{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "A random polytope constructed from tangents to the unit sphere is almost surely simple." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " PR\n", " \n", " \n", "\n", "\n", "\n", "\t\t
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       "polymake: used package lrs\n",
       "  Implementation of the reverse search algorithm of Avis and Fukuda.\n",
       "  Copyright by David Avis.\n",
       "  http://cgm.cs.mcgill.ca/~avis/C/lrs.html\n",
       "\n",
       "polymake: used package threejs\n",
       "   Three.js is a lightweight cross-browser JavaScript library/API used to create and display animated 3D computer graphics on a Web browser.\n",
       "   See http://github.com/mrdoob for the source code.\n",
       "\n",
       "
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\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "$R=rand_sphere(3,20);\n", "$PR=polarize($R);\n", "print $PR->SIMPLE, \"\\n\";\n", "$PR->VISUAL;" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Random sampling among its vertices yields a polytope which is almost surely neither simple nor simplicial." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "false false" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" }, { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", " Q\n", " \n", " \n", "\n", "\n", "\n", "\t\t
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