRU/z66haQmd2FhgUbURQWlPvcP0A5nuqLnvB21fuG+kD7kdIeRA9VK8cZagFOj/+vQTd4JXootiNru50MBtUpT+7fiIY/Xg95lsfn+/Kuuxp/x4yop1J/cT1q2eyJz310MYQ0rrMk6mld3J/qkMjPVh1fGnoqhcRujmdgf/x9kDoMasldgHrYUAjdOSgM7kjc4P+LhlrWxENWinIaFEqzDVnaUiieAxeg4YKZGck1HjgfDdQfjAfxn1ETvDS2PC7v+mvnN/RHkWH7UT/xDyrq6zrkMDopo2uNBC5BTe4daBTgx8ia/463hx42oICYwyHDRyrk+li8aE7Nu/6O+TvzFqDetng7tyJvJijh+VtvaeA9Uf5dlIfHjV/VzXI9xdtD624LOb6cd521I/eCkHNplBuQl34r3RhPjZxQTjnC6u+j/HdRnkG5tVQaItqJnGN1HeeduwD1tsXb+qz4uxQSuSUs28T43MTbQw8XkMHMknbkGgacG3/fHFZkW8jy7bzr7TgyN6AJEaXQw3ti/1zkfOrOazeihPVQDonchZrCt6BAjVJUWin4ZiYwO+96a2/r9UM+HcXdd3o5C8Z41FxeB8xDiy3NQ6FrB+P/uPvr7r6pm+Xa4+7PVch1FFmJccg5VY98DnU5dqA+4wTQEJm7r+nOC7v7YXd/MorjKPsCxiEn4cmoubyH8n1c5e4rulOuTMj7rVHvG+EgAX6EbvwvK/fXgVw3hlxfyLuuquQ7k3LssNVRff3vqK8lqK+bq1yd2ZKlbQcvD/p/B72V/61qf14MjM/G+OyRELoOcAh1LRqR0g5s+/DupeJ+3Yos/3ddIZF538cOk4IrCoiZPQBcjfplpwIvu/uZbZ/V85jZp1Dc8FI0EfxVdz8vX6mKT7K0BcHMRpvZe6PYgqzXdDQc9XRugrXNbSjwZAQaKlsLYGZnmNnJeQpWZJKlLQhm9jxKXLfA3RdFArlPAH/u7n+fr3THx8xOQ3HAS939RDNrRF7cgyg0Mz2AHSQpbR1jZjchRT2Igj9KCe3eRFFZe1F8bN0uhWJm/dCCyxeiSepjkAce1LzfCyxz90/kI2HxSEpbx0Tg+iRiUj7yeJ6KciTNRpkdH8lPwtowsykoy+QWpLRbkGNqCPIwN6JJFoVzCuVB6tPWNx9CD/QLwLtRiN6FyFK9twgKC+DuLSjj5Vw0PnplbP+Kglm+khS2dpKlrWOi/7cZzSAaXc/N4M5gZs2oJTHLuznYojeRLG1982tkidaj2ONeg5k9jqK6DCXjS9RIUtr6ZjTqzx4GTjWz9+QsTyaY2VTUzB+EQjFHxtpMiRpIzeM6x8xKkURr0KSB2a50LYUkvMnL0BjzXDTmfDgN/dROsrR1TKRPGYKcUY1oAnlvWB1wF+W0tQMoh2ImaiBZ2jrGzErTAXcgy/Qud/9dvlJ1HTMbj6Kj9sU2BK2qkDzINZAsbX2zCoUA9kNZIguvsADuvply1sMmtJxnUtgaSZa2jon+3xbkQR7lvS8b42YUbDEtxnITNZCyMdY3P0Ee5E3ArWa2Hi1MvB01K6+tduCY2f0oemoISgQ3CDWtW4H17v6hrgoVcc8nUV73Zh1a42ZL7LvmGHLdhxR0IIrs6o+mOo5DeZwu6apcfYVkaeuYCGOcgJR0PPK6noCyP84GPuju91QcvwCtFtcS55VyJjtyYA0Bmtx9Wxdk6kc5uftmlNFwMVqXdTlakf2T7n5bxTlzKK/AMAY5oiqda0OAMamJXCN5z8JP2/E3ZDHPRFkM9yPl+wdkpRYhZS4tizkYTSTYiazXf6GcXfASyotv/5rjLHhVo0x/G9/5GeQk242s+I2o7/0CUsqZcfxAtDLgXqTgH6+Q62oKkpepnrbcBUhbjTdKM2VaKa/Fens8+H8c5XOjvDDKA0LR10W5ETVlD1PjioDHkWN9KH9TlB+M654e5W9H+atRnh9yv0l5UavdaMmN3Ou1iFvuAqStxhslSzo//r6F8sLZW9GCy8spr+F6Vxw3g8iLjBZxdjTUsojOLbj8JOWV1hfFvgHAKfH39yhnidyCskYuo5yN8Wdx3FRgUt51WtQtDfkUBHff7+6LojgGWczVyFH1ftTH3YwcVGPinNXuviXOGYWa0wfQTJt3LHXSFmY2CyVOb0SKO9zMBrj7IXd/rUKuI2ioqiTXHKTEe2Mf7r7O3Td05PqJMskRVVDMrNHdD5vZPwNfAp5w90tL+493Tvy5Cq16Ps/d19ZwrX7oBTEJ9UE3AUf8GA9PhVx/C3wVeBFN4O9/PLkSHSNZ2oJSoQDfQxb0W1X730aERI5EzdkByCLv78Alt6LnZQRqqg9oR66b45xvuUgKmxHJ0vYRzGwxij7aivq6F7l7zdP9IvRwHWrmHkBe4cnufqgbxE20QbK0fYeVaJZQf+DpjigsvBV6+BgKPZwMrE4Kmw/J0vYRonlcChsc7+5bO/k921ATeXIocqKHSWGMfYefIOfTJuAHkerlYtRcPgBcXe1YiqToo1Af9g3Uj92JvMB3ks1C0IkOkixtH8HMVqJm7TYUyfQmcCLl0MPfd/cfVxx/Mopk2kB5vdfBKEDiEOrTjklN5J4nKW0fwcyGo7HcjUhRhwDfBL6ChmVmAae5+yozG4xWBpiIFPpqtLwHwBVoyOhoLcNFiexJjqg+grvvdvdXIqjhSRSh9G/ufhStADAcuD4OPwEpawuKbLodeY3Xu/tvXEtCJoXNiWRp+yCRd2qmuy8xs5uBz6KxXkMrqF+OpvRNRqGHN5hZE3Cgsw6sRHYkS9sHcfeD7r4kiqNRH3UVCr64DjWVtyDrOjLOaU4KWx8kS5vAzPq7+xEz+3/An6Fx3AtL+/OWL/F2kqVNUKGYN6Gx3G9U7U/UEcnSJhIFI1naRKJgJKVNJApGUtpEomAkpU0kCkZS2kSiYCSlTSQKRlLaRKJgJKVNJApGUtpEomAkpU0kCkZS2kSiYCSlTSQKRlLaRKJgJKVNJApGUtpEomAkpU0kCkZS2kSiYCSlTSQKRlLaRKJgJKVNJApGUtpEomAkpU0kCkZS2kSiYCSlTSQKRlLaRKJgJKVNJApGUtpEomAkpU0kCkZS2kSiYCSlTSQKRlLaRKJgJKVNJArG/wePInuqnjENQwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "koch_snowflake(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Iterations on a given side of the original triangle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sample_koch_curve(depth):\n",
    "    init_graphics()\n",
    "    koch_curve([0, 0], [1, 0], depth)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_koch_curve(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_koch_curve(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_koch_curve(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_koch_curve(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAACDCAYAAAB2tFtFAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvhp/UCwAAG+FJREFUeJztnXm8HFWxx7+VDRLIQsgCIcQk5GYPgZBAEEVBeD5QeYCAbCLKBxWQVQRFRRAURFREkF19bqjoU3mowAOECGFJiGDYYiDsJEQggCFAIKn3x+9M0hnmbrk90/fO1Pfz6c+902f6dHVPd3V1nTpV5u4EQRAE9UW3ogUIgiAI8ieUexAEQR0Syj0IgqAOCeUeBEFQh4RyD4IgqENCuQdBENQhodyDIAjqkFDuQRAEdUgo9yAIgjoklHsQBEEdEso9CIKgDgnlHgRBUIeEcg+CDGbWzcy+YGZWtCxB0BFCuQfBulwInAecWbQgQdARLFL+BoEws77AAmAAsBiY5O5vFCtVEKwfYbkHDYuZ9TKzPplVXwI2B04FRgMnZL67sZn1qLGIQbDehOUeNCRmthHwNLAcWARsCGwPPOLuE83sMWAkcDewEmgCegFbuPvKQoQOgnYQlnvQkLj7a8ASYDhS7KvS5zvSV2Yh18xqYCNgGPBMKPagqxCWe9CwmNkw4J/A/7n7Ps18x4CbgalAk7u/VEMRg2C9Ccs9qHvM7GIze8rM+mXXu/tzwDnA3mZ2ejObnw/sApxertjNbFMzW2xm51RF8CDoAKHcg7omDZgeBGwJfLDCV15Nfwc308Xgsu9l2QfYDDjczHp1RM4gyJtwywR1jZmdAXwNeAzoDkx099dT20DklpkP7OoVbgYz6wbcg6Joxrn78rS+X9r2JWACcLK7f6fqBxQEbSSUe1A3pDj1G4G30QDpBsAM4BrgCuQ7fxx4Km0yBg2Ubuvu97fQ77vRQOuTwBNp9Wj0NrA9cAawGzAXeAvoCRiwm7uvyOv4gqA9hFsmqCeagJnAu9LnIYADp7r7LcCjyI1i6NofDDzQkmIHcPfZaHLT0My2Q4CH3X0OcBLQg7UunFHADsCI3I4sCNpJWO5BXWFmVwEfR7NLF5rZKHd/PLUNAV5LYZCY2QB0DyxrQ7+DgTfc/d/pcz+gp7u/mD6PcvfHzWwbYB5wobuf0HyPQVBdQrkHdYWZbYZ84f9w9/fUeN8G/B3Fzje15aERBNUi3DJBXeHuS4DrgZ3MrNaW85koHv73odiDognLPagrzKw/stxfRpExq2q47w3SvgEmxGBqUCRhuQf1xlfQwObB7r6qfOJStTCzfu7+JnAYGkg9uRb7DYLmCMs9qBvMrAl4BFiKoluGIf/3CHd/oYr73QLF0T8OPA9MBDYBRrv709XabxC0RFjuQT3xGvAwUrIg18yGVL/wxrkoFPKV9HkResi8WeX9BkGzhOUe1DVmdiFwDLCNu8+vQv87ArOBb7r7l/PuPwjWl7Dcg06BmY0ys6Vm9tmcuz4DWAHcmGaw5oaZbQpch1IQ5Jo8zMxOTUnJhubZb9A4hHIPOguHoIHQz+VcnPp1VGyjL3Kd5MmGQB/kfsnNBWNm3YHPotm0++fVb9BYhHIPCidNPDoFuB+YBHwsx+5PBgYCH8o79