diff --git a/Apollonian Circle Packings.ipynb b/Apollonian Circle Packings.ipynb
index d6c06f7..92c155f 100644
--- a/Apollonian Circle Packings.ipynb	
+++ b/Apollonian Circle Packings.ipynb	
@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -30,7 +30,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -42,7 +42,7 @@
        "[ 0  0  0  1]"
       ]
      },
-     "execution_count": 2,
+     "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -62,7 +62,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 6,
    "metadata": {
     "scrolled": true
    },
@@ -113,7 +113,7 @@
        "[-2 -2 -2 -4]"
       ]
      },
-     "execution_count": 4,
+     "execution_count": 6,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -140,7 +140,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 7,
    "metadata": {
     "scrolled": true
    },
@@ -154,7 +154,7 @@
        "[-1/4 -1/4 -1/4  1/8]"
       ]
      },
-     "execution_count": 5,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -165,7 +165,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -197,7 +197,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -206,7 +206,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -218,7 +218,7 @@
        "[ 0  0  0  1]"
       ]
      },
-     "execution_count": 8,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -229,7 +229,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -238,7 +238,7 @@
        "b1^2 + b2^2 + b3^2 - b1*b_avg - b2*b_avg - b3*b_avg + 1/4*b_avg^2"
       ]
      },
-     "execution_count": 9,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -269,20 +269,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 12,
    "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",
+    "        scale_factor = (alpha.transpose() * matrix * alpha)[0][0]\n",
+    "        retval.append(identity_matrix(len(alphas)) - 2 * alpha * alpha.transpose() * matrix / scale_factor)\n",
     "    return retval"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -292,33 +293,56 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[  2   0   0  -1]\n",
+       "[  0   2   0  -1]\n",
+       "[  0   0   2  -1]\n",
+       "[ -1  -1  -1 1/2]"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "4 * M.inverse()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
    "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",
+       "[-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], [ 4  4  4 -1]\n",
        "]"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 15,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "S_i = weyl_generators(2 * M.inverse(), standard_basis(4))\n",
+    "S_i = weyl_generators(4 * M.inverse(), standard_basis(4))\n",
     "S_i"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -337,7 +361,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
@@ -346,7 +370,7 @@
        "(7, 2, 4, 6)"
       ]
      },
-     "execution_count": 14,
+     "execution_count": 17,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -358,7 +382,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
@@ -367,7 +391,7 @@
        "(-1, 4, 4, 6)"
       ]
      },
-     "execution_count": 15,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -378,7 +402,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [
     {
@@ -387,7 +411,7 @@
        "(-1, 2, 2, 6)"
       ]
      },
-     "execution_count": 16,
+     "execution_count": 19,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -396,36 +420,9 @@
     "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,
+   "execution_count": 20,
    "metadata": {},
    "outputs": [
     {
@@ -434,18 +431,13 @@
        "(-1, 2, 4, 14)"
       ]
      },
-     "execution_count": 18,
+     "execution_count": 20,
      "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"
+    "S4 * root"
    ]
   },
   {
@@ -459,7 +451,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -483,7 +475,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 22,
    "metadata": {},
    "outputs": [
     {
@@ -548,7 +540,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -565,7 +557,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [
     {
@@ -577,7 +569,7 @@
        "[   0    0    0    1 -1/2  1/2  1/2  1/2]"
       ]
      },
-     "execution_count": 22,
+     "execution_count": 24,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -594,7 +586,7 @@
     "    [0, 2, 0, -1],\n",
     "])\n",
     "\n",
-    "Wc.transpose().echelon_form()"
+    "Wc.transpose().rref()"
    ]
   },
   {
@@ -621,7 +613,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -635,7 +627,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 26,
    "metadata": {},
    "outputs": [
     {
@@ -647,7 +639,7 @@
        "[-3 -3 -3  1]"
       ]
      },
