Grid-Theory.ipynb

{
 "cells": [
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   "source": [
    "---\n",
    "# Grid Theory\n",
    "\n",
    "Konstantin Holzhausen (konstantin.holzhausen@fys.uio.no)\n",
    "\n",
    "7th July 2022\n",
    "\n",
    "---\n",
    "This notebook is commenting on the plane wave composition of [hexagonal Grid Cell ratemap patterns](https://onlinelibrary.wiley.com/doi/10.1002/hipo.20244). In particular, we analytically study the relationships between the symmetries of the pattern-generating plane waves and the symmetries of the generated pattern. We do that by comprehensively studying the relationship between the $\\vec{k}$-vectors (wave vectors) of the plane waves and the **Primitive Basis Vectors** of the hexagonal lattice formed by the peaks of the generated pattern.  "
   ]
  },
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   "source": [
    "## 2D Hexagonal Lattice, Basis Vectors, Primitive Basis, Unit Cells and the Wigner-Seitz-Cell\n",
    "\n",
    "In this section, we are going to review the fundamental definitions and notions of lattice theory with the example of a planar (two-dimensional) hexagonal lattice. To support our intuition, we start by creating a plot of a hexagonal lattice itself."
   ]
  },
  {
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   "execution_count": 1,
   "id": "1e4152c4-b614-40e2-94d2-23d88a8af0e0",
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   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "import matplotlib.patches as patches"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e529b24-2b93-402b-8f70-8fb08212cb9f",
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   "source": [
    "Each vertex $i$, or rather grid or lattice point, on a lattice $\\mathcal{L}$ can be represented by a **lattice vector** $\\vec{R}_{i}$. On a **hexagonal lattice**, all nearest neighbouring points $\\vec{R}_{i}, \\vec{R}_{j} \\in \\mathcal{L}$ have the same distance $d_{i,j} = \\left| \\vec{R}_{i} - \\vec{R}_{j} \\right|$ to each other. The **distances between neighbouring lattice points are called the lattice constants** $a$, $b$, ... . In case of the planar hexagonal lattice, the corresponding difference vectors enclose a $60^{\\circ}$ angle.\n",
    "\n",
    "Since the lattice is two-dimensional, it is spanned by a set of two basis vectors. The **choice of the basis is** first of all **arbitrary** and **ambiguous**. Consequently, there is no unique choice of basis vectors if we do not demand any further constraints. The only constraint so far is, that both vectors have to enclose  an integer multiple of a $60^{\\circ}$ angle $\\alpha$ that meets $0^{\\circ} \\not\\equiv \\alpha \\, \\textrm{mod}\\, 60^{\\circ}$.\n",
    "\n",
    "A **basis** that reaches every lattice, and thus **completely spans the lattice** is called a **primitive basis**.\n",
    "\n",
    "---\n",
    "#### Definition: Primitive Basis\n",
    "\n",
    "A set of basis vectors $\\{ a_{i} \\}_{i \\in I}$ that completely spans the lattice is called a primitive basis. Consequently, every lattice vector $\\vec{R}_{i} \\in \\mathcal{L}$ can be expressed as a linear combination of basis vectors with integer coefficients.\n",
    "\n",
    "$$\\vec{R}_{i} = \\sum_{i} \\, \\alpha_{i}\\, \\vec{a}_{i} \\quad \\textrm{with} \\quad \\alpha_{i} \\in \\mathbb{Z}$$\n",
    "\n",
    "The lengths of primitive basis vectors correspond to the respective lattice constants along the specified axis that is co-aligned with the direction of the primitive basis vector.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ebfb22f3-8f2c-4595-917e-512a875dacf1",
