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<!DOCTYPE html> <html lang="en"> <head> <title>skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="for computations with convex polyhedra, cones, and fans" href="index.html">Polyhedra</a> :: <a title="computes the k-skeleton of a Fan or PolyhedralComplex" href="_skeleton.html">skeleton</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_smallest__Face.html">next</a> | <a href="_simplex.html">previous</a> | <a href="_smallest__Face.html">forward</a> | <a href="_simplex.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">X = skeleton(k,F) </code></dd> <dd><code class="language-macaulay2">X = skeleton(k,PC)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">k</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li> <li><span><span class="tt">F</span>, <span>an instance of the type <a title="the class of all fans" href="___Fan.html">Fan</a></span></span></li> <li><span><span class="tt">PC</span>, <span>an instance of the type <a title="the class of all polyhedral complexes" href="___Polyhedral__Complex.html">PolyhedralComplex</a></span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">X</span>, <span>an instance of the type <a title="the class of all fans" href="___Fan.html">Fan</a></span>, or <span>an instance of the type <a title="the class of all polyhedral complexes" href="___Polyhedral__Complex.html">PolyhedralComplex</a></span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> <p></p> For a <a title="the class of all fans" href="___Fan.html">Fan</a><span class="tt"> F</span> and an integer <span class="tt">k</span> between 0 and the dimension of <span class="tt">F</span>, <span class="tt">skeleton</span> computes the <span class="tt">k</span>-skeleton of the <a title="the class of all fans" href="___Fan.html">Fan</a> <span class="tt">F</span>, i.e. the <a title="the class of all fans" href="___Fan.html">Fan</a> <span class="tt">F1</span> generated by all cones of dimension <span class="tt">k</span> in <span class="tt">F</span>. <p></p> For example, we can look at the 2-skeleton of the fan of projective 3-space: <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : P = convexHull matrix{{1,0,0,0},{0,1,0,0},{0,0,1,0}} o1 = P o1 : Polyhedron</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : F = normalFan P o2 = F o2 : Fan</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : F1 = skeleton(2,F) o3 = F1 o3 : Fan</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : raysF = rays F o4 = | 1 0 -1 0 | | 0 1 -1 0 | | 0 0 -1 1 | 3 4 o4 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : apply(maxCones F1, mc -> raysF_mc) o5 = {| 1 0 |, | 1 -1 |, | 1 0 |, | 0 -1 |, | 0 0 |, | -1 0 |} | 0 1 | | 0 -1 | | 0 0 | | 1 -1 | | 1 0 | | -1 0 | | 0 0 | | 0 -1 | | 0 1 | | 0 -1 | | 0 1 | | -1 1 | o5 : List</code></pre> </td> </tr> </table> <p></p> For a <a title="the class of all polyhedral complexes" href="___Polyhedral__Complex.html">PolyhedralComplex</a><span class="tt"> PC</span> and an integer <span class="tt">k</span> between 0 and the dimension of <span class="tt">PC</span>, <span class="tt">skeleton</span> computes the <span class="tt">k</span>-skeleton of the <a title="the class of all polyhedral complexes" href="___Polyhedral__Complex.html">PolyhedralComplex</a> <span class="tt">PC</span>, i.e. the <a title="the class of all polyhedral complexes" href="___Polyhedral__Complex.html">PolyhedralComplex</a> <span class="tt">PC1</span> generated by all polyhedra of dimension <span class="tt">k</span> in <span class="tt">PC</span>. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : PC = polyhedralComplex hypercube 3 o6 = PC o6 : PolyhedralComplex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : PC1 = skeleton(2,PC) o7 = PC1 o7 : PolyhedralComplex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : vertPC1 = vertices PC1 o8 = | -1 1 -1 1 -1 1 -1 1 | | -1 -1 1 1 -1 -1 1 1 | | -1 -1 -1 -1 1 1 1 1 | 3 8 o8 : Matrix QQ <-- QQ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : apply(maxPolyhedra PC1, mp -> vertPC1_(mp#0)) o9 = {| -1 1 |, | -1 -1 |, | -1 -1 |, | 1 1 |, | 1 1 |, | -1 1 |, | -1 | -1 -1 | | -1 1 | | -1 -1 | | -1 1 | | -1 -1 | | 1 1 | | 1 | -1 -1 | | -1 -1 | | -1 1 | | -1 -1 | | -1 1 | | -1 -1 | | -1 ------------------------------------------------------------------------ -1 |, | 1 1 |, | -1 1 |, | -1 -1 |, | 1 1 |, | -1 1 |} 1 | | 1 1 | | -1 -1 | | -1 1 | | -1 1 | | 1 1 | 1 | | -1 1 | | 1 1 | | 1 1 | | 1 1 | | 1 1 | o9 : List</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">skeleton</span>:</h2> <ul> <li><kbd>skeleton(ZZ,Fan)</kbd></li> <li><kbd>skeleton(ZZ,PolyhedralComplex)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="computes the k-skeleton of a Fan or PolyhedralComplex" href="_skeleton.html">skeleton</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Polyhedra/documentation/old_documentation.m2:2089:0</span>.</p> </div> </div> </div> </body> </html>