tzdnwUOQEnFjsmx6yNQFaiHgDPMbJMc+w4ahPC5BzXHzK5BmRRLESzjUK6WicAvgbFopudqZB2/Duzu7m+30u9GKHHYatYmDpuOJhUdkP+RrJmV+heU8z2bOMxR4rAWC2ynVME3obeKlenvNGAOcDxKZfAsisYBnad/uPuBuR9MUFeE5R7UlKQMZyKF3iMtg4C/u/sCpNA2RtY2wFbATkiptcYE4N2smzjM0FtBVUhpgk9k3cRhI5Mco9vQxZbpu6PS503RA+14d78PeAAlLOuOariORZkug6Bl3D2WWGq6oCn6q4AL0uehQL9M+5j0d1dkAZ/Wjr5/jPzfY7J91eCYSvvbOh3bhe3Y9tx0nDPKZUYPuU3T/z9C1n1NjimWrr2EWyaoGmbWEzjM3a+q0HYJcCQw093nVmjvgWLFu6Op/G+Ute8KvOhl6XrNbHOUAmCeu78vt4NpA+mtZC6y3N9Rb9XMZqBMkrPL1vdFMr8ITPEKN6WZ7QLcApzn7qdWaP80cJXXMN1C0LkJt0xQTa4GrjSzj1doOwddf9c1s+2VyCVzZQXF3ge4AfhreWSNuy9GPvCdzez4DsrfXs5E/vLfVVDs3YC/ArekB9caXGmEr0aDyd9tpu/fozeC88obUoK0y9ISBEAo96BKpPznuyEXyTFJuWX5FPKHH9tMF58HlgEfTaGBWY5DPu5NgH3L9tsP2BkV5ri4I8ewHpwDPAPsbma9y9o+AWyEBnk/k21Ig6p7otQFX22m75PRW8wnyrbtgc7lm8BHapVLJ+gCFO0XiqU+FjQI2CPz+VJU7u5LyJ98aKZtBIqAubqVPg9M2x6RWTcMWI4s2ftRqoHemfZ1/NcFnIfSOMGXM+v6AUvQTNabkftlYKb9xLTNnq30/SeU4mBIZt1RaduTUZTQeZm2XkCvoq+NWIpZwucedJg0i3IB8C8UtrcxsB1wX/r7HLKy70GuhVJira3c/alKfaZ+DVnCg4G707bjUXTNOPSQuIW1dVF7o0iSh919Ut7H2VbMbCGKlLkbDYBm662+gR5Kz6G3i56oJN/T7j6qYodr+x2HYt+fRz76HmnbJehczEfndg56eI4ABqAEZi/nepBBpyfvGXtBY7IUeBUNJD6PLPbngbvcfbWZ3YrCGUssQwq4WcUOCjM0s9uA7MDoy8BCd38MeMzM5iEFCVL+S1Ex6yL5G3rAlSynV4E5rnqrmNndyJovsRSY1Vqn7r7AzO5E4ZLZbW9L52o2evCtRlb7u4DHQ7E3JmG5B7lgZpORRXqJu3+uaHkaHTP7b+TWmuDui4qWJ6g9MaAa5IK7PwBcARxtZocVLU8jY2bHoKIhF4Rib1xCuQd58gKKgBlZsByNTmlm7L8KlSIolHDLBLmQqhH9E7jB3fdt7ftB9UgD0TeivDpNXsUqVEHnJSz3IC9uQtEqw8zsb2a2yMx+ULRQjYSZXWxmjwG3Af1RpMzNxUoVFEUo96DDpFDIrYCnUahfP5QIa9si5WpApiOXzMaoQMmzwDgzG1CoVEEhhFsmyAUzGw687O7L0+drURrcJndfUqhwDYCZjUS5eH7r7oemdQOADdz9+QJFCwoilHtQFcysCXgQ+Iu7/1fR8tQ7ZnYTsCMwzt2fKVqeoHjCLRNUBXdfiCbm7GVmUViiipjZ0cAHgOtDsQclwnIPqoKZDQIWIr/vVI9UtFUjkx55Y2Csu79asEhBJyAs9yAXzKxXWSrbr6Oi1B8LxV5dXOUHD0JFT04rrU+/Sc9mNwzqmrDcgw6TomUeQZNmnkOpbacD97v7NkXK1kiY2UMosdo9KGppSxQOuVXkl2k8wnIP8mApSsM7CiWjW40U/Z1FCtWA3I7Ou7M2cdjLodgbk7Dcg1wws6nAPOAid691BaSgDDP7EXAoMNHdHy1anqD2hOUe5IKrlulVwLFmdkjR8jQyZvZZ4JMocVgo9gYllHuQJ0tR4rCxRQvS4IxPf2PyUgMTbpkgF8xsGEocdnNMWiqWlDjsZmAqmiH8UiubBHVIKPcgFzKRGnNQseYtgD+7+7Fm9mM0e7KUhqAf8Iy771WIsF0YM7sO2AwNYANsDsxy9yPN7CLgP1HEUm8UsXSfu0eOnwYkyuwFHSaFQjaheqevo/qoo5HlCLAzitx4EXgrtY03s/7u/krtJe6amNkQYHek2OcDG6DzXppHMB0lcHs1Lc8BE81sQETMNB6h3IMO4+7Pm9kY4CV3/zessTB3NrP9kTL/krufm9q2Qxb+V4GTCxK7K3IWGifbyd0fATCzbwCnmdne6GF6tbsfnNoGAhuGYm9Mwi0TVAUzG4sSh/0bFcSe5O5vZNqvBD4BTHP3+cVI2XUwsx1R4e8L3P2kzPqN0VjHBsgVM87dny5GyqAzEdEyQVVw938CtyIXzUVZxZ44C+iOKgYFrfNnNDnsm9mVKcXylcBANMYRij0AQrkH1eVTwC+BShWZDkVhk6fWVKKuy+noYXhQhbazgV8DR9VUoqBTE26ZoOZk6q3+xd33KyUcSwmwggxm1gsNmK4GbgBmEHVRgzYQlntQBKV6q8PN7HbglbQueCe3Ai+j3PgDiLqoQRsJ5R7UlDTBZgnwBMpc+DaKi5+ZcsAHiVS6cAYKL12F6qI+iZKDBUGLhFsmKBwzmwj8A7jc3Y8uWp7Ogpn9HNgPGO/uTxQsTtDFCMu9IEz80cxuSNZsw+LuDwGXAkeZ2WFFy9MZMLNjgUOA8xtdsZtZdzObZWa/LFqWrkRMYiqO4cCHUMTIaOCxYsUpnFLtz60LlaLzUJrdG6GNMBF4L7DSzAa7e7il2kC4ZQrCzH4BfBT5Um90930KFqkwMvVW5wAf9LgoS2MTs1EBlIati5rOw03AdqhG7BXuHiGfbSCUew0ws0uA3VCxaFDirG1RfPIKNDHlfhQVATAMuMXdP1tjUWtCer2eDJSyFTahBFiTk4smAMxsBiqZ9zSwKK0ehJKBHVqYYFXEzC4H3gcsTqsGAZOAz6Hr5DjgXuC11D4cGUcxVlNG+NxrQxMwBl2ooAtyOXAu8D2U5Gl4ahuYvj+hxjLWhGSJvQ9lkOyOXINDUb3VUOwZ3H0O8BB68HUHeqLztlORclWZyageQP/0eQuUcO4y4EzWZhwFZcfcChhZWxG7BmG514A0SWce0Bcp7dXAmJIyM7PxKDRwFcr2Z8AUd19ZiMBVxsymIxfMee5+qpm9C3jV3ZcVLFqnI2WC7OHuz6U3wCPRtfFwwaJVBTPrAyxAhV9mABsCW7j7wtQ+GRVj740mwj2OEqmFIisjLPecMbM9zKx/dl2aeXkCsjDOdveVWSvV3R9JuVe+AowDTipX7Ga2kZl9uOoHkCNm1sPM9i1f7+5zgZ8AJ5rZRHd/MhR7Zdx9aVLsOwCfBi6upNjNbL+uFnVlZvukGbhrcPcVwCnANOAEd19RUuyp/YF0P52PLPfjyxW7mQ0xs92qfwSdHHePJacFRXo4cE8z7QtS+9YV2rZKbU+Q3qjK2m9N7TsWfZztOB9XJ5kPrNA2Er3BLC5azq6woPGJt4FBFdqOTuf5h0XL2Y7j+VCS+doKbYZKBK4GhlZof2/a9t5m+n4wbTuy6OMscgnLPV9OQhfVNmbWlG1I+c5HAn9DrpdyFqEMicMp87eb2QhgJnLbnJC71FUg5RL/D1Sc46gKVuVB6Cb+cq1l66KUsmh+LLvSzLoBRwArgb1TCuBOTboWjkUPq/cn11OW6cAQ4H+Qe6ac2WhQdXzKU5Tteypra8ieVL5hQ1H006UrL0CvzP/bI2viByiH+bVl3/1DWr95C/0NQrnPbyRjvQO/QlE130n72LmSDAWfix5A98zn76OH0VlJ5v0zbcPQgPLvi5a7qyzoQXgTsuA3zaz/VDq/p6e/ZzX3mxQsf/ZeKVnt30AP/8vLjvNOFC3Tt4X+RqP0FT8t2/avwAvAVejhMamSDI2wxIDqemJm89AFdl9atS26mTZDqVe/habUL0M5zbcmU42ohX6PQ4rxARQl0B/YBjgD+DbKy2LIcgGYAjzr7oVN/kll9h5E8i4G+qDBsPnovCxBg8n3oDeb8ehBNs7dG33yVptJg4n/QOXzHkWW/A4o18xwFFkzDp3nN9K6fihOvpBqTMlKfxhFgT2Ert0ZSb6hKGLsJHQ9L0cW+wTgk+7+k1b6/gZwGvB3FHFWCps8Cvgtmhi3Ap0zkB//YXffIbcD7MSEW2b9WYYU72B0wb4C/M1VZu77aMZpqbbl2+nzBW3o9xJ045bS365CLptvuwabZqF4+G7ohhmYZCmSpShEbRRSOI4U/Z3uvgq4DclYut6WA3eHYm8f7v4AijJakVZ1Q5b8bS4r7Q503kFhkyOB14pS7ABJrmXoPumPZF6GZH4LvdktQtcMyL30T+Cnbej+nLTt6vT5LTQZ7kpXSuQ7kNI3ZHT1Ze35qXvCcl9PzKw7Cm8cgBI7vV7j/fdC1v0qNED7Vi33X0GeUl3U77p71EUtGDO7DLlspniqt1qgLH2Q9b4M2C498Gu5//7ogbEQeK83iNILy309SRfo8cAI4LcFhKH9Ak12OrFoxQ7g7vei8MaTzGy/gsVpaMzscBQ2eVHRih3WhDd+AeXLubyW+04DzteiN4d3hE3WM6HcO8YT6JVwcgH7nopeZR8vYN/N8RR6Bd6uaEEanO3T3yeKFKKMUqz6tAL2PQG5OZ9t7Yv1RLhlOoCZXYNG/mtecT4NYi4EZrl74ZObMvLc4e57FC1PI5PeImehgeumIn3uGXluQ0q25vKY2VjkwvyZux9Ry30XSSj39SQp9v2QdfQkioh50N0PrvJ+f44ib14CtkQRO3/wgrNKmtl8FKkwBw2ubgZc7+7HFSlXI2FmPwA+iCKWNkJvUPPcvdA3KTP7E7An8nsvRlEtc9398Crv99coT80raJLgcODqat+jnYVwy6w/05ASexqdx/HArsnHVxWSBfR+1ibdegaVYCviVXcNyWqfgF57X0eDzE1EbvZasx067/1QRNJiYLKZDShKoHTNTkNZHBejcOHxwM5V3m9PlIl1HHIVPoGiabat5n47FUUH2nfVBSU0yk6QOBT5wA+v4j4PSPs4MrNuArBRJzgfY4AB6X8DbkAhm4OLlq0RFmSVvgZck1k3CHhXJ5CtFGtf+vz5dB3vUcV9Hpf2sVdm3RQaaCJTQ7plUnGIie4+K8c+u6G42jEo4+MrefWd+u+D4t+fB6Z7juFkZrYT8Ji7L8mxz1Jd1Gvc/aC8+g0qY2b/C+xOzvVWU+qLoa70w3n1WQrj7YHGq3KN9krpDBYCd5Nz8Rcz+wAwx7tA8ZRGdcvcCtyaUs3mgruvRvmmByGrNW