examples/3D Plotting a projection of the 4D permutahedron.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook we plot the [permutahedron](https://en.wikipedia.org/wiki/Permutohedron) of order 4 which is a 3-dimensional polyhedron living in a 4-dimensional space.\n",
    "It is defined as the convex hull of all permutations of $(0, 1, 2, 3)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "V-representation Polyhedra.PointsHull{Int64,Array{Int64,1},Int64}:\n",
       "24-element iterator of Array{Int64,1}:\n",
       " [0, 1, 2, 3]\n",
       " [0, 1, 3, 2]\n",
       " [0, 2, 1, 3]\n",
       " [0, 2, 3, 1]\n",
       " [0, 3, 1, 2]\n",
       " [0, 3, 2, 1]\n",
       " [1, 0, 2, 3]\n",
       " [1, 0, 3, 2]\n",
       " [1, 2, 0, 3]\n",
       " [1, 2, 3, 0]\n",
       " [1, 3, 0, 2]\n",
       " [1, 3, 2, 0]\n",
       " [2, 0, 1, 3]\n",
       " [2, 0, 3, 1]\n",
       " [2, 1, 0, 3]\n",
       " [2, 1, 3, 0]\n",
       " [2, 3, 0, 1]\n",
       " [2, 3, 1, 0]\n",
       " [3, 0, 1, 2]\n",
       " [3, 0, 2, 1]\n",
       " [3, 1, 0, 2]\n",
       " [3, 1, 2, 0]\n",
       " [3, 2, 0, 1]\n",
       " [3, 2, 1, 0]"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Combinatorics, Polyhedra\n",
    "v = vrep(collect(permutations([0, 1, 2, 3])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To plot a polyhedron, we need both the H-representation and V-representation so we will need a library to do representation conversion.\n",
    "We choose [CDD](https://github.com/JuliaPolyhedra/CDDLib.jl) in floating point arithmetic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Polyhedron CDDLib.Polyhedron{Float64}:\n",
       "24-element iterator of Array{Float64,1}:\n",
       " [0.0, 1.0, 2.0, 3.0]\n",
       " [0.0, 1.0, 3.0, 2.0]\n",
       " [0.0, 2.0, 1.0, 3.0]\n",
       " [0.0, 2.0, 3.0, 1.0]\n",
       " [0.0, 3.0, 1.0, 2.0]\n",
       " [0.0, 3.0, 2.0, 1.0]\n",
       " [1.0, 0.0, 2.0, 3.0]\n",
       " [1.0, 0.0, 3.0, 2.0]\n",
       " [1.0, 2.0, 0.0, 3.0]\n",
       " [1.0, 2.0, 3.0, 0.0]\n",
       " [1.0, 3.0, 0.0, 2.0]\n",
       " [1.0, 3.0, 2.0, 0.0]\n",
       " [2.0, 0.0, 1.0, 3.0]\n",
       " [2.0, 0.0, 3.0, 1.0]\n",
       " [2.0, 1.0, 0.0, 3.0]\n",
       " [2.0, 1.0, 3.0, 0.0]\n",
       " [2.0, 3.0, 0.0, 1.0]\n",
       " [2.0, 3.0, 1.0, 0.0]\n",
       " [3.0, 0.0, 1.0, 2.0]\n",
       " [3.0, 0.0, 2.0, 1.0]\n",
       " [3.0, 1.0, 0.0, 2.0]\n",
       " [3.0, 1.0, 2.0, 0.0]\n",
       " [3.0, 2.0, 0.0, 1.0]\n",
       " [3.0, 2.0, 1.0, 0.0]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using CDDLib\n",
    "p4 = polyhedron(v, CDDLib.Library())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The permutahedron lives in a 4-dimension space but is 3-dimensional as it is contained in the hyperplane $x_1 + x_2 + x_3 + x_4 = 0 + 1 + 2 + 3 = 6$.\n",
    "We choose an orthogonal basis of this hyperplane: $(1, -1, 0, 0)$, $(1, 1, -2, 0)$ and $(1, 1, 1, -3)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "v1 = [1, -1,  0,  0]\n",
    "v2 = [1,  1, -2,  0]\n",
    "v3 = [1,  1,  1, -3];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We project the polyhedron in this basis to obtain a full dimensional 3-dimensional polyhedron living in a 3-dimensional space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Polyhedron CDDLib.Polyhedron{Float64}:\n",
       "14-element iterator of HalfSpace{Float64,Array{Float64,1}}:\n",
       " HalfSpace([-0.0, 3.26599, -1.1547], 6.0)\n",
       " HalfSpace([1.41421, 0.816497, -1.1547], 3.9999999999999996)\n",
       " HalfSpace([-0.0, -0.0, -1.1547], 2.0)\n",
       " HalfSpace([-1.41421, 0.816497, -1.1547], 3.999999999999999)\n",
       " HalfSpace([-2.82843, -1.63299, -1.1547], 6.0)\n",
       " HalfSpace([-0.0, -1.63299, -1.1547], 4.000000000000001)\n",
       " HalfSpace([-1.41421, -0.816497, 1.1547], 4.000000000000002)\n",
       " HalfSpace([-0.0, -3.26599, 1.1547], 6.000000000000011)\n",
       " HalfSpace([-2.82843, 1.63299, 1.1547], 5.999999999999995)\n",
       " HalfSpace([2.82843, -1.63299, -1.1547], 6.0)\n",
       " HalfSpace([1.41421, -0.816497, 1.1547], 4.0000000000000036)\n",
       " HalfSpace([-0.0, -0.0, 1.1547], 2.0)\n",
       " HalfSpace([-0.0, 1.63299, 1.1547], 3.9999999999999964)\n",
       " HalfSpace([2.82843, 1.63299, 1.1547], 5.9999999999999964)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p3 = project(p4, [v1 v2 v3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get a plottable object, we transform the polyhedron into a mesh as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = Polyhedra.Mesh(p3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now plot this mesh with [MeshCat](https://github.com/rdeits/MeshCat.jl) or [Makie](https://github.com/JuliaPlots/Makie.jl) as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using MeshCat\n",
    "vis = Visualizer()\n",
    "setobject!(vis, m)\n",
    "IJuliaCell(vis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Makie\n",
    "mesh(m, color=:blue)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Makie\n",
    "wireframe(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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