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   "source": [
    "def rotmat(theta: float, degrees: bool=True):\n",
    "    if degrees:\n",
    "        theta = theta*np.pi / 180.\n",
    "    ct = np.cos(theta) # cos(theta)\n",
    "    st = np.sin(theta) # sin(theta)\n",
    "    return np.array([[ct, -st], [st, ct]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "af167eee-5364-4c87-9d40-cb31f852ed91",
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   "outputs": [],
   "source": [
    "# define matrix translating rotations around 60 degrees counter clock-wise\n",
    "rot60 = rotmat(60.)\n",
    "\n",
    "# define lattice constant\n",
    "a = 1.0\n",
    "\n",
    "# primitive basis vectors of hexagonal grid\n",
    "a_0 = a*np.array([1., 0.])\n",
    "a_1 = a*rot60 @ a_0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0824a451-1dbf-4007-97be-e77982f20089",
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   "source": [
    "---\n",
    "#### Unit Cell\n",
    "\n",
    "The unit cell of a lattice is the smallest building block of that lattice (in terms of volume or area). Repetition of the unit cell in every direction creates the lattice structure. In case of the planar hexagonal lattice, each unit cell tiles space perfectly (see plot below). Thus, because of the periodicity of the lattice, the unit cell shares and defines all properties of the lattice itself.\n",
    "\n",
    "Each unit cell contains in total exactly one lattice point, hence the name. The unit cell is not unambiguously defined either. Here, we review only two: the **primitive unit cell** and the **Wigner-Seitz** cell.\n",
    "\n",
    "The **primitive unit cell** is the parallelepiped which is spanned by the primitive basis vectors (see plot below). Thus, the volume of the primitive unit cell of the planar hexagonal lattice is given by\n",
    "\n",
    "$$V_{p} = \\left| \\vec{a}_{0} \\times \\vec{a}_{1} \\right| = a^{2} \\sin{\\left( 60^{\\circ} \\right)} = \\frac{\\sqrt{3}}{2}\\, a^{2} \\quad \\textrm{.}$$\n",
    "\n",
    "The unit cell of highest symmetry is the **Wigner-Seitz cell**. It is constructed by applying **Voronoi decomposition** to the lattice. This is particularly simple for lattices because of their regularity. We are not going to explain how this is done in pratice. We suggest the interested reader to make use of pertinent literature on this topic. As a result of the construction, space is euqally partitioned between lattice points. In case of the planar hexagonal lattice, the Wigner-Seitz cell is shaped like a hexagon itself. However, with respect to the hexagon of the lattice, the Wigner-Seitz cell is rotated by $30^{\\circ}$. It is noticed that the lattice point is located at the center of the Wigner-Seitz cell in contrast to the primitive unit cell, where the lattice points form the corners of the parallelepiped. In many situations, reviewing Wigner-Seitz cell instead of the primitive unit cell is beneficial because of its high symmetry.\n",
    "\n",
    "___"
   ]
  },
  {
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   "execution_count": 4,
   "id": "5a3ac494-74c4-431d-bb8d-09ce42b80a70",
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\n",
      "text/plain": [
       "<Figure size 1500x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# draw hexagonal grid\n",
    "\n",
    "x = np.arange(-4, 2 + 1)\n",
    "y = np.arange(-2, 2 + 1)\n",
    "xv, yv = np.meshgrid(x, y)\n",
    "hex_grid = xv[..., None]*a_0 + yv[..., None]*a_1\n",
    "\n",
    "gx, gy, no_dim = hex_grid.shape\n",
    "no_gridpoints = gx*gy\n",
    "hex_list = hex_grid.reshape(no_gridpoints, no_dim)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10., 4.), dpi=150)\n",
