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Sierpinski.ipynb

@@ -0,0 +1,306 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# John D. Cook's chaos game\n",
+    "\n",
+    "John D. Cook [proposed](https://www.johndcook.com/blog/2017/07/08/the-chaos-game-and-the-sierpinski-triangle/) an interesting \"game\": start at a vertex of an equilateral triangle. Then move to a new point halfway between the current point and one of the three vertexes, chosen at random. Repeat to create *N* points, and plot them. What do you get? \n",
+    "\n",
+    "I'll refactor Cook's code a bit and then we'll see:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import random\n",
+    "\n",
+    "triangle = ((0, 0), (0.5, (3**0.5)/2), (1, 0))\n",
+    "\n",
+    "def random_points(vertexes=triangle, N=4000):\n",
+    "    \"Create N points by repeatedly moving halfway towards a random vertex.\"\n",
+    "    points = [random.choice(vertexes)]\n",
+    "    for i in range(N-1):\n",
+    "        points.append(midpoint(points[-1], random.choice(vertexes)))\n",
+    "    return points\n",
+    "\n",
+    "def midpoint(p, q): return ((p[0] + q[0])/2, (p[1] + q[1])/2)\n",
+    "\n",
+    "def transpose(matrix): return zip(*matrix)\n",
+    "\n",
+    "def show(points):\n",
+    "    \"Display points in a scatter plot.\"\n",
+    "    X, Y = transpose(points)\n",
+    "    plt.scatter(X, Y, marker='.', color='r')\n",
+    "    plt.gca().set_aspect('equal')\n",
+    "    plt.axis('off')\n",
+    "    plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10fa247f0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(random_points())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Wow! The [Sierpinski Triangle](https://en.wikipedia.org/wiki/Sierpinski_triangle) (more or less). \n",
+    "\n",
+    "What happens if we start with a different set of vertexes, like a square:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAEACAYAAACUHkKwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJztfW1uLMeSXfNDswADBsn7vAADvNTYGyAf4AVI3sB4lto0\nQAq4BCThzcP88BLsP+TV+FfTHf7RPKqTJ09Uk7dkV5UQAQTI7q7KyqrKiIw4ERl5EhGboqKiom+l\n07k7UFRUtG4qJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJVJUVDSJSokUFRVNolIiRUVFk6iU\nSFFR0SQqJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR\n0SQqJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQq\nJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJVJU\nVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJVJUVDSJSokUFRVNolIiRUVFk6iUSFFR0SQqJfJH0cnJ\n6ebk5GJzcnLyjeedbU5Orn8//73t8XF6zlgb7Xlnm5OTm83JyefRax7OuXw75v3Xcr/3/b4U/tiz\n1Od3/PhTudbp5uTkqmujP+74/R360j7L976LNVJEFE/lzeY0NpttbDa7t7+nb3wRm80JHTN87s/b\nxWYTsdk8x2bzib5/iM3msjkvvy7/fyZtnI6c9/x2bfDWHH/59v3+jZ+Ta42di9+v6PMjtct92Hdt\n6XMc2r6ie3iJzeZcjrkw5/E199E+g+e3ezp7e3Z83LZ5H/393Utb9+94Pvex2bza+10Bz96BxbIT\n+uz3w18ogd3boOKBcf4mLBhEl+Y85lcRqn2nCA590Ovy/9fS9mMMyu1ajtXr72Kz+UzCt02Oc9fa\nvQn1NZ3L97M3bakC4bZZAajye0jO/S16xcpK73LkfsA3b89Mv99TO/xe0Y9X8y4/y/O5oHd4Jcdf\nzT72P8izd2CRfJg5WOjdbMgzz+doZ2kdfF+SQegsET4uhB+jncVO6PyHaGf3k7fvuD22ANT6YH6R\ne8mEHMeeSXs7aYe/d23tk+/1GWVKI2OnsKCY1BJ5keNUCWs/dyPvldtiS2sfh/HCFtFtd925x/8H\nefYOLI4PL5aFDzMtWx5X0c84LzTo9DcejDwIeZb9LjabX2SwPRkh5L7AFOZzHkjgVIncRDsjRvT9\n/Nl8d0xwf0na4/vdx0HZPpt2oay+yHUv4mA1vEdxPMdm86u5Np4HLJGHaBXT41u/TqOdOKCgcW24\nlXyevldW5E9v96PP75WuoVbMLsqdWTkfZtS/JoKAQfgY+SwOoecZ8DnamQjK5jBQI1VcGGAPb/+/\nUh9YIHQgO4UVpt86Y6L/qgwYM3k254DZetF7h8V2G60Su5PPl9EL8kV4JfLVCOBlDNbKgKkc2mDr\nku+TLYizaK2En+RZ3r0dA1cP98Z94Hbdexj7rnV1VsKzd2Ax3JvkH+Xn8L42gFL+/osMWMVG1GzO\nBHf3Jkx83pfk2Cf5fP3WBwjUU/Qu0BjesTV9gXCdvR0DYWTgE4IGbIIV1GX0gLRaBKyIWXFB6QAM\nbQHl4xbNMayCj1Nwmd03BxK/yvnOumsnlhXx7B1YDB8GvRs0biC43y6jnZ349+tocRL40zwDYuCz\nsLxGL/xskahJ/fjWllM6rIyeo52hce1ttO4Rm/X4HYoPM3t77PA82f27kb78mjwn3NOlnH8MCB0U\nQK+Q0V5m0ezpupll+BFWa+7f5Pe7OCgoxU2uY4UKpJQI82Gw8qx4K4NuF63P/UoDRsOrPMOGEU4V\ncigDdjnYd+Z+KT7DrpJGHvby21m0URNc00VOWDDPYohWuAgH4zuX0WIHcMf4fh2QzKwh5KtoLS5t\nh11DByizgtQ2XqNVoni2TsGhPf7OWRRP8pknIH3/+o4KE1k1t2b4mQwk5A44oO5KBPEyDi5MJpyZ\nVbM3v+3e2rqLVlHBHXLCrJGiNs+kP4f7iftDhEdzKvQ6LHx83QwTgLA4s3/smYxZEbg+KzEHWt6M\ntMP3pYoo4oCP/LOcD0Wv9wZLxlmL2bWH688tBx/k2TuwKG5neHVvEO93YNq1COJF9OYz/PinGMde\ndJA/Jtfk2VOVxvcxKCP0DzO1uiEP0bsljCk4V4ItmwzX0eNx73Ch/mru9Sv1w11TlRE/jyu5DyhP\ntRDH+qdgt7qSyozJPMSACWmSW/a+GUPpXcKV8OwdWAz3+Qj68p0JzrOgAoQOpHQmedAgVsvlS/SA\nIAQki3Jou5pZqqa+Wg56n48inCq4sNwUAI0YFNkL9VkzOsFPMVgSLkfjGDsc6OHtnoGJqBJBSB3P\nQrNJz8NbjezK8vV7t6R9f6oc8QwH5TO3HHwDz96BxbAH5P6SDCIMCAiQCjrM2PNoox+uHRz/OQYL\n4J6EThOs4C5pJmYGpu7kGseshW30ER70gy0xxWrO4qAEvh9pP8zzBCbB7hYUUva83ss7eVaK/9xE\nHzmBwjmNg3LO2jwz98LHMNCr7iIrm9YlnFsOvoFn78Bi2FsO+pmFFjPnWELUC7WjiDxYXYjHt2NZ\n+J/keAxyDb+6mZUF3SkkBWVd1ISfwTYOGM2P8tv123M8FuV6jl6xuXVB70lN5/tQK4+jSXw9dQP5\nHE4We07eKXCTz0k/oChciHofhwniJgaXtweyV8azd2BR3GYtKiC2i0PE5jwGZcCKBYPEzU4srG6w\n/yqf3aIvxil0gOqgh8vDyihTHk8xuF8a2XH34qyeF+kTuysuKUyttn4WHrdG9nRdCORptCFsYBMa\noobid7k5GYDLjHyaC/keE0W/XKJ/rmyptmH3uWXgG3j2DiyO8/ApWyBjg4wHpgqQY1Uqv0U/A+9i\ns/nHGNLLeYCeho86qPnO/EWugfwSvl+AoKykXN4KrsV9QkiYFd9YNObgovXvIlOCfB4rhyG6djj/\nPPqoFj83di0Ut3BW40u0oCm7SZfyHvZv/Ue0yOFA/O7LEvnTsF9qrm4DWyI6W3+lYyBAPFgZgHNp\n5kHHRAwKjH9X313DkTCpXdu/mXMQwdHjn+OwpucmWrdIZ3C0wdmqGkZWRafXujTvQSMXvJaox4na\nczVLmPvIILo+ayyQ04zVu2hD3lBKWNTolJwDpDNF2uJCK+LZO7Ao7gcIhzB51rmJg08MBcODAW6P\nmq9P0eMeLCS/JQLm3Iqv0hYfDwuC+xzCv8RBOfxK95od7xSduk38vHSBHRSAhrwVv7mgd+DwAs3o\n1fICnGbv7oPvMwtHjwGi1+Zcfdf6jPQ9og9uzdMqrZBSIsr9AIEAPcaQccoDDoObLQUXHubBgnNV\n0ahg6mfXDtqCZfQSbUEkrFvJAF38v5XzuMCPCsjXaPsGi8jl1oDhLmkmJ4OcbpZ3URU+X11NTfzj\nY9W6zJSdc1c4TH5qvncg7FP0lpTmkPC9rtIKKSWi3M7GilUcA9zALBzuPB18bpb/FC3gdh75Wo6v\n0So+jrB8GTlP+4R0+KvoIxj4H1gKC5UuloNwqqJDm7yKmV0yjXIBU3CJe8yaNarlE4aKYa0CYIXO\n65i45so2+vIJWWTsR3PP7NL21emOFb5aCc/egcXxYcZ0K2H3MkjGlIpaBywYP8jg+2La1Rk7S9fG\n79maljHm9rIZFUrJ4UKaM4KkLhXSiNYlcudyRKx1cXKrgJWRJtChdAJ/h/tQN/MX+Y7dJCiA1lKJ\nbsJBX9A3fVctANyPuVUrk9k7sCjuZ5iIwypMxkJeaWDsaYDwoFHF8VP4eiBuXcVOvnML2CJageFa\no2PHcruPcQALx7JDkfGp64WgrPgeFbPgZ6Fuigoz3ztjPe48pO1z+7pO6TIOikq/28p3vDQgE37+\nbqgsl68pug0PdON3jh65+rOrC/PO3oHFsPd1Mche6XtF21XolRGR4e+GweRBQLgoj+HzGfh/DD7F\nbJR/GjnXYQ4QYi1hwADjXtri/zVqo3VS9Tnims/yma+B9HAFmp9Cix33eMSxeiKv5j255335Nl4c\nOAuLRBU5JgtWwnBR+2c1tyx8kGfvwGK4HRQuIsLM7gpXLnPHarnB1+hrV1xGnzNwG32yFBbjuf5k\niWEQJO2HKgDtP9LcWVB+ln5zCFPdDJw3JFF5LIFzSdyiR53pgdPofSquwde4jnFcxbXDYK0qurO3\na8AVhQvK11Rs51Z+d6UcVwmwzt6BxbBfQJYNtAcZEFhExTMOz4KsINiXZ7CPB51GA3gW/94IlkYC\neGbL7sMVQXL3qlYZgF61oPZxwBdcG5qbwQqGw+ia7cpt8z3dvDEE90Wu59L7oXyy94tjUP6Qo06K\nuXDZRLhTKM3Iik7/5zHjn9PccvANPHsHFsU9KOkGotbX5EECFP494CaES81iHWwcwVAh2MegoJ5j\ns/kP0dZ51exSLqLkwox3ST9/kO94VuW+8HdtLVnvTrQYweEdnIVflaxRFwizHpsVjYayHXsfiEzx\nBABL0WEp2jfGqLL0emW4rqu0QkqJKLeug87uD9FWJ8uKJSsgqAP1OVrhctXJslwJbU9NYnZZXqPN\n+YgYsAEtY+hARw3jaqkDB7ayMkW7HBJVC0qBWs7jUIF3Lhf6iufyKu2yRajv6Wfz3F0FNvSfXTen\nLLk/XN6R69gqsP4Umqq/Qp69A4vjNhqhdTvCsM5wmF01igDhYwCQ11aw+YyB5bI22crgRWecL8LX\n/E/RRiEu5H5dRAoWkO7sdhO9snEKQN0XFZ6tXBOCqXu0QECPuZjK+xgqs7Oy03bxDLine truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x117a9eb70>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "square = ((0, 0), (0, 1), (1, 0), (1, 1))\n",
