{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculations to find the quadratic form of the octahedral packing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P$ is the matrix of the quadratic form corresponding to $h_1^2 + h_2^2 - \\tilde{b}b = 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = matrix([\n",
    "    [0, -1/2, 0, 0],\n",
    "    [-1/2, 0, 0, 0],\n",
    "    [0, 0, 1, 0],\n",
    "    [0, 0, 0, 1],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 0 -2  0  0]\n",
       "[-2  0  0  0]\n",
       "[ 0  0  1  0]\n",
       "[ 0  0  0  1]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.inverse()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For an Apollonian packing, you can make a matrix, $W$, where the rows are the coordinates of each of the circles in a quadruple. In an octahedral packing, this is impossible, as there are six circles in a unit. You can try creating a $6\\times 4$ matrix, but, later, we end up needing to invert $WPW^T$, the product of a $6\\times 4$, $4\\times 4$, and $4\\times 6$ matrix, which is singular. Fortunately, the sextuples come in three pairs of circles. Each pair consists of circles that aren't tangent to each other. It turns out that the average of the coordinates of the circles in a pair is the same across the sextuple. So, we can make a matrix where the first three rows are the coordinates of circles from different pairs, i.e. three mutually tangent circles, and the fourth is the average of the coordinates in a pair. From this, you can recover the coordinates for all the circles in a sextuple.\n",
    "\n",
    "Here $W$ is such a matrix computed for the $(0, 0, 1, 1, 2, 2)$ root sextuple."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "W = matrix([\n",
    "    [2, 0, 0, 1],\n",
    "    [2, 0, 0, -1],\n",
    "    [-1, 1, 0, 0],\n",
    "    [6, 2, 2*sqrt(2), 0]\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we have $$\n",
    "    WPW^T = \\left(\\begin{matrix}\n",
    "        1 & -1 & -1 & -1\\\\\n",
    "        -1 & 1 & -1 & -1\\\\\n",
    "        -1 & -1 & 1 & -1\\\\\n",
    "        -1 & -1 & -1 & -1\n",
    "    \\end{matrix}\\right) = M\n",
    "$$\n",
    "So, inverting both sides, we get $$\n",
    "    (WPW^T)^{-1} = M^{-1}\n",
    "$$ $$\n",
    "    (W^T)^{-1}P^{-1}W^{-1} = M^{-1}\n",
    "$$ $$\n",
    "    P^{-1} = W^TM^{-1}W\n",
    ".$$\n",
    "\n",
    "Like in the case with Apollonian packings, this is true for any sextuple $W$. So, we can substitute an arbitrary sextuple for $W$ and it must be equal to $P^{-1}$, letting us derive some useful quadratic forms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 1 -1 -1 -2]\n",
       "[-1  1 -1 -2]\n",
       "[-1 -1  1 -2]\n",
       "[-2 -2 -2 -4]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = W * P * W.transpose()\n",
    "M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$M^{-1}$ is the matrix of the quadratic form, although it is nice to normalize it to have ones along the diagonal, which is fine since it is equal to zero.\n",
    "$$\n",
    "\\left(\\begin{matrix}\n",
    "    1 & 0 & 0 & -1/2\\\\\n",
    "    0 & 1 & 0 & -1/2\\\\\n",
    "    0 & 0 & 1 & -1/2\\\\\n",
    "    -1/2 & -1/2 & -1/2 & 1/4\n",
    "\\end{matrix}\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 1/2    0    0 -1/4]\n",
       "[   0  1/2    0 -1/4]\n",
       "[   0    0  1/2 -1/4]\n",
       "[-1/4 -1/4 -1/4  1/8]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.inverse()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "bt1 = var('bt1')\n",
    "b1  = var('b1')\n",
    "h11 = var('h11')\n",
    "h12 = var('h12')\n",
    "bt2 = var('bt2')\n",
    "b2  = var('b2')\n",
    "h21 = var('h21')\n",
    "h22 = var('h22')\n",
    "bt3 = var('bt3')\n",
    "b3  = var('b3')\n",
    "h31 = var('h31')\n",
    "h32 = var('h32')\n",
    "b5_avg = var('b5_avg')\n",
    "b_avg  = var('b_avg')\n",
    "h1_avg = var('h1_avg')\n",
    "h2_avg = var('h2_avg')\n",
    "\n",
    "\n",
    "W2 = matrix([\n",
    "    [bt1, b1, h11, h12],\n",
    "    [bt2, b2, h21, h22],\n",
    "    [bt3, b3, h31, h32],\n",