    "\n",
    "# first primitive lattice vector\n",
    "ax.text(0.3*a_0[0], -0.2, \n",
    "        r\"$\\mathbf{a_0}$\",\n",
    "       )\n",
    "ax.arrow(0.0, 0.0, a_0[0], a_0[1],\n",
    "         color=\"orange\",\n",
    "         width=0.03,\n",
    "         head_width=.2,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_0\"\n",
    "        )\n",
    "\n",
    "# second primitive lattice vector\n",
    "ax.text(0., .3*a_1[1] + 0.2, \n",
    "        r\"$\\mathbf{a_1}$\",\n",
    "       )\n",
    "ax.arrow(0.0, 0.0, a_1[0], a_1[1],\n",
    "         color=\"orange\",\n",
    "         width=0.03,\n",
    "         head_width=.2,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "        )\n",
    "\n",
    "# primitive unit cell\n",
    "x = [0.0, a_0[0], a_0[0] + a_1[0], a_1[0]]\n",
    "y = [0.0, a_0[1], a_0[1] + a_1[1], a_1[1]]\n",
    "ax.text(0.5, 1.1, \n",
    "        r\"Primitive Unit Cell\",\n",
    "       )\n",
    "ax.add_patch(\n",
    "    patches.Polygon(\n",
    "        xy=list(zip(x,y)), \n",
    "        fill=True,\n",
    "        color='orange',\n",
    "        alpha=0.5,\n",
    "    )\n",
    ")\n",
    "\n",
    "# Wigner-Seitz cell\n",
    "# define radius of enclosing circle\n",
    "r_wsc = a / (2*np.cos(np.pi / 6))\n",
    "rot30 = rotmat(30.)\n",
    "\n",
    "# Wigner-Seitz cell vertices are rotated\n",
    "# create first vertex\n",
    "w_0 = r_wsc*(rot30 @ np.array([1.0, 0.0]))\n",
    "\n",
    "# all the others are created by rotation\n",
    "ws = [w_0]\n",
    "for i in range(5):\n",
    "    ws.append(rot60 @ ws[-1])\n",
    "ws = np.array(ws)\n",
    "    \n",
    "# plot the Wigner-Seitz cell\n",
    "r_offset = -2*a_0\n",
    "ws = ws + r_offset\n",
    "ax.text(\n",
    "    -2.75*a_0[0],\n",
    "    .6,\n",
    "    r\"Wigner-Seitz cell\"\n",
    ")\n",
    "ax.add_patch(\n",
    "    patches.Polygon(\n",
    "        xy=list(ws),\n",
    "        fill=True,\n",
    "        color=\"blue\",\n",
    "        alpha=0.2,\n",
    "    )\n",
    ")\n",
    "ax.text(\n",
    "    *ws[0].T,\n",
    "    r\"$\\mathbf{w_{0}}$\"\n",
    ")\n",
    "ax.arrow(r_offset[0],\n",
    "         r_offset[1],\n",
    "         ws[0, 0] - r_offset[0],\n",
    "         ws[0, 1] - r_offset[1],\n",
    "         color=\"blue\",\n",
    "         width=0.03,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "        )\n",
    "\n",
    "ax.set_title(r\"Hexagonal lattice\")\n",
    "ax.set_xlabel(r\"x\")\n",
    "ax.set_ylabel(r\"y\")\n",
    "ax.scatter(*hex_list.T)\n",
    "ax.axis(\"equal\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d22564d-5c1d-4ffc-b425-6c2f9d40cbe6",
   "metadata": {},
   "source": [
    "#### Relevant properties from construction\n",
    "\n",
    "The construction of the Wigner-Seitz cell around one lattice point can be summarized as follows:\n",
    "\n",
    "1. Draw the connecting lines between this lattice point and all of its surrounding nearest neighbours.\n",
    "2. Draw all of the lines perpendicular bisectors\n",
    "3. The intersections of these perpendicular bisectors are the vertices of the Wigner-Seitz cell. Consequently, the perpendicular bisectors form the faces of the Wigner-Setiz cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "78a0d6f5-0860-4757-881a-96c861956047",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1500x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# draw small hexagonal grid\n",
    "\n",
    "xs = np.arange(-1, 1 + 1)\n",
    "ys = np.arange(-1, 1 + 1)\n",
    "xvs, yvs = np.meshgrid(xs, ys)\n",
    "hex_grid_small = xvs[..., None]*a_0 + yvs[..., None]*a_1\n",
