diff --git a/Apollonian Circle Packings.ipynb b/Apollonian Circle Packings.ipynb
index 92c155f..7ee52b5 100644
--- a/Apollonian Circle Packings.ipynb	
+++ b/Apollonian Circle Packings.ipynb	
@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -30,7 +30,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [
     {
@@ -42,7 +42,7 @@
        "[ 0  0  0  1]"
       ]
      },
-     "execution_count": 4,
+     "execution_count": 2,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -62,7 +62,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 4,
    "metadata": {
     "scrolled": true
    },
@@ -113,7 +113,7 @@
        "[-2 -2 -2 -4]"
       ]
      },
-     "execution_count": 6,
+     "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -140,7 +140,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 5,
    "metadata": {
     "scrolled": true
    },
@@ -154,7 +154,7 @@
        "[-1/4 -1/4 -1/4  1/8]"
       ]
      },
-     "execution_count": 7,
+     "execution_count": 5,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -165,7 +165,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -197,7 +197,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -206,7 +206,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -218,7 +218,7 @@
        "[ 0  0  0  1]"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -229,7 +229,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 9,
    "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": 11,
+     "execution_count": 9,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -269,7 +269,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -283,7 +283,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -293,7 +293,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
@@ -305,7 +305,7 @@
        "[ -1  -1  -1 1/2]"
       ]
      },
-     "execution_count": 14,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -316,7 +316,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
@@ -330,7 +330,7 @@
        "]"
       ]
      },
-     "execution_count": 15,
+     "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -342,7 +342,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -361,7 +361,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
@@ -370,7 +370,7 @@
        "(7, 2, 4, 6)"
       ]
      },
-     "execution_count": 17,
+     "execution_count": 15,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -382,7 +382,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
@@ -391,7 +391,7 @@
        "(-1, 4, 4, 6)"
       ]
      },
-     "execution_count": 18,
+     "execution_count": 16,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -402,7 +402,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
@@ -411,7 +411,7 @@
        "(-1, 2, 2, 6)"
       ]
      },
-     "execution_count": 19,
+     "execution_count": 17,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -422,7 +422,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
@@ -431,7 +431,7 @@
        "(-1, 2, 4, 14)"
       ]
      },
-     "execution_count": 20,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -451,7 +451,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -475,7 +475,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 20,
    "metadata": {},
    "outputs": [
     {
@@ -540,7 +540,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -557,7 +557,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 22,
    "metadata": {},
    "outputs": [
     {
@@ -569,7 +569,7 @@
        "[   0    0    0    1 -1/2  1/2  1/2  1/2]"
       ]
      },
-     "execution_count": 24,
+     "execution_count": 22,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -613,7 +613,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -627,7 +627,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [
     {
@@ -639,7 +639,7 @@
        "[-3 -3 -3  1]"
       ]
      },
-     "execution_count": 26,
+     "execution_count": 24,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -651,7 +651,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [
     {
@@ -663,7 +663,7 @@
        "[-6/5 -6/5 -6/5    2]"
       ]
      },
-     "execution_count": 27,
+     "execution_count": 25,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -674,7 +674,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 26,
    "metadata": {},
    "outputs": [
     {
@@ -686,7 +686,7 @@
        "[bt4  b4 h14 h24]"
       ]
      },
-     "execution_count": 28,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -705,7 +705,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 27,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -714,7 +714,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 28,