    "    [b5_avg, b_avg, h1_avg, h2_avg],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "D = W2.transpose() * M.inverse() * W2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 0 -2  0  0]\n",
       "[-2  0  0  0]\n",
       "[ 0  0  1  0]\n",
       "[ 0  0  0  1]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.inverse()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "b1^2 + b2^2 + b3^2 - b1*b_avg - b2*b_avg - b3*b_avg + 1/4*b_avg^2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 * factor(simplify(D[1][1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, we end up deriving the quadratic form $$\n",
    "    b_1^2 + b_2^2 + b_3^2 + b_{\\text{sum}}^2/4 - b_{\\text{sum}}(b_1 + b_2 + b_3) = 0\n",
    ".$$\n",
    "\n",
    "This means that, given three mutually tangent circles with curvatures $b_1,b_2,b_3$, there are two solutions for $b_{\\text{avg}}$, allowing us to derive two new sets of three mutually tangent circles with curvatures $b_1' = 2b_{\\text{avg}} - b_1$ etc."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculations to find the generators for the \"Apollonian\" Group for the octahedral packing\n",
    "\n",
    "We can try the Weyl group, since that is one way to derive the generators for the Apollonian group and see if it works. My worry is that it won't since the coordinate system is so different."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def weyl_generators(matrix, alphas):\n",
    "    retval = []\n",
    "    for alpha in alphas:\n",
    "        retval.append(identity_matrix(len(alphas)) - 2 * alpha * alpha.transpose() * matrix)\n",
    "    return retval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def standard_basis(dim):\n",
    "    return [ matrix(dim, 1, [0] * i + [1] + [0] * (dim - i - 1)) for i in range(dim) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[\n",
       "[-1  0  0  1]  [ 1  0  0  0]  [ 1  0  0  0]  [  1   0   0   0]\n",
       "[ 0  1  0  0]  [ 0 -1  0  1]  [ 0  1  0  0]  [  0   1   0   0]\n",
       "[ 0  0  1  0]  [ 0  0  1  0]  [ 0  0 -1  1]  [  0   0   1   0]\n",
       "[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [  1   1   1 1/2]\n",
       "]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_i = weyl_generators(2 * M.inverse(), standard_basis(4))\n",
    "S_i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "S1 = S_i[0]\n",
    "S2 = S_i[1]\n",
    "S3 = S_i[2]\n",
    "S4 = S_i[3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can test this out on the packing on page 3 of the Guettler and Mallows. The root sextuple is (-1, 2, 2, 4, 4 7), so the coordinates would be (-1, 2, 4, 6). After applying $s_1$, it should swap out $s_1$ for its pair, resulting in (7, 2, 4, 6). Likewise for $s_2$ and $s_3$. Then, for $s_4$, it should give the average between the triple (-1, 2, 4) and the other tangent triple, i.e. (-1, 2, 4, 14)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(7, 2, 4, 6)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "root = vector([-1, 2, 4, 6])\n",
    "S1 * root"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, 4, 4, 6)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S2 * root"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, 2, 2, 6)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S3 * root"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For some reason, I have no idea why, $s_4$ doesn't seem to work. Instead this weird matrix does appear to work."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, 2, 4, 8)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S4 * root"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, 2, 4, 14)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix([\n",
    "    [1, 0, 0, 0],\n",
    "    [0, 1, 0, 0],\n",
    "    [0, 0, 1, 0],\n",
    "    [4, 4, 4, -1]\n",
    "]) * root"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculations to find the curvatures of the dual circles of the octahedral packing\n",
    "\n",
    "Here the basic idea was to work out the intersection points of the circles and let the computer algebraically find the radii of the circles. The end up coming out, rather anticlimactically, to $(0, 0, \\sqrt 2, \\sqrt 2, \\sqrt 2, \\sqrt 2, 2\\sqrt 2, 2\\sqrt 2)$. It's rather satisfying and a bit surprising that the dual of the root sextuple of the octahedral packing is the root of the cubic packing, since the dual polyhedron of the octahedron is the cube."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def circle_from_points(pta, ptb, ptc):\n",
