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   "source": [
    "# Random Constructions\n",
    "\n",
    "## Random points on the unit sphere\n",
    "\n",
    "The easiest way to randomly construct a polytope is by sampling points on the unit sphere.  The following chooses 100 points on the units sphere in 3-space.\n",
    "\n",
    "    \n"
   ]
  },
  {
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       "1\n",
       "\n"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "$p1=rand_sphere(3,100);\n",
    "print $p1->SIMPLICIAL;"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With probability one such polytopes are simplicial.\n",
    "\n",
    "## Random polytopes with are neither simplicial nor simple\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0 0\n",
       "30 166 264 128\n",
       "\n"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "($d,$m,$n) = (4,50,30);\n",
    "$p1=rand_sphere($d,$m);\n",
    "$p2=polarize($p1);\n",
    "$p3=new Polytope(POINTS=>rand_vert($p2->VERTICES,$n));\n",
    "print $p3->SIMPLICIAL, \" \", $p3->SIMPLE, \"\\n\", $p3->F_VECTOR;"
   ]
  }
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