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    "# freud.order.Hexatic: Hard Hexagons\n",
    "## Hexatic Order Parameter\n",
    "\n",
    "The hexatic order parameter measures how closely the local environment around a particle resembles perfect $k$-atic symmetry, *e.g.* how closely the environment resembles hexagonal/hexatic symmetry for $k=6$. The order parameter is given by:\n",
    "\n",
    "$$\n",
    "\\psi_k \\left( i \\right) = \\frac{1}{n} \\sum \\limits_j^n e^{k i \\theta_{ij}}\n",
    "$$\n",
    "\n",
    "where $\\theta_{ij}$ is the angle between the vector $\\vec{r}_{ij}$ and $\\left(1, 0\\right)$.\n",
    "\n",
    "The pseudocode is given below:\n",
    "\n",
    "```\n",
    "for each particle i:\n",
    "    neighbors = nearestNeighbors(i, n):\n",
    "    for each particle j in neighbors:\n",
    "        r_ij = position[j] - position[i]\n",
    "        theta_ij = arctan2(r_ij.y, r_ij.x)\n",
    "        psi_array[i] += exp(complex(0,k*theta_ij))\n",
    "```\n",
    "\n",
    "The data sets used in this example are a system of hard hexagons, simulated in the NVT thermodynamic ensemble in HOOMD-blue, for a dense fluid of hexagons at packing fraction $\\phi = 0.65$ and solids at packing fractions $\\phi = 0.75, 0.85$."
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     },
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   ],
   "source": [
    "import freud\n",
    "import numpy as np\n",
    "from bokeh.io import output_notebook\n",
    "from bokeh.plotting import figure, show\n",
    "\n",
    "# The following line imports this set of utility functions:\n",
    "# https://github.com/glotzerlab/freud-examples/blob/master/util.py\n",
    "import util\n",
    "\n",
    "output_notebook()"
   ]
  },
  {
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   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_hex_order_param(data_path, title):\n",
    "    # Create hexatic object\n",
    "    hex_order = freud.order.Hexatic(k=6)\n",
    "\n",
    "    # Load the data\n",
    "    box_data = np.load(f\"{data_path}/box_data.npy\")\n",
    "    pos_data = np.load(f\"{data_path}/pos_data.npy\")\n",
    "    quat_data = np.load(f\"{data_path}/quat_data.npy\")\n",
    "\n",
    "    # Grab data from last frame\n",
    "    box = box_data[-1].tolist()\n",
    "    points = pos_data[-1]\n",
    "    quats = quat_data[-1]\n",
    "    angles = 2 * np.arctan2(quats[:, 3], quats[:, 0])\n",
    "\n",
    "    # Compute hexatic order for 6 nearest neighbors\n",
    "    hex_order.compute(system=(box, points), neighbors={\"num_neighbors\": 6})\n",
    "    psi_k = hex_order.particle_order\n",
    "    avg_psi_k = np.mean(psi_k)\n",
    "\n",
    "    # Create hexagon vertices\n",
    "    verts = util.make_polygon(sides=6, radius=0.6204)\n",
    "    # Create array of transformed positions\n",
    "    patches = util.local_to_global(verts, points[:, :2], angles)\n",
    "    # Create an array of angles relative to the average\n",
    "    relative_angles = np.angle(psi_k) - np.angle(avg_psi_k)\n",
    "    # Plot in bokeh\n",
    "    p = figure(title=title)\n",
    "    p.patches(\n",
    "        xs=patches[:, :, 0].tolist(),\n",
    "        ys=patches[:, :, 1].tolist(),\n",
    "        fill_color=[util.cubeellipse(x) for x in relative_angles],\n",
    "        line_color=\"black\",\n",
    "    )\n",
    "    util.default_bokeh(p)\n",
    "    show(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"4fcc12dc-7ac3-4ffa-9127-6cb4c9b92e10\" data-root-id=\"1002\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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density\",\"text_font_size\":{\"value\":\"18pt\"}},\"id\":\"1003\",\"type\":\"Title\"},{\"attributes\":{\"callback\":null},\"id\":\"1005\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"1027\",\"type\":\"ResetTool\"},{\"attributes\":{},\"id\":\"1047\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1028\",\"type\":\"HelpTool\"},{\"attributes\":{},\"id\":\"1048\",\"type\":\"Selection\"},{\"attributes\":{\"active_drag\":\"auto\",\"active_inspect\":\"auto\",\"active_multi\":null,\"active_scroll\":\"auto\",\"active_tap\":\"auto\",\"tools\":[{\"id\":\"1023\",\"type\":\"PanTool\"},{\"id\":\"1024\",\"type\":\"WheelZoomTool\"},{\"id\":\"1025\",\"type\":\"BoxZoomTool\"},{\"id\":\"1026\",\"type\":\"SaveTool\"},{\"id\":\"1027\",\"type\":\"ResetTool\"},{\"id\":\"1028\",\"type\