Trig.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4ad1dbeb",
   "metadata": {},
   "source": [
    "[Home](Home.ipynb)\n",
    "\n",
    "# Trigonometry and Polyhedra\n",
    "\n",
    "We work backwards from the unit circle on the Argand Plane, and Euler's formula. Complex numbers make their appearance early on, as another number type, so we're ready for trig.\n",
    "\n",
    "\n",
    "$$\n",
    "e^{ \\pm i\\theta } = \\cos \\theta \\pm i\\sin \\theta\n",
    "$$\n",
    "\n",
    "$$\n",
    "e^{ i\\pi} = -1\n",
    "$$\n",
    "\n",
    "When we're not on the Complex Plane, but on an omni-triangulated sphere, or zoomed in to a beach (flat surface) a lot of the same concepts will apply.  Triangles and tetrahedra pop up everywhere.\n",
    "\n",
    "We practice using Python as a graphing calculator by doing plots with pandas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "407c6360",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b5550aa7",
   "metadata": {},
   "outputs": [],
   "source": [
    "domain = pd.Series(np.linspace(-2*np.pi, 2*np.pi, 200))\n",
    "\n",
    "sine   = pd.Series(np.sin(domain))\n",
    "cosine = pd.Series(np.cos(domain))\n",
    "\n",
    "table  = pd.DataFrame({\"Domain\":domain, \"Sine\":sine, \"Cosine\":cosine})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "39a073d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Domain</th>\n",
       "      <th>Sine</th>\n",
       "      <th>Cosine</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>-0.284164</td>\n",
       "      <td>-0.280355</td>\n",
       "      <td>0.959896</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>-0.221017</td>\n",
       "      <td>-0.219222</td>\n",
       "      <td>0.975675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>-0.157869</td>\n",
       "      <td>-0.157214</td>\n",
       "      <td>0.987565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>-0.094721</td>\n",
       "      <td>-0.094580</td>\n",
       "      <td>0.995517</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>-0.031574</td>\n",
       "      <td>-0.031569</td>\n",
       "      <td>0.999502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>0.031574</td>\n",
       "      <td>0.031569</td>\n",
       "      <td>0.999502</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>101</th>\n",
       "      <td>0.094721</td>\n",
       "      <td>0.094580</td>\n",
       "      <td>0.995517</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>102</th>\n",
       "      <td>0.157869</td>\n",
       "      <td>0.157214</td>\n",
       "      <td>0.987565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>103</th>\n",
       "      <td>0.221017</td>\n",
       "      <td>0.219222</td>\n",
       "      <td>0.975675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>104</th>\n",
       "      <td>0.284164</td>\n",
       "      <td>0.280355</td>\n",
       "      <td>0.959896</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       Domain      Sine    Cosine\n",
       "95  -0.284164 -0.280355  0.959896\n",
       "96  -0.221017 -0.219222  0.975675\n",
       "97  -0.157869 -0.157214  0.987565\n",
       "98  -0.094721 -0.094580  0.995517\n",
       "99  -0.031574 -0.031569  0.999502\n",
       "100  0.031574  0.031569  0.999502\n",
       "101  0.094721  0.094580  0.995517\n",
       "102  0.157869  0.157214  0.987565\n",
       "103  0.221017  0.219222  0.975675\n",
       "104  0.284164  0.280355  0.959896"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table[95:105]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ebb4bb53",
   "metadata": {},
   "outputs": [],
   "source": [
    "# ? table.plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "57a7fb5c",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "table.plot(x= \"Domain\", y=[\"Sine\", \"Cosine\"], grid=True, title=\"Trig Functions\");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3faf963",
   "metadata": {},
   "source": [
    "### The Omnitriangulated Sphere\n",
    "\n",
    "Starting with a spherical icosahedron, and subdividing each of the twenty triangles, with arcs parallel to each of triangle's edges, begets an omni-triangulated sphere.\n",
    "\n",
    "<a data-flickr-embed=\"true\" href=\"https://www.flickr.com/photos/kirbyurner/3859887274/in/dateposted-public/\" title=\"6-Frequency\"><img src=\"https://live.staticflickr.com/2570/3859887274_09d9387982.jpg\" width=\"375\" height=\"500\" alt=\"6-Frequency\"></a><script async src=\"//embedr.flickr.com/assets/client-code.js\" charset=\"utf-8\"></script>\n",
    "\n",
