This generates a Möbius strip using ipyvolume. (3D plots are not rendered inside this gist!)

moebius.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Möbius Strip\n",
    "\n",
    "This generates a Möbius strip using ipyvolume.\n",
    "\n",
    "The data points are calculated using the parametric form described in  \n",
    "https://jakevdp.github.io/PythonDataScienceHandbook/04.12-three-dimensional-plotting.html#Example:-Visualizing-a-M%C3%B6bius-strip\n",
    "\n",
    "Still desired: add a texture like it is available with `ipv.examples.klein_bottle(texture=image)`. Any help welcome!!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import ipyvolume as ipv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_points(num_theta=30, num_width=4):\n",
    "    \"Calculate x, y, and z vectors for a Möbius strip.\"\n",
    "    theta = np.linspace(0, 2 * np.pi, num_theta)\n",
    "    width = np.linspace(-0.4, 0.4, num_width)\n",
    "    width, theta = np.meshgrid(width, theta)\n",
    "    phi = 0.5 * theta\n",
    "    r = 1 + width * np.cos(phi)  # radius in x-y plane\n",
    "    x = np.ravel(r * np.cos(theta))\n",
    "    y = np.ravel(r * np.sin(theta))\n",
    "    z = np.ravel(width * np.sin(phi))\n",
    "    return x, y, z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Points only"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "069e572ad2e64a1c8cea62096493b371",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(Figure(camera=PerspectiveCamera(fov=46.0, position=(0.0, 0.0, 2.0), quaternion=(0.0, 0.0, 0.0, …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y, z = calculate_points()\n",
    "ipv.figure()\n",
    "ipv.scatter(x, y, z, size=4, color=np.array([x, y, z]).T, marker='sphere')\n",
    "ipv.squarelim()\n",
    "ipv.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate triangulation mesh manually"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "e13da38bf7a048e0a1cb14d2dc408ec7",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(Figure(camera=PerspectiveCamera(fov=46.0, position=(0.0, 0.0, 2.0), quaternion=(0.0, 0.0, 0.0, …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nt, nw = 30, 4\n",
    "x, y, z = calculate_points(num_theta=nt, num_width=nw)\n",
    "# works only for num_w == 4\n",
    "rng = range(0, nt*nw-nw, nw)\n",
    "triangles = [[i, i+1, i+4] for i in rng]\n",
    "triangles += [[i+1, i+4, i+5] for i in rng]\n",
    "triangles += [[i+1, i+2, i+5] for i in rng]\n",
    "triangles += [[i+2, i+5, i+6] for i in rng]\n",
    "triangles += [[i+2, i+3, i+6] for i in rng]\n",
    "triangles += [[i+2, i+3, i+6] for i in rng]\n",
    "triangles += [[i+3, i+6, i+7] for i in rng]\n",
    "\n",
    "ipv.figure()\n",
    "# ipv.scatter(x, y, z)\n",
    "# ipv.plot(x, y, z)\n",
    "ipv.plot_trisurf(x, y, z, triangles=triangles)\n",
    "ipv.squarelim()\n",
    "ipv.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate mesh automatically for any theta/width parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def tri(num_theta, num_width):\n",
    "    \"Triangulate a Möbius stripe into a list of index point triples (i, j, k) for mesh.\"\n",
    "    nt, nw = num_theta, num_width\n",
    "    for i in range(0, nt*nw-nw, nw):\n",
    "        for j in range(nw-1):\n",
    "            yield [i+j, i+j+1, i+j+nw]\n",
    "            yield [i+j+1, i+j+nw, i+j+nw+1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "64a99bf416474e22a75bf49c3fbd58fe",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(Figure(camera=PerspectiveCamera(fov=46.0, position=(0.0, 0.0, 2.0), quaternion=(0.0, 0.0, 0.0, …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nt, nw = 30, 8\n",
    "x, y, z = calculate_points(num_theta=nt, num_width=nw)\n",
    "ipv.figure()\n",
    "ipv.scatter(x, y, z, marker='sphere')\n",
    "# ipv.plot(x, y, z, color='orange')\n",
    "triangles = list(tri(nt, nw))\n",
    "ipv.plot_trisurf(x, y, z, triangles=triangles)\n",
    "ipv.squarelim()\n",
    "ipv.show()"
   ]
  }
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