    "metadata": {},
    "outputs": [
     {
@@ -723,7 +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": 30,
+     "execution_count": 28,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -750,7 +750,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 29,
    "metadata": {
     "scrolled": true
    },
@@ -771,7 +771,7 @@
        "]"
       ]
      },
-     "execution_count": 31,
+     "execution_count": 29,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -783,7 +783,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 30,
    "metadata": {
     "scrolled": true
    },
@@ -801,7 +801,7 @@
        "[   0    0    0    0    0    0    0    0]"
       ]
      },
-     "execution_count": 32,
+     "execution_count": 30,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -820,11 +820,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 31,
    "metadata": {},
    "outputs": [],
    "source": [
     "def quadform_from_root(root_matrix):\n",
+    "    n = root_matrix.dimensions()[1]\n",
     "    P = matrix([\n",
     "        [0, -1/2, 0, 0],\n",
     "        [-1/2, 0, 0, 0],\n",
@@ -833,8 +834,11 @@
     "    ])\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",
+    "    relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 1)]) * root_matrix.transpose().rref()\n",
+    "    relations = []\n",
+    "    for i, expr in enumerate(relations_temp):\n",
+    "        relations.append(var('b' + str(i + 1)) == expr)\n",
+    "        \n",
     "    # step 2: find matrix of quadratic form\n",
     "    W = root_matrix[-4:]\n",
     "    M = W * P * W.transpose()\n",
@@ -850,28 +854,131 @@
     "    ])\n",
     "    D = factor(simplify(expand(W2.transpose() * M.inverse() * W2)))\n",
     "    \n",
-    "    return relation, M.inverse(), D[1][1]"
+    "    return relations[4:], M.inverse(), D[1][1]"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "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/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[2*b5 == b1 + b2 + b3 - b4,\n",
+       " 2*b6 == b1 + b2 - b3 + b4,\n",
+       " 2*b7 == b1 - b2 + b3 + b4,\n",
+       " 2*b8 == -b1 + b2 + b3 + b4]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     },
     {
      "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "5 & -3 & -3 & -3 \\\\\n",
+       "-3 & 5 & -3 & -3 \\\\\n",
+       "-3 & -3 & 5 & -3 \\\\\n",
+       "-3 & -3 & -3 & 5\n",
+       "\\end{array}\\right)</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "5 & -3 & -3 & -3 \\\\\n",
+       "-3 & 5 & -3 & -3 \\\\\n",
+       "-3 & -3 & 5 & -3 \\\\\n",
+       "-3 & -3 & -3 & 5\n",
+       "\\end{array}\\right)\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[ 5 -3 -3 -3]\n",
+       "[-3  5 -3 -3]\n",
+       "[-3 -3  5 -3]\n",
+       "[-3 -3 -3  5]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, b_{1}^{2} - 6 \\, b_{1} b_{2} + 5 \\, b_{2}^{2} - 6 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 5 \\, b_{3}^{2} - 6 \\, b_{1} b_{4} - 6 \\, b_{2} b_{4} - 6 \\, b_{3} b_{4} + 5 \\, b_{4}^{2}</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, b_{1}^{2} - 6 \\, b_{1} b_{2} + 5 \\, b_{2}^{2} - 6 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 5 \\, b_{3}^{2} - 6 \\, b_{1} b_{4} - 6 \\, b_{2} b_{4} - 6 \\, b_{3} b_{4} + 5 \\, b_{4}^{2}\n",
+       "\\end{math}"
+      ],
+      "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"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
+       "\\end{array}\\right)\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
+       "\\end{array}\\right)\\right]\n",
+       "\\end{math}"
+      ],
       "text/plain": [
        "[\n",
        "[ -1 6/5 6/5 6/5]  [  1   0   0   0]  [  1   0   0   0]\n",
@@ -886,39 +993,138 @@
        "]"
       ]
      },
-     "execution_count": 34,
      "metadata": {},
-     "output_type": "execute_result"
+     "output_type": "display_data"
     }
    ],
    "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))"
+    "show([2 * eq for eq in relation])\n",
+    "show(32 * mat)\n",
+    "show(32 * equation)\n",
+    "show(weyl_generators(32 * mat, standard_basis(4)))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "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/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{1} - b_{2} + b_{4}, b_{6} = b_{1} - b_{3} + b_{4}\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{1} - b_{2} + b_{4}, b_{6} = b_{1} - b_{3} + b_{4}\\right]\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[b5 == b1 - b2 + b4, b6 == b1 - b3 + b4]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     },