    "\n",
    "gx_s, gy_s, no_dim_s = hex_grid_small.shape\n",
    "no_gridpoints_s = gx_s*gy_s\n",
    "hex_list_small = hex_grid_small.reshape(no_gridpoints_s, no_dim_s)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10., 4.), dpi=150)\n",
    "\n",
    "# draw Wigner-Seitz cell\n",
    "ax.add_patch(\n",
    "    patches.Polygon(\n",
    "        xy=list(ws - r_offset),\n",
    "        fill=True,\n",
    "        color=\"blue\",\n",
    "        alpha=0.2,\n",
    "    )\n",
    ")\n",
    "# draw lines\n",
    "O = np.array([0.0, 0.0])\n",
    "ax.plot(\n",
    "    *np.array([O, a_0]).T,\n",
    "    color='black',\n",
    ")\n",
    "ax.text(\n",
    "    *(3.*a_0 / 4. + np.array([0., -0.15])),\n",
    "    r\"$\\frac{a}{2}$\",\n",
    "    color=\"black\"\n",
    ")\n",
    "ax.text(\n",
    "    *(a_0 / 4. + np.array([0., -0.15])),\n",
    "    r\"$\\frac{a}{2}$\",\n",
    "    color=\"black\"\n",
    ")\n",
    "ax.plot(\n",
    "    *np.array([a_1, rot60.T @ a_0]).T,\n",
    "    color=\"black\",\n",
    "    linestyle=\"dashed\",\n",
    ")\n",
    "ax.plot(\n",
    "    *np.array([O, w_0]).T,\n",
    "    color=\"blue\"\n",
    ")\n",
    "ax.text(\n",
    "    *(w_0 / 2. + np.array([-0.1, 0.1])).T,\n",
    "    r\"$R_{WS}$\",\n",
    "    color=\"blue\",\n",
    ")\n",
    "\n",
    "# draw hex-grid\n",
    "ax.scatter(*hex_list_small.T)\n",
    "ax.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71118a64-613a-499a-9cdf-a3b90a6d241f",
   "metadata": {},
   "source": [
    "Now, that we know the geometric construction of the Wigner-Seitz cell, we know that the primitive basis vector $\\vec{a}_0$ of the hexagonal lattice and the first vertex vector of the Wigner-Seitz cell $\\vec{w}_{0}$ enclose a $30^{\\circ}$ angle\n",
    "\n",
    "$$\\angle\\; \\left( \\vec{a}_0,\\, \\vec{w}_0 \\right) = 30^{\\circ} = \\frac{\\pi}{6} \\quad \\textrm{.}$$\n",
    "\n",
    "Furthermore, we know that the faces of the Wigner-Seitz cell bisect the line segments connecting the nearest neighbours. Since the length of these line segments is the definition of the lattice constant $a$, each of the bisectors has to be $\\frac{a}{2}$ long.\n",
    "\n",
    "Based on these informations, we can calculate the length $\\| \\vec{w}_{0} \\| = R_{WS}$\n",
    "\n",
    "$$\\cos{\\left(30^{\\circ}\\right)} = \\frac{a / 2}{R_{WS}} \\\\ \n",
    "\\Rightarrow \\; R_{WS} = \\frac{a}{2\\, \\cos{\\left(30^{\\circ}\\right)}}  = \\frac{a}{2\\, \\cos{\\left( \\pi / 6 \\right)}} = \\frac{a}{\\sqrt{3}} \\quad \\textrm{.}$$\n",
    "\n",
    "Knowing this, it is trivial to calculate the volume of the Wigner-Seitz cell $V_{WSC}$\n",
    "\n",
    "$$V_{WSC} = 3 \\, \\| \\vec{w}_{0} \\times \\vec{w}_{1} \\| = 3\\, R_{WS}^{2} \\,\\sin{\\left( 60^{\\circ} \\right)} = 3\\, \\frac{a^{2}}{3}\\, \\frac{\\sqrt{3}}{2} = \\frac{\\sqrt{3}}{2}\\, a^{2}.$$\n",
    "\n",
    "Thus, by construction the Wigner-Seitz cell and the primitive unit cell both contain exactly one lattice point in total and take the exact same amount of volume $V_{p} = V_{WSC} = \\frac{\\sqrt{3}}{2}\\, a^{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab5f4413-524c-4bbe-aac6-bce1302d5e47",
   "metadata": {},
   "source": [
    "## Generated Grid Cells\n",
    "\n",
    "According to [Solstad et al.](https://onlinelibrary.wiley.com/doi/10.1002/hipo.20244), grid cell like ratemaps can be represented by a superposition of three planar waves with wave vectors $\\{\\vec{k}_{i}\\}_{i \\in \\{1, 2, 3\\}}$. As a consequence, the ratemap $\\Phi_{\\mu}$ of a grid cell $\\mu \\in \\mathcal{G}$ out of the set of all grid cells $\\mathcal{G}$ with phase $\\vec{\\rho}_{\\mu}$ on a domain $\\Omega$ can ideally be expressed as \n",