+uQ3m9z85ZsQ9Fg2+359UnrKmLehtwoJntn2ffwbqY2aeBDwN/zlOxJ+4E7jSzvnl16Mp4ej6a9PabvPrNcDN6W/hazop9MkoBcWNefVaThlPuZjaFtYm58k7CtT+aVHRKzv2S+nwLlebLk+PRdbBVqvyTC8nPOxXNzP1jXv0GFfkZGn/ZPr3h5YKZ7YLyAHUHjsmr38ReKAVBNRLHfRG5ZA7Iud8TUejzNDPbKue+86dov1C1F6Bn5n9DT/UXgR8jZTmh0nfXo+/p6If/dhWP5Ux00b4nJ5lHo0Hhq5G7ZzbrJixrc99IAXTLfP5uOh/bFn0NNMICvCddG2dk1vXI/ibtvDZ6oNKPT6BJQMuBYetzbVToe48k6+ereD4urdL9/UMqJL1rb9+1WOra525m05ALZiHKyJitx/gbZO28xrqJhR5y95lt6PtepBzvT6tKicOGec7+9sw+N0JJuGDdxGHPuPvUNmx/O7Km56GbaxLK9zEKhdBdhaaJL0WhdOOB3d39rlb6HcraRGdLUOKw6Sg0dEo7DjHoAGa2AI353MPacnR9kR++xdhyM/st8BGUCO91ZLE3obfR+1D1oxfS324oMuced9+llX4NJZUbxLqJw95EvvyqVBszs8FU6f5G+fO/ydrEgAPQPbSjd6JSkfXullmILMox6XMpgddl7v4vNAD6b9ZNLNTWJFwvox91UNr+VTSBpyqKHcDdX0P54F9h3cRhbZ0UsgxVkN88fV6Okns9i1IHLETWDuic9UCxya2xNG03Gp3v0r5mt1GuIB9uR+fdUOKwUcj10ZZr8iX025XcDSuRIv+duz+KHhjZotR9WFvgvFlc1uOrrJs47GWUOKxqZSSrfH9/DyUsW5XaSvqlM80Wbwi3zOHISj2khe/0QxbnOm6JVvrtgZ7cjwO9CziuXujmW0Abw7vQhXgLsrAHtvC9fdM5O7od8kxP25xX9G8ei4Oyi75Nxi3Rhm1KvurdW/hOKRf/H9rRbx80H2QeBeSXr+b9jdILO/DRon/zd8hWtAA5/XjdUYTKVRXauqHXyhXo9bRS+y3pB5pRoX0mmoE6pULbrmm7a9t60eR4zL9K+96zQltTknmXCm1bI9/hHajwcnn76HTzPtJM+8XI1fUOHyOy/lcBexd9TTTyAhyWfuPvV2jbELgLOLeZtqeQdfuOojLJoLgPvaWNqdC+V1KGIyq0HZiu18vbezwdPBdVu7+Ra+pN9DZtZW0G/BoFE9RUN5SWenHLjEUxvoek8m5rcIUoPoOqpU9sZvsp6IJ9skLbF1Dmx+MrtD1KGjRcP7E7RMlSXlih7XNI5kpRO8+gC3JSM/2ORf72xa5CxGsws37oTeh9zWy/CN1M725d/KCK7ISUy6MV2maiAh+fMbPeZW1vIr96f6BSmHB3ZDi8wdqxnyxfRDnkK+VvKRX1LqJQRrXu76nogfe0J42eYXP0BvxhKp/L6lO0lZHDk9mA65FPbBVwYVn7DKQEz2+hj/GUlftK67dBP+6DyAfZVNb+KzT49A5LpQbHvRnyJ/6xbP2IJNOD6bh3KGsvlb+b2kLfZ1NWzi+t/1ZavwzFsGcja4YgP+UNFGSpxLLmtzDkgnge6J9ZX6pBsDT9jl8u265U/u6UFvreibJyfmn9bmn9I8jHP7RMnr+iB8cmBZyPqt3fKIXxOuX80vqfoIfl68BvirgOuly0jJldgH6UEgPRk/lEZFUcBcxlbbWaNeXvXFWSmuv3u6mPUrkv0n42SH/nowfIA6mtVP7uTHc/o6PHtT6Y2SmsW84P9HZSKus3B12881JbKYrlMnc/qoV++/DOcn6907Y/Ra/1l6Loh3+l9rGobNpkd3+YoFBS8ZS5yNe9KK3eDOVaORglvdsLXSMr0W+9NXogbO0tDHaa2S9Qoe65yIo3dG28jUIy70v9lAbjS1FqR7v7JbkdZDuo1v2dZruXl/PriyJzvpXWn4nOSXZg+z53r2qx+3pwy4xAiu1iVCR4JcqWCPqB+qFR7mYVe+Lr6Clb2rZv2n62uy9AU5k3RQqS9L3lwHn5HMZ68X10wZRk3hBFJcxNCvYOdBylxFHD0fk5vaVOXcUVbkeRNZuk1Vsgi/804EpkhZX22xMpjvtDsXcOXMVTHkQDoD3S6i1R/dVfIXeEod8VpIAHoOIvrUWxnIqs9NK2A9F9Mcvd5yNFthlyWYDu0ReAKzp2VB2iKve3q5zfXeg+2Tiz7QpUAPx89Ia9ZcUOqkkRrws5v3L1Q3liSp+3Zd1K6zNo40QO0pM883k6aVARPeG3zbT1ocIgTAHHPwnYOPN5CtAn/d8N2D7T1guY1sZ+rWzbnqhEWunzOFKisMznmr9yx9LibzgEGJ35PIjMQGi6N7pnPu/Qjr63IzPgjnzpJU9AqfRkqa0/FYIZCjgfVbm/K9wbG5Jxe2bvyVouXc4tEwRBELROPbhlgiAIgjJCuQdBENQhodyDIAjqkFDuQRAEdUgo9yAIgjoklHsQBEEdEso9CIKgDgnlHgRBUIeEcg+CIKhDQrkHQRDUIaHcgyAI6pBQ7kEQBHVIKPcgCII6JJR7EARBHRLKPQiCoA4J5R4EQVCHhHIPgiCoQ0K5B0EQ1CGh3IMgCOqQUO5BEAR1SCj3IAiCOiSUexAEQR0Syj0IgqAOCeUeBEFQh4RyD4IgqENCuQdBENQh/w+ugUXOq23vnAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_koch_curve(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Sierpinski Triangle\n",