     {
      "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "1 & -2 & -2 & -1 \\\\\n",
+       "-2 & 4 & 0 & -2 \\\\\n",
+       "-2 & 0 & 4 & -2 \\\\\n",
+       "-1 & -2 & -2 & 1\n",
+       "\\end{array}\\right)</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "1 & -2 & -2 & -1 \\\\\n",
+       "-2 & 4 & 0 & -2 \\\\\n",
+       "-2 & 0 & 4 & -2 \\\\\n",
+       "-1 & -2 & -2 & 1\n",
+       "\\end{array}\\right)\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[ 1 -2 -2 -1]\n",
+       "[-2  4  0 -2]\n",
+       "[-2  0  4 -2]\n",
+       "[-1 -2 -2  1]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 4 \\, b_{1} b_{2} + 4 \\, b_{2}^{2} - 4 \\, b_{1} b_{3} + 4 \\, b_{3}^{2} - 2 \\, b_{1} b_{4} - 4 \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 4 \\, b_{1} b_{2} + 4 \\, b_{2}^{2} - 4 \\, b_{1} b_{3} + 4 \\, b_{3}^{2} - 2 \\, b_{1} b_{4} - 4 \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + b_{4}^{2}\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "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"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & 4 & 4 & 2 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "1 & -1 & 0 & 1 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "1 & 0 & -1 & 1 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "2 & 4 & 4 & -1\n",
+       "\\end{array}\\right)\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & 4 & 4 & 2 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "1 & -1 & 0 & 1 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "1 & 0 & -1 & 1 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "2 & 4 & 4 & -1\n",
+       "\\end{array}\\right)\\right]\n",
+       "\\end{math}"
+      ],
       "text/plain": [
        "[\n",
        "[-1  4  4  2]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
@@ -928,9 +1134,8 @@
        "]"
       ]
      },
-     "execution_count": 35,
      "metadata": {},
-     "output_type": "execute_result"
+     "output_type": "display_data"
     }
    ],
    "source": [
@@ -943,31 +1148,131 @@
     "    [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))"
+    "show(relation)\n",
+    "show(8 * mat)\n",
+    "show(8 * equation)\n",
+    "show(weyl_generators(8 * mat, standard_basis(4)))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "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/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     },
     {
      "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "1 & -1 & -1 & -1 \\\\\n",
+       "-1 & 1 & -1 & -1 \\\\\n",
+       "-1 & -1 & 1 & -1 \\\\\n",
+       "-1 & -1 & -1 & 1\n",
+       "\\end{array}\\right)</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "1 & -1 & -1 & -1 \\\\\n",
+       "-1 & 1 & -1 & -1 \\\\\n",
+       "-1 & -1 & 1 & -1 \\\\\n",
+       "-1 & -1 & -1 & 1\n",
+       "\\end{array}\\right)\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[ 1 -1 -1 -1]\n",
+       "[-1  1 -1 -1]\n",
+       "[-1 -1  1 -1]\n",
+       "[-1 -1 -1  1]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "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"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & 2 & 2 & 2 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "2 & -1 & 2 & 2 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "2 & 2 & -1 & 2 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "2 & 2 & 2 & -1\n",
+       "\\end{array}\\right)\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & 2 & 2 & 2 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "2 & -1 & 2 & 2 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "2 & 2 & -1 & 2 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "2 & 2 & 2 & -1\n",
+       "\\end{array}\\right)\\right]\n",
+       "\\end{math}"
+      ],
       "text/plain": [
        "[\n",
        "[-1  2  2  2]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
@@ -977,9 +1282,8 @@
        "]"
       ]
      },
-     "execution_count": 36,
      "metadata": {},
-     "output_type": "execute_result"
+     "output_type": "display_data"
     }
    ],
    "source": [
@@ -990,17 +1294,382 @@
     "    [-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))"
+    "show(relation)\n",
+    "show(4 * mat)\n",
+    "show(4 * equation)\n",
+    "show(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."