    "\n",
    "$$\n",
    "    \\Phi_{\\mu}(\\vec{r}) \\; \\propto \\; \\sum_{k = 1}^{3}\\, \\varphi(\\vec{r}) = \\sum_{k = 1}^{3} \\, \\cos{\\left( 2 \\pi\\, f \\, \\langle \\vec{r} - \\vec{\\rho}_{\\mu},\\, \\vec{k}_{k}\\rangle  \\right)} \\quad \\textrm{on} \\quad \\Omega \\quad \\textrm{.}\n",
    "$$\n",
    "\n",
    "$f$ denotes the spatial frequency of the planar waves if $\\|\\vec{k}_{i}\\| = 1$, and relates to the angular frequency $\\omega = 2\\, \\pi \\, f$. As we will see within this section, it turns out that $f$ **is not the spatial frequency of the resulting grid pattern**, but it is related to that. The three k-vectors are required to pairwise enclose a $60^{\\circ}$ angle if the resulting grid pattern is supposed to be hexagonal. \n",
    "\n",
    "Arbitrary orientations of the grid patterns can either be generated by rotating the k-vectors or the coordinate system. In the following, we consider that $\\vec{k}_{0}$ is aligned with the cardinal x-Axis\n",
    "\n",
    "$$\n",
    " \\vec{k}_{0} = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\quad \\textrm{.}\n",
    "$$\n",
    "\n",
    "Consequently, the other two k-vectors are given by\n",
    "\n",
    "$$\n",
    " \\vec{k}_{1} = \\begin{pmatrix} \\cos{(60^{\\circ})} \\\\ \\sin{(60^{\\circ})} \\end{pmatrix} = \\frac{1}{2}\\, \\begin{pmatrix} 1 \\\\ \\sqrt{3} \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    " \\vec{k}_{1} = \\begin{pmatrix} \\cos{(120^{\\circ})} \\\\ \\sin{(120^{\\circ})} \\end{pmatrix} = \\frac{1}{2}\\, \\begin{pmatrix} -1 \\\\ \\sqrt{3} \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "respectively. The following figure shows the resulting pattern for an angular frequency of $\\omega\\,=\\,2 \\pi$, thus $f=1$, and zero phase with the three k-vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d041516e-43df-4774-b30e-38ead735f2a0",
   "metadata": {},
   "outputs": [],
   "source": [
    "# define k vectors of planar waves\n",
    "k_1 = np.array([1.0, 0.0])\n",
    "k_2 = rot60 @ k_1\n",
    "k_3 = rot60 @ k_2\n",
    "ks = np.array([k_1, k_2, k_3])\n",
    "\n",
    "def Phi(r, ks, f=1.0, rho=np.array([0.0, 0.0])):\n",
    "    res = 0.0\n",
    "    for k in ks:\n",
    "        res += np.cos(2*np.pi*f*(r - rho) @ k)\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "986c3d05-f025-4b32-a1c6-683cdfeb9118",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot synthetic grid cells\n",
    "min_val = -2.2\n",
    "max_val = 2.2\n",
    "res = 100\n",
    "\n",
    "x_gc = np.linspace(min_val, max_val, num=res)\n",
    "y_gc = np.linspace(min_val, max_val, num=res)\n",
    "\n",
    "xv_gc, yv_gc = np.meshgrid(x_gc, y_gc)\n",
    "rv_gc = np.stack([xv_gc, yv_gc], axis=-1)\n",
    "\n",
    "phi_vals = Phi(rv_gc, ks)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(4., 4.), dpi=150)\n",
    "ax.imshow(phi_vals, extent=[min_val, max_val, min_val, max_val], alpha=1., zorder=0)\n",
    "# plot k=vectors\n",
    "# k_1\n",
    "ax.arrow(*O.T,\n",
    "         *k_1.T,\n",
    "         color=\"white\",\n",
    "         width=0.03,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=2.,\n",
    "        )\n",
    "ax.arrow(*O.T,\n",
    "         *k_1.T,\n",
    "         color=\"black\",\n",
    "         width=0.08,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=1.,\n",
    "        )\n",
    "ax.text(*(k_1 / 2. + np.array([-0.1, 0.1])).T, r\"$\\mathbf{\\vec{k}_1}$\", color='white')\n",
    "# k_2\n",
    "ax.arrow(*O.T,\n",
    "         *k_2.T,\n",
    "         color=\"white\",\n",
    "         width=0.03,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=2.,\n",