":\"HelpTool\"}]},\"id\":\"1029\",\"type\":\"Toolbar\"},{\"attributes\":{\"axis_label_text_font_size\":{\"value\":\"14pt\"},\"formatter\":{\"id\":\"1044\",\"type\":\"BasicTickFormatter\"},\"major_label_text_font_size\":{\"value\":\"12pt\"},\"major_tick_in\":10,\"major_tick_out\":0,\"minor_tick_in\":5,\"minor_tick_out\":0,\"ticker\":{\"id\":\"1019\",\"type\":\"BasicTicker\"}},\"id\":\"1018\",\"type\":\"LinearAxis\"},{\"attributes\":{},\"id\":\"1011\",\"type\":\"LinearScale\"},{\"attributes\":{\"bottom_units\":\"screen\",\"fill_alpha\":{\"value\":0.5},\"fill_color\":{\"value\":\"lightgrey\"},\"left_units\":\"screen\",\"level\":\"overlay\",\"line_alpha\":{\"value\":1.0},\"line_color\":{\"value\":\"black\"},\"line_dash\":[4,4],\"line_width\":{\"value\":2},\"render_mode\":\"css\",\"right_units\":\"screen\",\"top_units\":\"screen\"},\"id\":\"1046\",\"type\":\"BoxAnnotation\"},{\"attributes\":{},\"id\":\"1009\",\"type\":\"LinearScale\"},{\"attributes\":{\"ticker\":{\"id\":\"1014\",\"type\":\"BasicTicker\"}},\"id\":\"1017\",\"type\":\"Grid\"},{\"attributes\":{\"callback\":null},\"id\":\"1007\",\"type\":\"DataRange1d\"},{\"attributes\":{\"axis_label_text_font_size\":{\"value\":\"14pt\"},\"formatter\":{\"id\":\"1042\",\"type\":\"BasicTickFormatter\"},\"major_label_text_font_size\":{\"value\":\"12pt\"},\"major_tick_in\":10,\"major_tick_out\":0,\"minor_tick_in\":5,\"minor_tick_out\":0,\"ticker\":{\"id\":\"1014\",\"type\":\"BasicTicker\"}},\"id\":\"1013\",\"type\":\"LinearAxis\"},{\"attributes\":{},\"id\":\"1014\",\"type\":\"BasicTicker\"}],\"root_ids\":[\"1002\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"a47c89c4-2433-48a1-9873-9c10b72d7e24\",\"roots\":{\"1002\":\"4fcc12dc-7ac3-4ffa-9127-6cb4c9b92e10\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_hex_order_param(\"data/phi065\", \"Hexatic Order Parameter, 0.65 density\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the density increases to $\\phi=0.75$, the shapes are forced to align more closely so that they may tile space effectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"d458864c-23f2-48f6-8926-1613b9e3b678\" data-root-id=\"1097\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Order Parameter, 0.75 density\",\"text_font_size\":{\"value\":\"18pt\"}},\"id\":\"1098\",\"type\":\"Title\"},{\"attributes\":{\"dimension\":1,\"ticker\":{\"id\":\"1114\",\"type\":\"BasicTicker\"}},\"id\":\"1117\",\"type\":\"Grid\"}],\"root_ids\":[\"1097\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"c4733e35-6073-4b4b-b868-a41c5f7d68e3\",\"roots\":{\"1097\":\"d458864c-23f2-48f6-8926-1613b9e3b678\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1097"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_hex_order_param(\"data/phi075\", \"Hexatic Order Parameter, 0.75 density\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the density increases to $\\phi=0.85$, the alignment becomes even stronger and defects are no longer visible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"43dce315-be98-44b3-8445-70bc253b302c\" data-root-id=\"1200\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Order Parameter, 0.85 density\",\"text_font_size\":{\"value\":\"18pt\"}},\"id\":\"1201\",\"type\":\"Title\"},{\"attributes\":{\"overlay\":{\"id\":\"1260\",\"type\":\"BoxAnnotation\"}},\"id\":\"1223\",\"type\":\"BoxZoomTool\"},{\"attributes\":{\"axis_label_text_font_size\":{\"value\":\"14pt\"},\"formatter\":{\"id\":\"1258\",\"type\":\"BasicTickFormatter\"},\"major_label_text_font_size\":{\"value\":\"12pt\"},\"major_tick_in\":10,\"major_tick_out\":0,\"minor_tick_in\":5,\"minor_tick_out\":0,\"ticker\":{\"id\":\"1217\",\"type\":\"BasicTicker\"}},\"id\":\"1216\",\"type\":\"LinearAxis\"},{\"attributes\":{},\"id\":\"1256\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{},\"id\":\"1209\",\"type\":\"LinearScale\"},{\"attributes\":{},\"id\":\"1225\",\"type\":\"ResetTool\"},{\"attributes\":{},\"id\":\"1258\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{},\"id\":\"1212\",\"type\":\"BasicTicker\"}],\"root_ids\":[\"1200\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"acc10e55-21c1-4bfe-997f-f84cc8be5b75\",\"roots\":{\"1200\":\"43dce315-be98-44b3-8445-70bc253b302c\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1200"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_hex_order_param(\"data/phi085\", \"Hexatic Order Parameter, 0.85 density\")"
   ]
  }
 ],
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