<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>dualGrading -- The dual vertices of a polytope.</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_dualize.html">next</a> | <a href="_dual__Face.html">previous</a> | <a href="_dualize.html">forward</a> | <a href="_dual__Face.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>dualGrading -- The dual vertices of a polytope.</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>dualGrading(C)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, <span>an <a href="___Complex.html">embedded complex</a></span>, a polytope</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p>Returns the dual grading of (i.e., a matrix whose row are) the dual polytope of C. The rows are sorted according to the <a href="_polytopal__Facets.html" title="The facets of a polytope.">polytopalFacets</a> of C.</p>
<div/>
<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : C=simplex(R)
o2 = 4: x x x x x
0 1 2 3 4
o2 : complex of dim 4 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0</pre>
</td></tr>
<tr><td><pre>i3 : grading C
o3 = | -1 -1 -1 -1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
5 4
o3 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i4 : dA=dualGrading C
o4 = | -1 -1 -1 4 |
| -1 -1 4 -1 |
| -1 4 -1 -1 |
| 4 -1 -1 -1 |
| -1 -1 -1 -1 |
5 4
o4 : Matrix QQ <--- QQ</pre>
</td></tr>
<tr><td><pre>i5 : dA===grading dualize C
o5 = true</pre>
</td></tr>
<tr><td><pre>i6 : dA===C.dualComplex.simplexRing.grading
o6 = true</pre>
</td></tr>
<tr><td><pre>i7 : pf=polytopalFacets C
o7 = {x x x x , x x x x , x x x x , x x x x , x x x x }
0 1 2 3 0 1 2 4 0 1 3 4 0 2 3 4 1 2 3 4
o7 : List</pre>
</td></tr>
<tr><td><pre>i8 : coordinates pf#0
o8 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}
o8 : List</pre>
</td></tr>
<tr><td><pre>i9 : (dualGrading C)^{0}
o9 = | -1 -1 -1 4 |
1 4
o9 : Matrix QQ <--- QQ</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div><div>This just returns C.dualComplex.grading. If this data has not beed computed use <a href="_vertices__Dual__Polytope.html" title="The dual vertices of a polytope.">verticesDualPolytope</a>. Integrate into this verticesDualPolytope.</div>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_vertices__Dual__Polytope.html" title="The dual vertices of a polytope.">verticesDualPolytope</a> -- The dual vertices of a polytope.</span></li>
<li><span><a href="_grading.html" title="The grading of a first order deformation or complex or polynomial ring.">grading</a> -- The grading of a first order deformation or complex or polynomial ring.</span></li>
<li><span><a href="_polytopal__Facets.html" title="The facets of a polytope.">polytopalFacets</a> -- The facets of a polytope.</span></li>
<li><span><a href="_dualize.html" title="The dual of a face or complex.">dualize</a> -- The dual of a face or complex.</span></li>
<li><span><a href="_is__Polytope.html" title="Check whether a complex is a polytope.">isPolytope</a> -- Check whether a complex is a polytope.</span></li>
<li><span><a href="_facets.html" title="The maximal faces of a complex.">facets</a> -- The maximal faces of a complex.</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>dualGrading</tt> :</h2>
<ul><li>dualGrading(Complex)</li>
</ul>
</div>
</div>
</body>
</html>