    "        )\n",
    "ax.arrow(*O.T,\n",
    "         *k_2.T,\n",
    "         color=\"black\",\n",
    "         width=0.08,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=1.,\n",
    "        )\n",
    "ax.text(*(k_2 / 2. + np.array([-0.3, +0.1])).T, r\"$\\mathbf{\\vec{k}_2}$\", color='white')\n",
    "# k_3\n",
    "ax.arrow(*O.T,\n",
    "         *k_3.T,\n",
    "         color=\"white\",\n",
    "         width=0.03,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=2.,\n",
    "        )\n",
    "ax.arrow(*O.T,\n",
    "         *k_3.T,\n",
    "         color=\"black\",\n",
    "         width=0.08,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=1.,\n",
    "        )\n",
    "ax.text(*(k_3 / 2. + np.array([-0.3, -0.2])).T, r\"$\\mathbf{\\vec{k}_3}$\", color='white')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66858d63-be61-4133-a903-ccfa03efd861",
   "metadata": {},
   "source": [
    "With the aid of the upper plot we notice the following: \n",
    "\n",
    "1. Because we chose $\\cos$ instead of $\\sin$, the pattern is centred around and starts with a peak in the origin of our coordinate system, where the $k$-vectors start. \n",
    "2. **The direction of the $k$-vectors is not the direction of what appears to be the primitive basis of the appearing hexagonal pattern lattice!**\n",
    "\n",
    "The second point is of particular interest for us. In following the direction of any of the $k$-vectors, we do not approach the nearest neighbouring lattice point from the centre, but the second nearest one. The latter one appears to be $2 \\cdot f^{-1}$ (that is the two-fold of the **spatial period** $\\mathbf{T = f^{-1}}$   of the plane waves) distant from the centre. And indeed, it is relatively easy to show that the pattern is $2 \\cdot f$-periodic in the direction of $\\vec{k_{1}}$. In order to show this, we study the pattern under the translation operator\n",
    "\n",
    "$$\\mathcal{T}_{\\vec{k}_{0}}(2\\, f): \\vec{r} \\; \\mapsto \\; \\vec{r} + 2 \\cdot f\\, \\vec{k}_{0}$$\n",
    "\n",
    "$$\n",
    "    \\mathcal{T}_{\\vec{k}_{0}}(2\\, f)\\, \\Phi(\\vec{r}) \\quad = \\quad \\sum_{k = 1}^{3} \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} + 2\\, f^{-1}\\, \\vec{k}_{1} - \\vec{\\rho}_{0},\\; \\vec{k}_{k} \\rangle \\right)} \\\\\n",
    "    \\quad = \\quad \\sum_{k = 1}^{3} \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{k} \\rangle + 2 \\pi \\cdot 2\\,  \\langle \\vec{k}_{1},\\; \\vec{k}_{k} \\rangle \\right)} \\quad {.}\n",
    "$$\n",
    "\n",
    "Since \n",
    "\n",
    "$$\n",
    "    \\langle k_{1}, k_{i} \\rangle = \\begin{cases} \n",
    "        1 \\quad \\textrm{if}\\; i=1 \\\\\n",
    "        \\cos{\\left( \\frac{\\pi}{3} \\right)} = \\frac{1}{2} \\quad \\textrm{if}\\, i=2 \\\\\n",
    "        \\cos{\\left( \\frac{2\\, \\pi}{3} \\right)} = - \\frac{1}{2} \\quad \\textrm{if}\\,  i=3\n",
    "    \\end{cases} \\quad \\textrm{,}\n",
    "$$\n",
    "\n",
    "$$\n",
    "    \\mathcal{T}_{\\vec{k}_{0}}(2\\, f)\\, \\Phi(\\vec{r}) \\quad = \\quad \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{1} \\rangle + 4\\, \\pi \\right)} + \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{2} \\rangle + 2\\, \\pi \\right)} + \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{3} \\rangle - 2\\, \\pi \\right)} \\\\\n",
    "    \\quad = \\quad \\sum_{k = 1}^{3} \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{k} \\rangle \\right)} \\\\\n",
    "    \\quad = \\quad \\Phi(\\vec{r}) \\quad \\textrm{.}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "59cf9150-65b2-462f-940b-90ee6a81a0d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot synthetic grid cells\n",