    "\n",
    "The Sierpinski Triangle construction starts with a filled equilateral triangle. This triangle is divided into 4 smaller equilateral triangles and the inner one is removed.\n",
    "\n",
    "This process is then iterated on the remaining smaller triangles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fill_triangle(A, B, C):\n",
    "    plt.fill([A[0], B[0], C[0]], [A[1], B[1], C[1]], \"black\")\n",
    "\n",
    "def remove_triangle(A, B, C):\n",
    "    plt.fill([A[0], B[0], C[0]], [A[1], B[1], C[1]], \"white\")\n",
    "\n",
    "def get_inner_triangle(A, B, C):\n",
    "    AA = [(B[0] + C[0])/2, (B[1] + C[1])/2]\n",
    "    BB = [(A[0] + C[0])/2, (A[1] + C[1])/2]    \n",
    "    CC = [(A[0] + B[0])/2, (A[1] + B[1])/2]\n",
    "    return [AA, BB, CC]\n",
    "\n",
    "def sierpinski_for_triangle(A, B, C, depth):\n",
    "    if depth == 0:\n",
    "        return\n",
    "    [AA, BB, CC] = get_inner_triangle(A, B, C)\n",
    "    remove_triangle(AA, BB, CC)\n",
    "    sierpinski_for_triangle(A, BB, CC, depth - 1)\n",
    "    sierpinski_for_triangle(AA, B, CC, depth - 1)\n",
    "    sierpinski_for_triangle(AA, BB, C, depth - 1)\n",
    "    \n",
    "def sierpinski(depth):\n",
    "    init_graphics()\n",
    "    A = [0, 0]\n",
    "    B = [1, 0]\n",
    "    C = complete_triangle(A, B)\n",
    "    fill_triangle(A, B, C)\n",
    "    sierpinski_for_triangle(A, B, C, depth)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's run a few iterations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sierpinski(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sierpinski(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sierpinski(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sierpinski(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Jumping to depth 7:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sierpinski(8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Mandelbrot Set\n",
    "\n",
    "Note: the fractals in this section may take several minutes to render.\n",
    "\n",
    "The Mandelbrot set is defined as the set of complex numbers $c$ such that the sequence:\n",
    "* $z_0 = 0$\n",
    "* for $n \\ge 0$, $z_{n+1} = z_n^2 + c$\n",
    "\n",
    "is bounded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy\n",
    "\n",
    "# Returns true if the sequence is (probably) bounded.\n",
    "def is_in_mandelbrot_set(c):\n",
    "    z = 0\n",
    "    for i in range(200):\n",