    "    a = var('a')\n",
    "    b = var('b')\n",
    "    r = var('r')\n",
    "    x = var('x')\n",
    "    y = var('y')\n",
    "    \n",
    "    circle_func = (x - a)^2 + (y - b)^2 == r^2\n",
    "    \n",
    "    eq1 = circle_func.subs(x == pta[0]).subs(y == pta[1])\n",
    "    eq2 = circle_func.subs(x == ptb[0]).subs(y == ptb[1])\n",
    "    eq3 = circle_func.subs(x == ptc[0]).subs(y == ptc[1])\n",
    "    \n",
    "    res = solve([eq1, eq2, eq3], a, b, r)[1]\n",
    "    \n",
    "    return (res[0].rhs(), res[1].rhs(), res[2].rhs())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1/2*sqrt(2), 1)\n",
      "sqrt(2)\n",
      "(1/4*sqrt(2), 0)\n",
      "2*sqrt(2)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 14 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "c1 = line([(-3, 1), (3, 1)])\n",
    "c2 = line([(-3, -1), (3, -1)])\n",
    "c3 = circle((-sqrt(2), 0), 1)\n",
    "c4 = circle((sqrt(2), 0), 1)\n",
    "c5 = circle((0, 1/2), 1/2)\n",
    "c6 = circle((0, -1/2), 1/2)\n",
    "\n",
    "b1 = line([(-sqrt(2), -3), (-sqrt(2), 3)], rgbcolor=(1, 0, 0))\n",
    "b2 = line([(sqrt(2), -3), (sqrt(2), 3)], rgbcolor=(1, 0, 0))\n",
    "\n",
    "x, y, r = circle_from_points((sqrt(2) / 3, 1 / 3), (0, 1), (sqrt(2), 1))\n",
    "print('({}, {})'.format(x, y))\n",
    "print(1 / r)\n",
    "\n",
    "b3 = circle(( x,  y), r, rgbcolor=(1, 0, 0))\n",
    "b4 = circle((-x,  y), r, rgbcolor=(1, 0, 0))\n",
    "b5 = circle(( x, -y), r, rgbcolor=(1, 0, 0))\n",
    "b6 = circle((-x, -y), r, rgbcolor=(1, 0, 0))\n",
    "\n",
    "x, y, r = circle_from_points((sqrt(2) / 3, 1 / 3), (0, 0), (sqrt(2) / 3, -1 / 3))\n",
    "print('({}, {})'.format(x, y))\n",
    "print(1 / r)\n",
    "\n",
    "b7 = circle(( x, y), r, rgbcolor=(1, 0, 0))\n",
    "b8 = circle((-x, y), r, rgbcolor=(1, 0, 0))\n",
    "\n",
    "show(c1 + c2 + c3 + c4 + c5 + c6 + b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The quadratic form for the cubical packing\n",
    "\n",
    "We first made a matrix, $W_c$, whose rows are the abbc coordinates of the circles in the root octuple. Then we found the row echelon form of that matrix, resulting in a system of linear relations the coordinates must satisfy. Then, from there, we could derive the rest of the coordinates from the first four (we chose the first four to be the \"cubicle\" from the Stange), allowing us to derive the quadratic form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def abbc_coords(b, h1, h2):\n",
    "    return [(h1^2 + h2^2 - 1) / b, b, h1, h2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The idea here is that by multiplying $W_c$ by an arbitrary vector in $\\mathbf{R}^8$ and setting that equal to $\\vec{0}$, we can find a basis for the null space of $W_c$, which will end up giving us a bunch of linear relations the curvatures must satisfy. The thing to notice is that $W_c\\vec{v} = 0$ for some $\\vec{v}\\in\\mathbf{R}^8$ gives the system of equations with coefficient matrix $W^T$. So finding the row echelon form of $W^T$ will give us the coefficients in the simplified system of linear relations, with each row equal to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[   1    0    0    0  1/2  1/2  1/2 -1/2]\n",
       "[   0    1    0    0  1/2  1/2 -1/2  1/2]\n",
       "[   0    0    1    0  1/2 -1/2  1/2  1/2]\n",
       "[   0    0    0    1 -1/2  1/2  1/2  1/2]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Wc = matrix([\n",
    "    [4, 0, 0, 1],\n",
    "    [0, 2, 0, 1],\n",
    "    [2, 1, -sqrt(2), -1],\n",
    "    [2, 1, sqrt(2), -1],\n",
    "    [2, 1, -sqrt(2), 1],\n",
    "    [2, 1, sqrt(2), 1],\n",
    "    [4, 0, 0, -1],\n",
    "    [0, 2, 0, -1],\n",
    "])\n",
    "\n",
    "Wc.transpose().echelon_form()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From here we can conclude that $$\n",
    "\\begin{align*}\n",
    "    2b_1 &= -b_5 - b_6 - b_7 + b_8\\\\\n",
    "    2b_2 &= -b_5 - b_6 + b_7 - b_8\\\\\n",