    "min_val = -2.2\n",
    "max_val = 2.2\n",
    "res = 100\n",
    "f = 1.\n",
    "\n",
    "x_gc = np.linspace(min_val, max_val, num=res)\n",
    "y_gc = np.linspace(min_val, max_val, num=res)\n",
    "\n",
    "xv_gc, yv_gc = np.meshgrid(x_gc, y_gc)\n",
    "rv_gc = np.stack([xv_gc, yv_gc], axis=-1)\n",
    "\n",
    "phi_vals = Phi(rv_gc, ks)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(4., 4.), dpi=150)\n",
    "# ratemap\n",
    "ax.imshow(phi_vals, extent=[min_val, max_val, min_val, max_val], alpha=.7, zorder=0)\n",
    "# k_1\n",
    "ax.arrow(*O.T,\n",
    "         *k_1.T,\n",
    "         color=\"white\",\n",
    "         width=0.03,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=2.,\n",
    "        )\n",
    "ax.text(*(k_1 / 2. + np.array([-0.1, -0.3])).T, r\"$\\mathbf{\\vec{k}_1}$\", color='white')\n",
    "ax.plot(\n",
    "    *np.array([O, 2*k_1]).T,\n",
    "    color='black',\n",
    "    linestyle='dashed',\n",
    ")\n",
    "ax.text(\n",
    "    *(k_1 + np.array([+.1, -.3])).T,\n",
    "    r\"$2\\, f^{-1}$\",\n",
    ")\n",
    "ax.text(\n",
    "    *(k_1 / 2. + np.array([-.1, +.05])).T,\n",
    "    r\"$f^{-1}$\",\n",
    ")\n",
    "ax.plot(\n",
    "    *np.array([2*k_2, -2*k_3]).T,\n",
    "    color='black',\n",
    "    linestyle='dotted',\n",
    ")\n",
    "# primitive basis vector\n",
    "r = 1 / (f*np.cos(np.pi / 6))\n",
    "a_0 = r* rot30 @ k_1\n",
    "ax.arrow(*O.T,\n",
    "         *a_0.T,\n",
    "         color=\"orange\",\n",
    "         width=0.03,\n",
    "         head_width=.15,\n",
    "         length_includes_head=True,\n",
    "         label=r\"a_1\",\n",
    "         zorder=2.,\n",
    "        )\n",
    "ax.text(\n",
    "    *(a_0/2. + np.array([-.3, +.1])).T,\n",
    "    r\"$\\mathbf{\\vec{a}_{0}}$\",\n",
    "    color='orange',\n",
    ")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "How do we find the primitive basis vectors? First, it makes sense to notify what we know: the planar wave vectors $\\vec{k}_{i}$! In addition to that, we know that the period of $\\Phi$ in the direction of $\\vec{k}_{1}$ is $T_{\\vec{k}_{1}} = 2\\, f^{-1}$. That means, that the line segment, which connects the centre lattice point with the next peak in $\\vec{k}_{1}$-direction is $2\\, f^{-1}$ long. In the figure above, this line segment is depicted as a black dashed line.\n",
    "\n",
    "Next, we notice that the perpendicular bisector of this line segment, marked by the black dotted line, also goes through every other lattice point with another period, including two of the actual nearest neighbours of the centre point we are interested in. Thus, finding the primitive basis vectors of the grid cell lattice appears visually identical to the construction of the Wigner-Seitz cell vertices. Based on that, we know that\n",
    "\n",
    "$$\n",
    "    \\angle \\left( \\vec{a}_{0},\\, \\vec{k}_{1} \\right) = \\frac{\\pi}{6}\n",
    "$$\n",
    "\n",
    "and that the lattice constant is\n",
    "\n",
    "$$\n",
    "    a_{\\textrm{GC}} = \\frac{2\\, f^{-1}}{2\\, \\cos{\\left(\\pi / 6\\right)}} = \\frac{2}{\\sqrt{3}}\\, f^{-1} \\quad \\textrm{.}\n",
    "$$\n",
    "\n",
    "Consequently, the primitive lattice vector $\\vec{a}_{0}$ of the grid cell pattern lattice can be expressed as\n",
    "\n",
    "$$\n",
    "    \\vec{a}_{0} \\quad = \\quad a_{GC} \\, \\frac{1}{2}\\begin{pmatrix} \\sqrt{3} \\\\ 1 \\end{pmatrix} = f^{-1} \\begin{pmatrix} 1 \\\\ \\frac{1}{\\sqrt{3}} \\end{pmatrix} \\quad \\textrm{.}\n",
    "$$\n",
    "\n",