+    "# $n$-Gon Base Pyramid\n",
+    "\n",
+    "The goal here is to find the quadratic form for an arbitrary $n$-gon base pyramid. The key to the whole process is a magic formula Dylan found for the bilinear form between two circles in an $n$-gon base pyramidal packing, namely $$\n",
+    "    \\frac{1 - \\cos\\left(\\frac{2\\pi}{n}\\right) - 4\\sin^2\\left(\\frac{p\\pi}{n}\\right)}{1-\\cos\\left(\\frac{2\\pi}{n}\\right)}\n",
+    "$$\n",
+    "\n",
+    "where $p$ is how many circles are between the two circles in question. If we are finding the bilinear form between the central circle and any other circle it will always be $-1$ so we don't need to worry about that case."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# function to compute W^T*P*W for n-gon pyramidal packing\n",
+    "def wmatrix(n):\n",
+    "    vals = []\n",
+    "    for i in range(n+1):\n",
+    "        row = []\n",
+    "        for j in range(n+1):\n",
+    "            if i == j:                  # same vertex bilinear form'd with itself, so 1\n",
+    "                row.append(1)\n",
+    "            elif i ==0 or j == 0:       # vertex bilinear form'd with special point, so tangent and therefore -1\n",
+    "                row.append(-1)\n",
+    "            else:\n",
+    "                p = abs(i - j)          # otherwise Dylan's crazy formula\n",
+    "                row.append(\n",
+    "                    (1 - cos(2 * pi / n) - 4 * sin(pi * p / n)^2) / (1 - cos(2 * pi / n))\n",
+    "                )\n",
+    "        vals.append(row)\n",
+    "    return matrix(vals)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# just convenience function for quadratic forms\n",
+    "def qform(matrix, vector):\n",
+    "    return vector * matrix * vector"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# function to compute the linear relations and quadratic formula for an ngon base pyramid\n",
+    "# the circles will be numbered 1 in the center, then 2 through n winding around the circle by step, so\n",
+    "# for n = 4 with step = 1, we have\n",
+    "#              b3\n",
+    "#          b4  b1  b2\n",
+    "#              b5\n",
+    "# and the quadratic form is in terms of b1, b2, b3, and b4\n",
+    "# and for n = 6 with step = 2, we have\n",
+    "#          b4     b3\n",
+    "#       b5     b1     b2\n",
+    "#          b6     b7\n",
+    "# and the quadratic form is in terms of b1, b3, b5, and b7\n",
+    "def ngon_linear_relations_and_quadratic_form(n, step=1):\n",
+    "    mat = wmatrix(n)\n",
+    "    \n",
+    "    # work out initial relations\n",
+    "    relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 2) ]) * mat.transpose().rref()\n",
+    "    relations = []\n",
+    "    for i in range(4, n + 1):\n",
+    "        relations.append(var('b' + str(i + 1)) == relations_temp[i])\n",
+    "    \n",
+    "    # rewrite the relations in terms of the variables we care about, depends on the step\n",
+    "    shuffled = (list(range(1, n + 2)) * step)[::step]\n",
+    "    targets = [ var('b' + str(i)) for i in shuffled[4:] ]\n",
+    "    relations = solve(relations, *targets)[0]\n",
+    "    \n",
+    "    # find the matrix corresponding to the quadratic form, picking the appropriate rows from the matrix\n",
+    "    Q = mat[:4*step:step,:4*step:step].inverse()\n",
+    "    \n",
+    "    # find the quadratic form in variables; proper units will satisfy this being equal to zero\n",
+    "    nqform = qform(Q, vector([ var('b' + str(i)) for i in range(1, 5) ]))\n",
+    "    \n",
+    "    return relations, Q, expand(nqform)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{5} = b_{2} - b_{3} + b_{4}</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{5} = b_{2} - b_{3} + b_{4}\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "b5 == b2 - b3 + b4"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "4 & -2 & 0 & -2 \\\\\n",
+       "-2 & 1 & -2 & -1 \\\\\n",
+       "0 & -2 & 4 & -2 \\\\\n",
+       "-2 & -1 & -2 & 1\n",
+       "\\end{array}\\right)</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "4 & -2 & 0 & -2 \\\\\n",
+       "-2 & 1 & -2 & -1 \\\\\n",
+       "0 & -2 & 4 & -2 \\\\\n",
+       "-2 & -1 & -2 & 1\n",