-     "execution_count": 24,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -659,30 +651,30 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 27,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "[ 5 -3 -3 -3]\n",
-       "[-3  5 -3 -3]\n",
-       "[-3 -3  5 -3]\n",
-       "[-3 -3 -3  5]"
+       "[   2 -6/5 -6/5 -6/5]\n",
+       "[-6/5    2 -6/5 -6/5]\n",
+       "[-6/5 -6/5    2 -6/5]\n",
+       "[-6/5 -6/5 -6/5    2]"
       ]
      },
-     "execution_count": 25,
+     "execution_count": 27,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "32 * m.inverse()"
+    "32 * m.inverse() * 2/5"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 28,
    "metadata": {},
    "outputs": [
     {
@@ -694,7 +686,7 @@
        "[bt4  b4 h14 h24]"
       ]
      },
-     "execution_count": 26,
+     "execution_count": 28,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -713,7 +705,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 29,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -722,7 +714,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 30,
    "metadata": {},
    "outputs": [
     {
@@ -731,7 +723,7 @@
        "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,
+     "execution_count": 30,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -758,7 +750,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 31,
    "metadata": {
     "scrolled": true
    },
@@ -767,26 +759,31 @@
      "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",
+       "[ -1 6/5 6/5 6/5]  [  1   0   0   0]  [  1   0   0   0]\n",
+       "[  0   1   0   0]  [6/5  -1 6/5 6/5]  [  0   1   0   0]\n",
+       "[  0   0   1   0]  [  0   0   1   0]  [6/5 6/5  -1 6/5]\n",
+       "[  0   0   0   1], [  0   0   0   1], [  0   0   0   1],\n",
+       "\n",
+       "[  1   0   0   0]\n",
+       "[  0   1   0   0]\n",
+       "[  0   0   1   0]\n",
+       "[6/5 6/5 6/5  -1]\n",
        "]"
       ]
      },
-     "execution_count": 29,
+     "execution_count": 31,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "S_i = weyl_generators(m, standard_basis(4))\n",
+    "S_i = weyl_generators(m.inverse(), standard_basis(4))\n",
     "S_i"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 32,
    "metadata": {
     "scrolled": true
    },
@@ -794,23 +791,23 @@
     {
      "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]"
+       "[   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]\n",
+       "[   0    0    0    0    0    0    0    0]\n",
+       "[   0    0    0    0    0    0    0    0]\n",
+       "[   0    0    0    0    0    0    0    0]\n",
+       "[   0    0    0    0    0    0    0    0]"
       ]
      },
-     "execution_count": 30,
+     "execution_count": 32,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "Wc * P * Wc.transpose()"
+    "(Wc * P * Wc.transpose()).rref()"
    ]
   },
   {
@@ -823,7 +820,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 33,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -839,7 +836,7 @@
     "    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",
+    "    W = root_matrix[-4:]\n",
     "    M = W * P * W.transpose()\n",
     "    \n",
     "    # step 3: repeat with arbitrary matrix\n",
@@ -858,7 +855,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 34,
    "metadata": {},
    "outputs": [
     {
@@ -877,14 +874,19 @@
      "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",
+       "[ -1 6/5 6/5 6/5]  [  1   0   0   0]  [  1   0   0   0]\n",
+       "[  0   1   0   0]  [6/5  -1 6/5 6/5]  [  0   1   0   0]\n",
+       "[  0   0   1   0]  [  0   0   1   0]  [6/5 6/5  -1 6/5]\n",
+       "[  0   0   0   1], [  0   0   0   1], [  0   0   0   1],\n",
+       "\n",
+       "[  1   0   0   0]\n",
+       "[  0   1   0   0]\n",
+       "[  0   0   1   0]\n",
+       "[6/5 6/5 6/5  -1]\n",
        "]"
       ]
      },
-     "execution_count": 32,
+     "execution_count": 34,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -900,7 +902,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 35,
    "metadata": {},
    "outputs": [
     {
@@ -920,13 +922,13 @@
       "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  1  0  0]  [ 1 -1  0  1]  [ 0  1  0  0]  [ 0  1  0  0]\n",
+       "[ 0  0  1  0]  [ 0  0  1  0]  [ 1  0 -1  1]  [ 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,
+     "execution_count": 35,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -949,7 +951,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 36,
    "metadata": {},
    "outputs": [
     {
@@ -975,7 +977,7 @@
        "]"
       ]
      },
-     "execution_count": 34,
+     "execution_count": 36,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1019,7 +1021,8 @@
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
    "version": "3.9.2"
-  }
+  },
+  "name": "Apollonian Circle Packings.ipynb"
  },
  "nbformat": 4,
  "nbformat_minor": 4