    "        z = z * z + c\n",
    "        if numpy.absolute(z) > 2:\n",
    "            return False\n",
    "    return True\n",
    "\n",
    "def mandelbrot(x0, x1, y0, y1):\n",
    "    init_graphics()\n",
    "    # Note: choose 100 instead of 200 for faster (though less precise) rendering.\n",
    "    delta = min((x1 - x0) / 200, (y1 - y0) / 200)\n",
    "    for x in numpy.arange(x0 , x1, delta):\n",
    "        for y in numpy.arange(y0, y1, delta):\n",
    "            if is_in_mandelbrot_set(x + y*1j):\n",
    "                plt.fill([x, x + delta, x + delta, x], \n",
    "                         [y, y, y + delta, y + delta], 'black')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's display the full set and then let's zoom in. \n",
    "\n",
    "Note: the actual Mandelbrot set is connected, even if graphics and computation limitations make the following renderings appear disconnected at times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mandelbrot(-2, 1, -1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mandelbrot(-1, 0, 0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mandelbrot(-0.66, -0.4, 0.5, 0.72)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mandelbrot(-0.57, -0.50, 0.57, 0.63)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mandelbrot(-0.56, -0.535, 0.615, 0.635)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Hilbert Curve\n",
    "\n",
    "The Hilbert curve is a continuous curve that fills an entire square. It can be seen as the limit of the Hilbert curve or order $n$, which is defined recursively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Draw a line between the points (i, j) and (i+1, j)\n",
    "def draw_horizontal_line(i, j):\n",
    "    plt.plot([i, i + 1], [j, j], 'black')\n",
    "\n",
    "# Draw a line between the points (i, j) and (i, j+1)\n",
    "def draw_vertical_line(i, j):\n",
    "    plt.plot([i, i], [j, j + 1], 'black')\n",
    "\n",
    "# Draws the Hilbert curve of given order. The curve is drawn with the\n",
    "# opposite corners (i, j) and (i + 2**order, j + 2**order).\n",
    "# The orientation marks the \"opening\" of the curve.\n",
    "def draw_hilbert(i, j, order, orientation):\n",
    "    if order <= 0:\n",
    "        return\n",
    "    # Maps orientation to the orientations of the sub-quadrants, in the\n",
    "    # following order: \n",
    "    #   23\n",
    "    #   14\n",
    "    orientations_map = {'S': 'WSSE', 'N': 'NWEN', 'W': 'SNWW', 'E': 'EENS'}\n",
    "    orientations = orientations_map[orientation]\n",
    "    # Each sub-quadrant is within a square of side length L.\n",
    "    L = 2 ** (order - 1)\n",
