andreasgrv/look-at-cube.ipynb

## look-at-cube.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "6e803f0a-2d12-4e12-b572-fd6a03c33f49",
   "metadata": {},
   "source": [
    "# Viewing a cube from the direction of a vertex\n",
    "\n",
    "The code below [answers this question](https://x.com/adad8m/status/1845809571321061610).\n",
    "\n",
    "\n",
    "## The case of the 3d cube\n",
    "\n",
    "What happens if we view a 3d cube from the direction of one of the vertices? What do we see?\n",
    "\n",
    "\n",
    "### Answer: Cancel the \"viewing\" direction\n",
    "The idea is that we can simply \"cancel out\" the direction we are viewing the cube from.\n",
    "I.e. we project the vertices of the cube from d dimensions to (d-1) dimensions along the viewing direction.\n",
    "This means we get a projection of the cube into one dimension lower, i.e. we get the hexagon below.\n",
    "The internal point is the overlapping projection of the $(1,1,1)$ and $(-1,-1,-1)$ vertices (which get mapped to 0 - seen in red)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "55170e3b-bafc-4aa8-84d6-cd351b90655e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1ef9bb81b72643df9001cac8c86053ef",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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",
      "text/html": [
       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=640.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib widget\n",
    "\n",
    "import itertools\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "def cube_vertices(d):\n",
    "    return np.array(list(itertools.product([-1, 1], repeat=d)))\n",
    "\n",
    "\n",
    "D = 3\n",
    "\n",
    "# Cube vertices (cube centered at origin)\n",
    "cc = cube_vertices(D)\n",
    "\n",
    "# If we look from the all ones direction\n",
    "vv = np.ones(D).reshape(1, -1)\n",
    "\n",
    "# We only retain cube information\n",
    "# in the directions perpendicular to the viewing direction\n",
    "W = sp.linalg.null_space(vv)\n",
    "\n",
    "# Coordinates of cube vertices as viewed from vv\n",
    "projection = cc.dot(W)\n",
    "\n",
    "colors = ['red' if np.allclose(pp, np.zeros(D-1)) else 'blue' for pp in projection]\n",
    "\n",
    "# Plot vertices\n",
    "ax = plt.figure().add_subplot()\n",
    "ax.scatter(*projection.T, color=colors)\n",
    "ax.set_aspect('equal')\n",
    "plt.axis('off')\n",
    "plt.suptitle('Projection of 3d cube along $\\mathbf{v} = (1,1,1)$', fontsize=20)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8b4e3b4-78ce-4437-b3fe-4fae61f6f6f3",
   "metadata": {},
   "source": [
    "## The case of the 4d-cube\n",
    "\n",
    "In the 4d case, instead of a hexagon we get a [Rhombic Dodecahedron](https://en.wikipedia.org/wiki/Rhombic_dodecahedron#Related_polytope)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "1220092e-d124-4c3a-a1ff-cc348b9f88ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "46b12a3b6a5c44cdb50cba0863b62e8a",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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",
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       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=640.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "D = 4\n",
    "\n",
    "# Cube vertices (cube centered at origin)\n",
    "cc = cube_vertices(D)\n",
    "\n",
    "# If we look from the all ones direction\n",
    "vv = np.ones(D).reshape(1, -1)\n",
    "\n",
    "# the direction we look from is in the null space\n",
    "W = sp.linalg.null_space(vv)\n",
    "\n",
    "projection = cc.dot(W)\n",
    "\n",
    "colors = ['red' if np.allclose(pp, np.zeros(D-1)) else 'blue' for pp in projection]\n",
    "\n",
    "# The plot below can be rotated when run in jupyter lab (i.e. 3dplot)\n",
    "ax = plt.figure().add_subplot(projection='3d')\n",
    "ax.scatter(*projection.T, color=colors)\n",
    "ax.set_aspect('equal')\n",
    "plt.axis('off')\n",
    "plt.suptitle('Projection of 4d cube along $\\mathbf{v} = (1,1,1,1)$', fontsize=20)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}
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