{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Introduction to polytopes in Sage\n", "\n", "This tutorial aims to showcase some of the possibilities of Sage concerning polytopes." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $V$-representation\n", "\n", "First, let’s define a polyhedron object as the convex hull of a set of points and some rays." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "P1 = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1]])\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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vvNLnMaxcuVL169d3+ShGUqnPA+vY05jfjh07nGFTt25dGWP03XffuRwqJZ1d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\n", "text/plain": [ "A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The string representation already gives a lot of information:\n", "\n", "* the dimension of the polyhedron (the smallest affine space containing it)\n", "* the dimension of the space in which it is defined\n", "* the base ring ($\\mathbb{Z}^2$) over which the polyhedron lives (this specifies the parent class)\n", "* the number of vertices\n", "* the number of rays\n", "\n", "Of course, you want to know what this object looks like:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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ioGLEEwUmoKgUEQUVIZ4oOAFF5YgoKDnxRAkIKCpJREFJiSdKQkBRWSIKSkY8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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P1.plot()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also add a lineality space." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see all the methods available by pressing after the dot." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "ename": "SyntaxError", "evalue": "invalid syntax (, line 1)", "output_type": "error", "traceback": [ "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m P1.\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" ] } ], "source": [ "P1." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "P2 = Polyhedron(vertices = [[1/2, 0, 0], [0, 1/2, 0]],rays = [[1, 1, 0]],lines = [[0, 0, 1]])" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 2 vertices, 1 ray, 1 line (use the .plot() method to plot)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P2" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Polyhedra in ZZ^2" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P1.parent()" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Polyhedra in QQ^3" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P2.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $H$-representation\n", "\n", "If a polyhedron object was constructed via a $V$-representation, Sage can provide the $H$-representation of the object." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "An inequality (1, 1) x - 1 >= 0\n", "An inequality (1, -1) x + 1 >= 0\n", "An inequality (-1, 1) x + 1 >= 0\n" ] } ], "source": [ "for h in P1.Hrepresentation():\n", " print(h)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Each line gives a row of the matrix $A$ and an entry of the vector $b$. The variable $x$ is a vector in the ambient space where $P1$ is defined. The $H$-representation may contain equations:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The construction of a polyhedron object via its $H$-representation, requires a precise format. Each inequality $(a_{i1},\\ldots,a_{id})⋅x+bi\\geq 0$ must be written as $[b_i,a_{i1}, \\ldots, a_{id}]$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is worth using the parameter eqns to shorten the construction of the object. In the following example, the first four rows are the negative of the second group of four rows." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As we learned from lectures, examples are very important. Sage has implemented several examples: \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Representation of objects\n", "\n", "Many objects are related to the $H$- and $V$-representations. Sage has classes implemented for them." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $H$-representation\n", "\n", "You can store the $H$-representation in a variable and use the inequalities and equalities as objects." