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   "source": [
    "# Can two-dimensional topological voids exist in two dimensions?\n",
    "\n",
    "#### License: Apache 2.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The classic example of a two-dimensional homology class is the \"void\" surrounded by a sphere in three-dimensional space.\n",
    "Challenge question: **Can two-dimensional topological voids arise from point clouds in two-dimensional space?**\n",
    "We will answer this question programmatically by computing Vietoris-Rips persistence homology of random point clouds in the square $[0, 1] \\times [0, 1] \\subset \\mathbb{R}^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from giotto.homology import VietorisRipsPersistence as VR\n",
    "import itertools\n",
    "\n",
    "import matplotlib.pyplot as plt\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initializing the Vietoris-Rips transformer\n",
    "vr = VR(homology_dimensions=(2,), max_edge_length=np.inf)\n",
    "n_samples = 15000\n",
    "n_points = 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create n_samples point clouds of n_points points\n",
    "PCS = np.random.random((n_samples, n_points, 2))  \n",
    "# Compute persistence diagrams of all point clouds\n",
    "DGMS = vr.fit_transform(PCS)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "diffs = np.nan_to_num(DGMS[:, :, 1] - DGMS[:, :, 0])  # Compute lifetimes\n",
    "indices = np.argwhere(diffs != 0)  # Indices with non-zero lifetime\n",
    "print('There are {} persistent homology classes in dimension 2 across all samples!'.format(len(indices[:, 0])))\n",
    "print('There are {} different point clouds with at least one persistent homology class in dimension 2.'.format(len(np.unique(indices[:, 0]))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now plot the edges which exist when these persistent homology classes are born.\n",
    "What do the clique complexes of the resulting graphs remind you of?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in indices[:, 0]:\n",
    "    for e in itertools.combinations(PCS[i], 2):\n",
    "        if np.linalg.norm(e[0] - e[1]) < DGMS[i, 0, 1] - 0.00001:\n",
    "            edge = np.stack([e[0], e[1]])\n",
    "            plt.plot(edge[:, 0], edge[:, 1])\n",
    "    plt.show()"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
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   "source": []
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