    "    2b_3 &= -b_5 + b_6 - b_7 - b_8\\\\\n",
    "    2b_4 &= b_5 - b_6 - b_7 - b_8\n",
    "\\end{align*}\n",
    "$$\n",
    "This differs slightly from Stange's system of equations because we put the circles in a slightly different order."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In any case, this gives us the tools to derive the full octuple from just four coordinates, letting us use those four coordinates to represent the entire octuple, and finally making $WPW^T$ nonsingular, allowing us to find the quadratic form the same way we did for the Descartes quadratic form and the octahedral quadratic form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "W = matrix([\n",
    "    [4, 0, 0, 1],\n",
    "    [0, 2, 0, 1],\n",
    "    [2, 1, -sqrt(2), -1],\n",
    "    [2, 1, sqrt(2), -1],\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 1 -3 -3 -3]\n",
       "[-3  1 -3 -3]\n",
       "[-3 -3  1 -3]\n",
       "[-3 -3 -3  1]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = W * P * W.transpose()\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 5 -3 -3 -3]\n",
       "[-3  5 -3 -3]\n",
       "[-3 -3  5 -3]\n",
       "[-3 -3 -3  5]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "32 * m.inverse()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[bt1  b1 h11 h21]\n",
       "[bt2  b2 h12 h22]\n",
       "[bt3  b3 h13 h23]\n",
       "[bt4  b4 h14 h24]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W2 = matrix([\n",
    "    [\n",
    "        var('bt' + str(i)),\n",
    "        var('b' + str(i)),\n",
    "        var('h1' + str(i)),\n",
    "        var('h2' + str(i)),\n",
    "    ] for i in range(1, 5)\n",
    "])\n",
    "W2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "D = W2.transpose() * m.inverse() * W2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5*b1^2 - 6*b1*b2 + 5*b2^2 - 6*b1*b3 - 6*b2*b3 + 5*b3^2 - 6*b1*b4 - 6*b2*b4 - 6*b3*b4 + 5*b4^2"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "32 * simplify(expand(D[1][1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This spits out the lovely quadratic form $$\n",
    "    8(b_1^2 + b_2^2 + b_3^2 + b_4^2) = 3(b_1 + b_2 + b_3 + b_4)^2\n",
    ".$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the generators for the Weyl group. They might or might not be generators for the packing itself, however."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[\n",
       "[-1  6  6  6]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
       "[ 0  1  0  0]  [ 6 -1  6  6]  [ 0  1  0  0]  [ 0  1  0  0]\n",
       "[ 0  0  1  0]  [ 0  0  1  0]  [ 6  6 -1  6]  [ 0  0  1  0]\n",
       "[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 6  6  6 -1]\n",
       "]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S_i = weyl_generators(m, standard_basis(4))\n",
    "S_i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ 1 -3 -3 -3 -1 -1 -1 -5]\n",
       "[-3  1 -3 -3 -1 -1 -5 -1]\n",
       "[-3 -3  1 -3 -1 -5 -1 -1]\n",
       "[-3 -3 -3  1 -5 -1 -1 -1]\n",
       "[-1 -1 -1 -5  1 -3 -3 -3]\n",
       "[-1 -1 -5 -1 -3  1 -3 -3]\n",
       "[-1 -5 -1 -1 -3 -3  1 -3]\n",
       "[-5 -1 -1 -1 -3 -3 -3  1]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Wc * P * Wc.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generalization of this method\n",
    "This method seems remarkably general. Given a matrix representing a root unit of a packing, we can find the linear relation between the coordinates, and thus represent the packing with only four coordinates, allowing us to find the quadratic form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def quadform_from_root(root_matrix):\n",
    "    P = matrix([\n",
    "        [0, -1/2, 0, 0],\n",
    "        [-1/2, 0, 0, 0],\n",
    "        [0, 0, 1, 0],\n",
    "        [0, 0, 0, 1],\n",
    "    ])\n",
    "    \n",
    "    # step 1: find linear relation between coords\n",
    "    relation = root_matrix.transpose().rref() * vector([ var('b' + str(i)) for i in range(1, root_matrix.dimensions()[0] + 1)])\n",
    "    \n",
    "    # step 2: find matrix of quadratic form\n",
    "    W = root_matrix[0:4]\n",
    "    M = W * P * W.transpose()\n",
    "    \n",