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "P3_QQ = Polyhedron(vertices = [[0.5, 0], [0, 0.5]], base_ring=QQ)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": true }, "outputs": [], "source": [ "HRep = P3_QQ.Hrepresentation()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "An equation (2, 2) x - 1 == 0" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H1 = HRep[0]; H1" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "An inequality (0, -2) x + 1 >= 0" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2 = HRep[1]; H2" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(2, 2)" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ " H1.A()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-1" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H1.b()" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H1.is_equation()" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H1.is_inequality()" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H1.contains(vector([0,0]))" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2.contains(vector([0,0]))" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H1.is_incident(H2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is possible to obtain the different objects of the H-representation as follows." ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(An equation (2, 2) x - 1 == 0,)" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P3_QQ.equations()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(An inequality (0, -2) x + 1 >= 0, An inequality (0, 1) x + 0 >= 0)" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P3_QQ.inequalities()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### $V$-representation\n", "\n", "Similarly, you can access vertices, rays and lines of the polyhedron." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(A line in the direction (0, 0, 1),\n", " A vertex at (0, 1/2, 0),\n", " A vertex at (1/2, 0, 0),\n", " A ray in the direction (1, 1, 0))" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "VRep = P2.Vrepresentation(); VRep" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "A line in the direction (0, 0, 1)" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L = VRep[0]; L" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "A vertex at (0, 1/2, 0)" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V = VRep[1]; V" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "A ray in the direction (1, 1, 0)" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R = VRep[3]; R" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L.is_line()" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L.is_incident(V)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R.is_incident(L)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0, 0, 1)" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L.vector()" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0, 1/2, 0)" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V.vector()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is possible to obtain the different objects of the V-representation as follows." ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(A vertex at (0, 1/2, 0), A vertex at (1/2, 0, 0))" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P2.vertices()" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(A ray in the direction (1, 1, 0),)" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P2.rays()" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(A line in the direction (0, 0, 1),)" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P2.lines()" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[ 0 1/2]\n", "[1/2 0]\n", "[ 0 0]" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ " P2.vertices_matrix()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "** Exercise A **\n", "\n", "In this exercise, you will construct the cuboctahedron in several different ways and verify that you get the same combinatorial type.\n", "\n", "1) The first options is to check if it is in the library: . It is! So we can just use the provided construction." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ " CO = polytopes.cuboctahedron()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's see if it looks like what we expect:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CO.plot()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can ask the $f$-vector of the cuboctahedron:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "Img = CO.projection().tikz([-470,-1,790],36,opacity=0.1)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "proj = CO.projection()" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "proj.tikz?" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "latex.add_to_preamble(\"\\\\usepackage{tikz}\")" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\\begin{tikzpicture}%\n", "\t[x={(0.858944cm, 0.505253cm)},\n", "\ty={(-0.505040cm, 0.809017cm)},\n", "\tz={(-0.084559cm, 0.300352cm)},\n", "\tscale=2.000000,\n", "\tback/.style={loosely dotted, thin},\n", "\tedge/.style={color=blue!95!black, thick},\n", "\tfacet/.style={fill=blue!95!black,fill opacity=0.100000},\n", "\tvertex/.style={inner sep=1pt,circle,draw=green!25!black,fill=green!75!black,thick,anchor=base}]\n", "%\n", "%\n", "%% Coordinate of the vertices:\n", "%%\n", "\\coordinate (-1.00000, -1.00000, 0.00000) at (-1.00000, -1.00000, 0.00000);\n", "\\coordinate (-1.00000, 0.00000, -1.00000) at (-1.00000, 0.00000, -1.00000);\n", "\\coordinate (-1.00000, 0.00000, 1.00000) at (-1.00000, 0.00000, 1.00000);\n", "\\coordinate (-1.00000, 1.00000, 0.00000) at (-1.00000, 1.00000, 0.00000);\n", "\\coordinate (0.00000, -1.00000, -1.00000) at (0.00000, -1.00000, -1.00000);\n", "\\coordinate (0.00000, -1.00000, 1.00000) at (0.00000, -1.00000, 1.00000);\n", "\\coordinate (0.00000, 1.00000, -1.00000) at (0.00000, 1.00000, -1.00000);\n", "\\coordinate (0.00000, 1.00000, 1.00000) at (0.00000, 1.00000, 1.00000);\n", "\\coordinate (1.00000, -1.00000, 0.00000) at (1.00000, -1.00000, 0.00000);\n", "\\coordinate (1.00000, 0.00000, -1.00000) at (1.00000, 0.00000, -1.00000);\n", "\\coordinate (1.00000, 0.00000, 1.00000) at (1.00000, 0.00000, 1.00000);\n", "\\coordinate (1.00000, 1.00000, 0.00000) at (1.00000, 1.00000, 0.00000);\n", "%%\n", "%%\n", "%% Drawing edges in the back\n", "%%\n", "\\draw[edge,back] (-1.00000, 0.00000, -1.00000) -- (0.00000, -1.00000, -1.00000);\n", "\\draw[edge,back] (-1.00000, 0.00000, -1.00000) -- (0.00000, 1.00000, -1.00000);\n", "\\draw[edge,back] (-1.00000, 1.00000, 0.00000) -- (0.00000, 1.00000, -1.00000);\n", "\\draw[edge,back] (0.00000, -1.00000, -1.00000) -- (1.00000, 0.00000, -1.00000);\n", "\\draw[edge,back] (0.00000, 1.00000, -1.00000) -- (1.00000, 0.00000, -1.00000);\n", "\\draw[edge,back] (0.00000, 1.00000, -1.00000) -- (1.00000, 1.00000, 0.00000);\n", "\\draw[edge,back] (1.00000, -1.00000, 0.00000) -- (1.00000, 0.00000, -1.00000);\n", "\\draw[edge,back] (1.00000, 0.00000, -1.00000) -- (1.00000, 1.00000, 0.00000);\n", "%%\n", "%%\n", "%% Drawing vertices in the back\n", "%%\n", "\\node[vertex] at (0.00000, 1.00000, -1.00000) {};\n", "\\node[vertex] at (1.00000, 0.00000, -1.00000) {};\n", "%%\n", "%%\n", "%% Drawing the facets\n", "%%\n", "\\fill[facet] (0.00000, -1.00000, 1.00000) -- (-1.00000, -1.00000, 0.00000) -- (-1.00000, 0.00000, 1.00000) -- cycle {};\n", "\\fill[facet] (-1.00000, 1.00000, 0.00000) -- (-1.00000, 0.00000, -1.00000) -- (-1.00000, -1.00000, 0.00000) -- (-1.00000, 0.00000, 1.00000) -- cycle {};\n", "\\fill[facet] (0.00000, 1.00000, 1.00000) -- (-1.00000, 0.00000, 1.00000) -- (-1.00000, 1.00000, 0.00000) -- cycle {};\n", "\\fill[facet] (1.00000, -1.00000, 0.00000) -- (0.00000, -1.00000, -1.00000) -- (-1.00000, -1.00000, 0.00000) -- (0.00000, -1.00000, 1.00000) -- cycle {};\n", "\\fill[facet] (1.00000, 0.00000, 1.00000) -- (0.00000, -1.00000, 1.00000) -- (-1.00000, 0.00000, 1.00000) -- (0.00000, 1.00000, 1.00000) -- cycle {};\n", "\\fill[facet] (1.00000, 0.00000, 1.00000) -- (0.00000, -1.00000, 1.00000) -- (1.00000, -1.00000, 0.00000) -- cycle {};\n", "\\fill[facet] (1.00000, 1.00000, 0.00000) -- (0.00000, 1.00000, 1.00000) -- (1.00000, 0.00000, 1.00000) -- cycle {};\n", "%%\n", "%%\n", "%% Drawing edges in the front\n", "%%\n", "\\draw[edge] (-1.00000, -1.00000, 0.00000) -- (-1.00000, 0.00000, -1.00000);\n", "\\draw[edge] (-1.00000, -1.00000, 0.00000) -- (-1.00000, 0.00000, 1.00000);\n", "\\draw[edge] (-1.00000, -1.00000, 0.00000) -- (0.00000, -1.00000, -1.00000);\n", "\\draw[edge] (-1.00000, -1.00000, 0.00000) -- (0.00000, -1.00000, 1.00000);\n", "\\draw[edge] (-1.00000, 0.00000, -1.00000) -- (-1.00000, 1.00000, 0.00000);\n", "\\draw[edge] (-1.00000, 0.00000, 1.00000) -- (-1.00000, 1.00000, 0.00000);\n", "\\draw[edge] (-1.00000, 0.00000, 1.00000) -- (0.00000, -1.00000, 1.00000);\n", "\\draw[edge] (-1.00000, 0.00000, 1.00000) -- (0.00000, 1.00000, 1.00000);\n", "\\draw[edge] (-1.00000, 1.00000, 