    "The respective other primitive basis vector belonging to the primitive basis that contains $\\vec{a}_{0}$ can be obtained by rotating $\\vec{a}_{0}$ by $60^{\\circ}$ in any direction.\n",
    "\n",
    "In order to prove that $\\vec{a}_{0}$ is, indeed, a primitive basis vector of the lattice formed by the maxima in $\\Phi$ with lattice constant $a_{GC}$, it is necessary and sufficient to demonstrate that $\\Phi$ is $a_{GC}$-periodic in the direction $\\vec{a}_{0}$. In order to do that, we study the effect of the translation operator $\\mathcal{T}_{\\vec{a}_{0}}(a_{GC})$\n",
    "\n",
    "$$\n",
    "    \\mathcal{T}_{\\vec{a}_{0}}(a_{GC})\\, \\Phi \\left( \\vec{r} \\right) \\quad =  \\quad \\Phi \\left( \\vec{r} + \\vec{a}_{0} \\right) \\\\\n",
    "    = \\quad \\sum_{k = 1}^{3} \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} + \\vec{a}_{0} - \\vec{\\rho}_{0},\\; \\vec{k}_{k} \\rangle \\right)} \\\\\n",
    "    \\quad \\sum_{k = 1}^{3} \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{k} \\rangle  + 2\\, \\pi\\, \\langle \\begin{pmatrix} 1 \\\\ 1 / \\sqrt{3} \\end{pmatrix}, \\vec{k}_{k} \\rangle \\right)} \\quad \\textrm{.}\n",
    "$$\n",
    "\n",
    "By explicitely calculating the respective scalar products\n",
    "\n",
    "$$\n",
    "    \\langle \\begin{pmatrix} 1 \\\\ 1 / \\sqrt{3} \\end{pmatrix}, \\vec{k}_{0} \\rangle = \\left(1, 1/\\sqrt{3} \\right) \\cdot \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = 1 \\\\\n",
    "    \\langle \\begin{pmatrix} 1 \\\\ 1 / \\sqrt{3} \\end{pmatrix}, \\vec{k}_{0} \\rangle = \\left(1, 1/\\sqrt{3} \\right) \\cdot \\begin{pmatrix} 1 / 2 \\\\ \\sqrt{3} / 2 \\end{pmatrix} = \\frac{1}{2} + \\frac{1}{2} = 1 \\\\\n",
    "    \\langle \\begin{pmatrix} 1 \\\\ 1 / \\sqrt{3} \\end{pmatrix}, \\vec{k}_{0} \\rangle = \\left(1, 1/\\sqrt{3} \\right) \\cdot \\begin{pmatrix} -1 / 2 \\\\ \\sqrt{3} / 2 \\end{pmatrix} = -\\frac{1}{2} + \\frac{1}{2} = 0 \\quad \\textrm{,}\n",
    "$$\n",
    "\n",
    "we see\n",
    "\n",
    "$$\n",
    "    \\mathcal{T}_{\\vec{a}_{0}}(a_{GC})\\, \\Phi \\left( \\vec{r} \\right) \\quad =  \\quad \\Phi \\left( \\vec{r} + \\vec{a}_{0} \\right) \\\\\n",
    "    = \\quad \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{1} \\rangle + 2\\, \\pi \\right)} + \n",
    "    \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{2} \\rangle + 2\\, \\pi \\right)} +\n",
    "    \\cos{\\left( 2 \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0},\\; \\vec{k}_{3} \\rangle \\right)} \\\\\n",
    "    = \\quad \\sum_{k=1}^{3} \\, \\cos{\\left( 2\\, \\pi\\, f\\, \\langle \\vec{r} - \\vec{\\rho}_{0}, \\vec{k}_{k} \\rangle \\right)} = \\Phi(\\vec{r}) \\quad \\textrm{,}\n",
    "$$\n",
    "\n",
    "which proves our point, that $\\Phi$ is $a_{GC}$-periodic in the direction of $\\vec{a}_{0}$ which means that $\\vec{a}_{0}$ is a primitive basis vector of the lattice formed by the maxima of $\\Phi$ that we call the grid cell ratemap pattern and that $a_{GC}$ is its lattice constant!\n",
    "\n",
    "To sum things up:\n",
    "1. $\\vec{a}_{0}$ of the grid lattice is oriented at an $30^{\\circ}$ offset with respect to $\\vec{k}_{1}$, a wave vector of the plane waves generating $\\Phi$\n",
    "2. The lattice constant of the grid cell pattern $a_{GC}$ relates to the spatial frequency of the planar waves $f$ as $a_{GC} = \\sqrt{3} / 2 \\; f^{-1}$.\n",
    "\n",
    "If we want to do statistics with these kinds of patterns or analyse its composition and periodicity, we have to keep in mind that **$f$ is not $a_{GC}$, but we know how to transform between the two**, especially when constructing the Wigner-Seitz cell of the grid cell pattern!"
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