Convert to simplicial at the end of demo

demo/simplification.ipynb

@@ -238,11 +238,12 @@
    "id": "b3c894c1-0895-404b-8e8a-172429882dea",
    "metadata": {},
    "source": [
-    "The cell below shows the resulting matrix, which we'll call $e_1$.\n",
-    "(I'm showing the transpose so that we don't have to spill lines.)\n",
-    "You might notice that, for example, rows 0 and 1 are identical.\n",
-    "Any two rows that are scalar multiples of each other can be merged.\n",
-    "The adjacency of higher-dimensional cells can then be coalesced into adjacency to the merged cell."
+    "The code below shows the resulting matrix, which we'll call $e_1$.\n",
+    "You might notice that, for example, columns 0 and 1 are identical.\n",
+    "(So are columns 4 and 6 and columns 3 and 13.)\n",
+    "From a geometric viewpoint, this means that we have two edges that start at the same vertex and end at the same vertex.\n",
+    "This is undesirable.\n",
+    "The new 2-cells are the same as the old for now."
    ]
   },
   {
@@ -252,16 +253,24 @@
    "metadata": {},
    "outputs": [],
    "source": [
+    "e_0 = d_0\n",
     "e_1 = P @ d_1\n",
-    "e_1.T"
+    "e_2 = d_2\n",
+    "print(e_1)"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "2ca5ef8d-739d-4440-a4ea-17db3ff03725",
    "metadata": {},
    "source": [
-    "The function below creates the matrices that we will multiply by $e_1$ and $d_2$ in order to merge these cells."
+    "We might also have columns that are not exact duplicates by scalar multiples of each other.\n",
+    "For example, an edge that goes from vertex 0 to vertex 1, and another edge that goes from vertex 1 to vertex 0.\n",
+    "We can clean things up by:\n",
+    "1. replacing all 1-cells that are multiples of each other with a single representative\n",
+    "2. for any 2-cell that was incident upon any of the degenerate 1-cells, replace it with a 2-cell that is incident upon the new representative 1-cell.\n",
+    "\n",
+    "The function below creates the matrices that we will multiply by $e_1$ and $e_2$ in order to merge these cells."
    ]
   },
   {
@@ -325,15 +334,16 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "print(e_1 @ A)"
+    "f_1 = e_1 @ A\n",
+    "print(f_1)"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "1e546417-668c-4420-836b-7d3a640de149",
    "metadata": {},
    "source": [
-    "And the columns of $B\\cdot d_2$ have their adjacencies to redundant columns summed together."
+    "And the columns of $B\\cdot e_2$ have their adjacencies to redundant columns summed together."
    ]
   },
   {
@@ -343,7 +353,19 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "print(B @ d_2)"
+    "print(B @ e_2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6a6f55e7-411d-4c84-9a16-a09a75d6de15",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "empty_cell_ids = np.flatnonzero(np.count_nonzero(B @ e_2, axis=0) < 3)\n",
+    "f_2 = np.delete(B @ e_2, empty_cell_ids, axis=1)\n",
+    "print(f_2)"
    ]
   },
   {
@@ -353,6 +375,27 @@
    "source": [
     "You'll also note that some 2-cells are now empty, which we expect."
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "cc4032ba-f62b-4c53-b6cb-0ba439ebd4cb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "ltriangles = zmsh.polytopal.to_simplicial([e_0, f_1, f_2])\n",
+    "print(ltriangles)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2a875059-817e-422b-ac3c-451c9caa151d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "vertex_ids[ltriangles]"
+   ]
   }
  ],
  "metadata": {