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<head><title>faces -- computes all faces of a certain codimension of a Cone or Polyhedron</title>
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<div><h1>faces -- computes all faces of a certain codimension of a Cone or Polyhedron</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> L = faces(k,C) </tt><br/><tt>L = faces(k,P)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>k</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
<li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
<li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, containing the faces of codimension <tt>k</tt></span></li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p/>
<tt>faces</tt> computes the faces of codimension <tt>k</tt> of the given <a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a>
or <a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a>, where <tt>k</tt> must be between 0 and the dimension of the second
argument. The faces will be of the same class as the original convex object.<p/>
For example, we can look at the edges of the cyclicPolytope with 5 vertices in 3 space<table class="examples"><tr><td><pre>i1 : P = cyclicPolytope(3,5)
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 6
number of rays => 0
number of vertices => 5
o1 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i2 : L = faces(2,P)
o2 = {{ambient dimension => 3 }, {ambient dimension => 3
dimension of lineality space => 0 dimension of lineality space =>
dimension of polyhedron => 1 dimension of polyhedron => 1
number of facets => 2 number of facets => 2
number of rays => 0 number of rays => 0
number of vertices => 2 number of vertices => 2
------------------------------------------------------------------------
}, {ambient dimension => 3 }, {ambient dimension => 3
0 dimension of lineality space => 0 dimension of lineality space
dimension of polyhedron => 1 dimension of polyhedron => 1
number of facets => 2 number of facets => 2
number of rays => 0 number of rays => 0
number of vertices => 2 number of vertices => 2
------------------------------------------------------------------------
}, {ambient dimension => 3 },
=> 0 dimension of lineality space => 0
dimension of polyhedron => 1
number of facets => 2
number of rays => 0
number of vertices => 2
------------------------------------------------------------------------
{ambient dimension => 3 }, {ambient dimension => 3
dimension of lineality space => 0 dimension of lineality space =>
dimension of polyhedron => 1 dimension of polyhedron => 1
number of facets => 2 number of facets => 2
number of rays => 0 number of rays => 0
number of vertices => 2 number of vertices => 2
------------------------------------------------------------------------
}, {ambient dimension => 3 }, {ambient dimension => 3
0 dimension of lineality space => 0 dimension of lineality space
dimension of polyhedron => 1 dimension of polyhedron => 1
number of facets => 2 number of facets => 2
number of rays => 0 number of rays => 0
number of vertices => 2 number of vertices => 2
------------------------------------------------------------------------
}}
=> 0
o2 : List</pre>
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Since this is only a list of polyhedra we look at their vertices:<table class="examples"><tr><td><pre>i3 : apply(L,vertices)
o3 = {| 0 2 |, | 1 2 |, | 0 1 |, | 0 3 |, | 2 3 |, | 3 4 |, | 0 4 |, | 2
| 0 4 | | 1 4 | | 0 1 | | 0 9 | | 4 9 | | 9 16 | | 0 16 | | 4
| 0 8 | | 1 8 | | 0 1 | | 0 27 | | 8 27 | | 27 64 | | 0 64 | | 8
------------------------------------------------------------------------
4 |, | 1 4 |}
16 | | 1 16 |
64 | | 1 64 |
o3 : List</pre>
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<div class="waystouse"><h2>Ways to use <tt>faces</tt> :</h2>
<ul><li>faces(ZZ,Cone)</li>
<li>faces(ZZ,Polyhedron)</li>
</ul>
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