+       "\\end{array}\\right)\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[ 4 -2  0 -2]\n",
+       "[-2  1 -2 -1]\n",
+       "[ 0 -2  4 -2]\n",
+       "[-2 -1 -2  1]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, b_{1}^{2} - 4 \\, b_{1} b_{2} + b_{2}^{2} - 4 \\, b_{2} b_{3} + 4 \\, b_{3}^{2} - 4 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, b_{1}^{2} - 4 \\, b_{1} b_{2} + b_{2}^{2} - 4 \\, b_{2} b_{3} + 4 \\, b_{3}^{2} - 4 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + b_{4}^{2}\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "4*b1^2 - 4*b1*b2 + b2^2 - 4*b2*b3 + 4*b3^2 - 4*b1*b4 - 2*b2*b4 - 4*b3*b4 + b4^2"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & 1 & 0 & 1 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "4 & -1 & 4 & 2 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 1 & -1 & 1 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "4 & 2 & 4 & -1\n",
+       "\\end{array}\\right)\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & 1 & 0 & 1 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "4 & -1 & 4 & 2 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 1 & -1 & 1 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "4 & 2 & 4 & -1\n",
+       "\\end{array}\\right)\\right]\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[\n",
+       "[-1  1  0  1]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]\n",
+       "[ 0  1  0  0]  [ 4 -1  4  2]  [ 0  1  0  0]  [ 0  1  0  0]\n",
+       "[ 0  0  1  0]  [ 0  0  1  0]  [ 0  1 -1  1]  [ 0  0  1  0]\n",
+       "[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 4  2  4 -1]\n",
+       "]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "relations, Q, nqform = ngon_linear_relations_and_quadratic_form(4)\n",
+    "\n",
+    "show(relations)\n",
+    "show(8 * Q)\n",
+    "show(8 * nqform)\n",
+    "show(weyl_generators(8 * Q, standard_basis(4)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{2} = 2 \\, b_{3} - b_{5} + 2 \\, b_{7}, 3 \\, b_{4} = 2 \\, b_{3} + 2 \\, b_{5} - b_{7}, 3 \\, b_{6} = -b_{3} + 2 \\, b_{5} + 2 \\, b_{7}\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{2} = 2 \\, b_{3} - b_{5} + 2 \\, b_{7}, 3 \\, b_{4} = 2 \\, b_{3} + 2 \\, b_{5} - b_{7}, 3 \\, b_{6} = -b_{3} + 2 \\, b_{5} + 2 \\, b_{7}\\right]\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[3*b2 == 2*b3 - b5 + 2*b7, 3*b4 == 2*b3 + 2*b5 - b7, 3*b6 == -b3 + 2*b5 + 2*b7]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "9 & -1 & -1 & -1 \\\\\n",
+       "-1 & 1 & -1 & -1 \\\\\n",
+       "-1 & -1 & 1 & -1 \\\\\n",
+       "-1 & -1 & -1 & 1\n",
+       "\\end{array}\\right)</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
+       "9 & -1 & -1 & -1 \\\\\n",
+       "-1 & 1 & -1 & -1 \\\\\n",
+       "-1 & -1 & 1 & -1 \\\\\n",
+       "-1 & -1 & -1 & 1\n",
+       "\\end{array}\\right)\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[ 9 -1 -1 -1]\n",
+       "[-1  1 -1 -1]\n",
+       "[-1 -1  1 -1]\n",
+       "[-1 -1 -1  1]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "9*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"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "2 & -1 & 2 & 2 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "2 & 2 & -1 & 2 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "2 & 2 & 2 & -1\n",
+       "\\end{array}\\right)\\right]</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
+       "-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "2 & -1 & 2 & 2 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "2 & 2 & -1 & 2 \\\\\n",
+       "0 & 0 & 0 & 1\n",
+       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
+       "1 & 0 & 0 & 0 \\\\\n",
+       "0 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 \\\\\n",
+       "2 & 2 & 2 & -1\n",
+       "\\end{array}\\right)\\right]\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "[\n",
+       "[ -1 2/9 2/9 2/9]  [ 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",
+       "]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "relations, Q, nqform = ngon_linear_relations_and_quadratic_form(6, 2)\n",
+    "\n",
+    "show([3 * eq for eq in relations])\n",
+    "show(12 * Q)\n",
+    "show(12 * nqform)\n",
+    "show(weyl_generators(12 * Q, standard_basis(4)))"
    ]
   }
  ],