0.00000) -- (0.00000, 1.00000, 1.00000);\n", "\\draw[edge] (0.00000, -1.00000, -1.00000) -- (1.00000, -1.00000, 0.00000);\n", "\\draw[edge] (0.00000, -1.00000, 1.00000) -- (1.00000, -1.00000, 0.00000);\n", "\\draw[edge] (0.00000, -1.00000, 1.00000) -- (1.00000, 0.00000, 1.00000);\n", "\\draw[edge] (0.00000, 1.00000, 1.00000) -- (1.00000, 0.00000, 1.00000);\n", "\\draw[edge] (0.00000, 1.00000, 1.00000) -- (1.00000, 1.00000, 0.00000);\n", "\\draw[edge] (1.00000, -1.00000, 0.00000) -- (1.00000, 0.00000, 1.00000);\n", "\\draw[edge] (1.00000, 0.00000, 1.00000) -- (1.00000, 1.00000, 0.00000);\n", "%%\n", "%%\n", "%% Drawing the vertices in the front\n", "%%\n", "\\node[vertex] at (-1.00000, -1.00000, 0.00000) {};\n", "\\node[vertex] at (-1.00000, 0.00000, -1.00000) {};\n", "\\node[vertex] at (-1.00000, 0.00000, 1.00000) {};\n", "\\node[vertex] at (-1.00000, 1.00000, 0.00000) {};\n", "\\node[vertex] at (0.00000, -1.00000, -1.00000) {};\n", "\\node[vertex] at (0.00000, -1.00000, 1.00000) {};\n", "\\node[vertex] at (0.00000, 1.00000, 1.00000) {};\n", "\\node[vertex] at (1.00000, -1.00000, 0.00000) {};\n", "\\node[vertex] at (1.00000, 0.00000, 1.00000) {};\n", "\\node[vertex] at (1.00000, 1.00000, 0.00000) {};\n", "%%\n", "%%\n", "\\end{tikzpicture}" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Img" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(1, 12, 24, 14, 1)" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "CO.f_vector()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "2) The second construction uses the fact that we can truncate the vertices by new hyperplanes that cross the edges exactly at the mid-point.\n", "\n" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": true }, "outputs": [], "source": [ "Cube = polytopes.cube()" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Applications/SageMath/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:1: DeprecationWarning: edge_truncation is deprecated. Please use truncation instead.\n", "See http://trac.sagemath.org/18128 for details.\n", " from ipykernel.kernelapp import IPKernelApp\n" ] } ], "source": [ "EC = Cube.edge_truncation(1/2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Just to make a sanity check we can compute the f-vector:\n", "\n" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(1, 12, 24, 14, 1)" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EC.f_vector()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "But as we know, two different polytopes can have the same f-vector, so we should check if they are combinatorially isomorphic:\n", "\n" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EC.is_combinatorially_isomorphic(CO)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is great!\n", "\n", "Just to understand the truncation construction better, visualize the usual truncation:\n" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "collapsed": true }, "outputs": [], "source": [ "TC = Cube.truncation()" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "TC.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "3) The third construction will use some nice trick that we can do in sage.\n", "\n", "We can take the polar dual of the cube, the octahedron, and scale it by a factor 2:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "collapsed": true }, "outputs": [], "source": [ "OC = 2*Cube.polar()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Then, we can take the intersection with the cube again :" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [], "source": [ "ECCO = Cube & OC" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What is ECCO?" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ECCO.is_combinatorially_isomorphic(EC)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That's it! You found 3 ways to construct the cuboctahedron.\n", "\n", "* Question: do they all have the same vertices?\n", "* Question: which polytope do we get if we dilate the octahedron by only 3/2 instead of 2?\n", "* Can you find this polytope in the library and check if it is isomorphic?" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "** Exercise B **\n", "\n", "Choose five examples discussed in lecture from the list:\n", "\n", "And find their $V$-representation, $H$-representation, and $f$-vector\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.3.beta1", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 2 }