    "A first concept to visit at this point is that of Descartes' Deficit.  As the number of triangles increases, the degrees around each vertex approaches 360, but never gets there, because each vertex contributes a tiny amount to the 720 \"missing degrees\" overall.  \n",
    "\n",
    "In other words, the number of degrees less than 360, around every vertex, add up to 720.  Why not confirm this with a cube, octahedron, tetrahedron (which latter also has 720 degrees).\n",
    "\n",
    "Each vertex of a regular tetrahedron has 60 + 60 + 60 = 180 degrees, with is 180 less than 360.  Each of four vertexes contributes the \"missing 180\" adding up to 720.\n",
    "\n",
    "Other identities apply regarding our sphere, such as:\n",
    "\n",
    "* $V + F = E + 2$\n",
    "* $N:F:E = 1:2:3$ where $N = V-2$\n",
    "* $V = 10\\ F^{2}\\ +\\ 2$ where $F$ = Frequency"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5932cc7d",
   "metadata": {},
   "source": [
    "*Recommended Reading:*\n",
    "\n",
    "[Divided Spheres](http://www.dividedspheres.com/) by Ed Popko"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5a21b53",
   "metadata": {},
   "source": [
    "### The Goldberg Polyhedra\n",
    "\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/5/54/Goldberg_polyhedron_6_4.png\" width=\"300\" height=\"300\">\n",
    "\n",
    "Omnitriangulated spheres of specific frequencies (number of intervals along each major edge) may be turned into polyhedrons with all hexagonal and twelve pentagonal facets.  These are known as Goldberg Polyhedra, named for Michael Goldberg."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8d319d1",
   "metadata": {},
   "source": [
    "Resources:\n",
    "\n",
    "* [Goldberg Polyhedra](https://en.wikipedia.org/wiki/Goldberg_polyhedron)\n",
    "* [Antiprism](http://antiprism.com)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e678d526",
   "metadata": {},
   "source": [
    "### The Waterman Polyhedra\n",
    "\n",
    "<a data-flickr-embed=\"true\" href=\"https://www.flickr.com/photos/kirbyurner/51395771856/in/dateposted-public/\" title=\"Waterman Polyhedron\"><img src=\"https://live.staticflickr.com/65535/51395771856_d197097fff_w.jpg\" width=\"389\" height=\"400\" alt=\"Waterman Polyhedron\"></a><script async src=\"//embedr.flickr.com/assets/client-code.js\" charset=\"utf-8\"></script>\n",
    "\n",
    "Start your polyhedron with its body center, and all starting vertexes, equally distant from that center, at the centers of CCP spheres.  See [Quadrays](Quadrays.ipynb) for more about the CCP lattice.\n",
    "\n",
    "In other words, start with a maximal radius sweeping out points of contact with maximally equi-distant CCP spheres. \n",
    "\n",
    "Now compute the maximum convex hull incorporating all these vertexes *and any others of lesser distance* nevertheless at CCP ball centers. This process defines the Waterman Polyhedron for that radius.\n",
    "\n",
    "The simplest Waterman would therefore be the cuboctahedron of twelve CCP balls around one.  As the radius increases, the polyhedron becomes more spherical.\n",
    "\n",
    "Watermans all have whole number tetravolumes given they may be subdivided into all-IVM-corner tetrahedra (each corner is a CCP ball center). Those have [whole number tetravolume](Quadrays.ipynb), so their sum must as well."
   ]
  },
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    "Resources:\n",
    "\n",
    "* [Waterman Polyhedron](https://en.wikipedia.org/wiki/Waterman_polyhedron)\n",
    "* [Paul Bourke on Watermans](http://paulbourke.net/geometry/waterman/)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "920c4f1e",
   "metadata": {},
   "source": [
    "### Space-filling Tetrahedra\n",
    "\n",
    "This topic connects several mathematicians, including Michael Goldberg, to Buckminster Fuller's invented vocabulary of A, B, S, T & E modules, Mite (AAB), Sytes, Kites and so on.\n",
    "\n",
    "<a data-flickr-embed=\"true\" href=\"https://www.flickr.com/photos/kirbyurner/4073112337/in/photolist-25JufgG-NYVxaA-ujipN3-f75zUP-8ryECF-8ryEix-7pQH9f-7pLP7a-7mcmne-7cVLoM-5zY9gA-5zTRjp-7k4Eid-7jZLe2-7k4Ejf-7k4Em5-7jZLhp\" title=\"A &amp; B Modules\"><img src=\"https://live.staticflickr.com/3508/4073112337_4d0bb5b0b4_o.gif\" width=\"320\" height=\"240\" alt=\"A &amp; B Modules\"></a><script async src=\"//embedr.flickr.com/assets/client-code.js\" charset=\"utf-8\"></script>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "81a42c71",
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import YouTubeVideo"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a52f7c39",
   "metadata": {},
   "source": [
    "Curate your own Youtubes?"
   ]
  }
 ],
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