    "    # step 3: repeat with arbitrary matrix\n",
    "    W2 = matrix([\n",
    "        [\n",
    "            var('bt' + str(i)),\n",
    "            var('b' + str(i)),\n",
    "            var('h1' + str(i)),\n",
    "            var('h2' + str(i)),\n",
    "        ] for i in range(1, 5)\n",
    "    ])\n",
    "    D = factor(simplify(expand(W2.transpose() * M.inverse() * W2)))\n",
    "    \n",
    "    return relation, M.inverse(), D[1][1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(2*b1 + b5 + b6 + b7 - b8, 2*b2 + b5 + b6 - b7 + b8, 2*b3 + b5 - b6 + b7 + b8, 2*b4 - b5 + b6 + b7 + b8)\n",
      "[ 5 -3 -3 -3]\n",
      "[-3  5 -3 -3]\n",
      "[-3 -3  5 -3]\n",
      "[-3 -3 -3  5]\n",
      "5*b1^2 - 6*b1*b2 + 5*b2^2 - 6*b1*b3 - 6*b2*b3 + 5*b3^2 - 6*b1*b4 - 6*b2*b4 - 6*b3*b4 + 5*b4^2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[\n",
       "[-9  6  6  6]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
       "[ 0  1  0  0]  [ 6 -9  6  6]  [ 0  1  0  0]  [ 0  1  0  0]\n",
       "[ 0  0  1  0]  [ 0  0  1  0]  [ 6  6 -9  6]  [ 0  0  1  0]\n",
       "[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 6  6  6 -9]\n",
       "]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# cubical\n",
    "relation, mat, equation = quadform_from_root(Wc)\n",
    "print(2 * relation)\n",
    "print(32 * mat)\n",
    "print(32 * equation)\n",
    "weyl_generators(32 * mat, standard_basis(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(b1 + b5 + b6, b2 - b5, b3 - b6, b4 + b5 + b6)\n",
      "[ 1 -2 -2 -1]\n",
      "[-2  4  0 -2]\n",
      "[-2  0  4 -2]\n",
      "[-1 -2 -2  1]\n",
      "b1^2 - 4*b1*b2 + 4*b2^2 - 4*b1*b3 + 4*b3^2 - 2*b1*b4 - 4*b2*b4 - 4*b3*b4 + b4^2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[\n",
       "[-1  4  4  2]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
       "[ 0  1  0  0]  [ 4 -7  0  4]  [ 0  1  0  0]  [ 0  1  0  0]\n",
       "[ 0  0  1  0]  [ 0  0  1  0]  [ 4  0 -7  4]  [ 0  0  1  0]\n",
       "[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 2  4  4 -1]\n",
       "]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# octahedral\n",
    "relation, mat, equation = quadform_from_root(matrix([\n",
    "    [2, 0, 0, 1],\n",
    "    [2, 0, 0, -1],\n",
    "    [-1, 1, 0, 0],\n",
    "    [4, 2, 2*sqrt(2), -1],\n",
    "    [4, 2, 2*sqrt(2), 1],\n",
    "    [7, 1, 2*sqrt(2), 0],\n",
    "]))\n",
    "print(relation)\n",
    "print(8 * mat)\n",
    "print(8 * equation)\n",
    "weyl_generators(8 * mat, standard_basis(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(b1, b2, b3, b4)\n",
      "[ 1 -1 -1 -1]\n",
      "[-1  1 -1 -1]\n",
      "[-1 -1  1 -1]\n",
      "[-1 -1 -1  1]\n",
      "b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[\n",
       "[-1  2  2  2]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
       "[ 0  1  0  0]  [ 2 -1  2  2]  [ 0  1  0  0]  [ 0  1  0  0]\n",
       "[ 0  0  1  0]  [ 0  0  1  0]  [ 2  2 -1  2]  [ 0  0  1  0]\n",
       "[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 2  2  2 -1]\n",
       "]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# tetrahedral\n",
    "relation, mat, equation = quadform_from_root(matrix([\n",
    "    [2, 0, 0, 1],\n",
    "    [2, 0, 0, -1],\n",
    "    [-1, 1, 0, 0],\n",
    "    [3, 1, 2, 0]\n",
    "]))\n",
    "print(relation)\n",
    "print(4 * mat)\n",
    "print(4 * equation)\n",
    "weyl_generators(4 * mat, standard_basis(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I suspect that it is highly dependent on the order of the circles in the root. Interestingly, it looks like the only relation it was able to deduce for the tetrahedral packing is $b_1, b_2, b_3, b_4 = 0$, meaning there is no null space, as we'd expect. The octahedral quadratic form we get is very different, which isn't surprising, I guess, since this is a very different coordinate system, but I'm not sure if it's right. It feels very strange, since it's only really two dimensional, rather than four, and it doesn't quite look like the equations hold up. But it does absolutely work for the tetrahedral packing, which is awesome."
   ]
  }
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