+    "\n",
+    "show(random_points(square))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "There doesn't seem to be any structure there. Let's try again and compare the two:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAEACAYAAACUHkKwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x117ab8b70>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(random_points(square))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "I'm still not seeing anything but random points. How about a right triangle?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAEACAYAAACUHkKwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10fa24be0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "right_triangle = ((0, 0), (0, 1), (1, 0))\n",
+    "\n",
+    "show(random_points(right_triangle))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We get a squished Serpinski triangle. How about a pentagon? (I'm lazy so I had Wolfram Alpha [compute the vertexes](https://www.wolframalpha.com/input/?i=vertexes+of+regular+pentagon).)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASoAAAEACAYAAAAEDTt/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1181532e8>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "pentagon = ((0.5, -0.688), (0.809, 0.262), (0., 0.850), (-0.809, 0.262), (-0.5, -0.688))\n",
+    "\n",
+    "show(random_points(pentagon))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "To clarify, let's try again, and with a different number of points:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASoAAAEACAYAAAAEDTt/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10f3e1828>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(random_points(pentagon, 6000))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASoAAAEACAYAAAAEDTt/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11835df28>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(random_points(pentagon, 8000))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "I definitely see a central hole, and five secondary holes surrounding that, and then, maybe 15 holes surrounding those? Or maybe not 15; hard to tell. Is a Sierpinski Pentagon a thing? I hadn't heard of it but a [quick search](https://www.google.com/search?q=sierpinski+pentagon) reveals that yes indeed, it is [a thing](http://ecademy.agnesscott.edu/~lriddle/ifs/pentagon/sierngon.htm), and it does have 15 holes surrounding the 5 holes. Let's try the hexagon:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAEACAYAAABVtcpZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11836ff60>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "hexagon = ((0.5, -0.866), (1, 0), (0.5, 0.866), (-0.5, 0.866), (-1, 0), (-0.5, -0.866))\n",
+    "\n",
+    "show(random_points(hexagon))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAEACAYAAABVtcpZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x117ae8470>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(random_points(hexagon, 10000))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can see a little of the six-fold symmetry, but it is not as clear as the triangle and pentagon."
+   ]
+  }
+ ],
+ "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": "3.5.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}