    "    # Recurse over the 4 sub-quadrants.\n",
    "    draw_hilbert(i,     j,     order - 1, orientations[0])\n",
    "    draw_hilbert(i,     j + L, order - 1, orientations[1])\n",
    "    draw_hilbert(i + L, j + L, order - 1, orientations[2])\n",
    "    draw_hilbert(i + L, j,     order - 1, orientations[3])\n",
    "    # Connect the 4 sub Hilbert curves with 3 lines.\n",
    "    if orientation == 'S':\n",
    "        draw_horizontal_line(i + L - 1,   j + L)\n",
    "        draw_vertical_line(  i,           j + L - 1)\n",
    "        draw_vertical_line(  i + 2*L - 1, j + L - 1)\n",
    "    if orientation == 'N':\n",
    "        draw_horizontal_line(i + L - 1,   j + L - 1)\n",
    "        draw_vertical_line(  i,           j + L - 1)\n",
    "        draw_vertical_line(  i + 2*L - 1, j + L - 1)\n",
    "    if orientation == 'W':\n",
    "        draw_vertical_line(  i + L,       j + L - 1)\n",
    "        draw_horizontal_line(i + L - 1,   j)\n",
    "        draw_horizontal_line(i + L - 1,   j + 2*L - 1)\n",
    "    if orientation == 'E':\n",
    "        draw_vertical_line(  i + L - 1,   j + L - 1)\n",
    "        draw_horizontal_line(i + L - 1,   j)\n",
    "        draw_horizontal_line(i + L - 1,   j + 2*L - 1)\n",
    "\n",
    "# Draws the Hilbert curve of given order.\n",
    "def hilbert(order, orientation = 'N'):\n",
    "    init_graphics()\n",
    "    draw_hilbert(0, 0, order, orientation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hilbert(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hilbert(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hilbert(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hilbert(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hilbert(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In general, we get $H_n$ by starting with 4 copies of $H_{n-1}$ appropriately rotated. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def H_3(show_connections):\n",
    "    init_graphics()\n",
    "    draw_hilbert(0, 0, 2, 'N')\n",
    "    draw_hilbert(0, 4, 2, 'W')\n",
    "    draw_hilbert(4, 0, 2, 'N')\n",
    "    draw_hilbert(4, 4, 2, 'E')\n",
    "    if show_connections:\n",
    "        plt.plot([0, 0], [3, 4], c='black', linestyle=':', linewidth=4)\n",
    "        plt.plot([3, 4], [3, 3], c='black', linestyle=':', linewidth=4)\n",
    "        plt.plot([7, 7], [3, 4], c='black', linestyle=':', linewidth=4)\n",
    "\n",
    "H_3(False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then connect appropriately the different copies:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H_3(True)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}