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<head><title>faces -- the i-faces of a simplicial complex </title>
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<div><h1>faces -- the i-faces of a simplicial complex </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>faces(i,D)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>i</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the dimension of the faces</span></li>
<li><span><tt>D</tt>, <span>a <a href="___Simplicial__Complex.html">simplicial complex</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with one row, whose entries are squarefree monomials representing the faces of dimension <tt>i</tt> of <tt>D</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>In Macaulay2, every <a href="___Simplicial__Complex.html">simplicial complex</a> is equipped with a polynomial ring, and the matrix of i-faces is defined over this ring.<table class="examples"><tr><td><pre>i1 : loadPackage "SimplicialComplexes";</pre>
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This triangulation of the real projective plane has 6 vertices, 15 edges and 10 triangles.<table class="examples"><tr><td><pre>i2 : R = ZZ[a..f]
o2 = R
o2 : PolynomialRing</pre>
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<tr><td><pre>i3 : D = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,
b*c*d,b*d*e,b*e*f,c*d*f,c*e*f)
o3 = | def aef bdf bcf acf cde bce abe acd abd |
o3 : SimplicialComplex</pre>
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<tr><td><pre>i4 : faces(-1,D)
o4 = | 1 |
1 1
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : faces(0,D)
o5 = | a b c d e f |
1 6
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : faces(1,D)
o6 = | ab ac ad ae af bc bd be bf cd ce cf de df ef |
1 15
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : faces(2,D)
o7 = | abd abe acd acf aef bce bcf bdf cde def |
1 10
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : fVector D
o8 = HashTable{-1 => 1}
0 => 6
1 => 15
2 => 10
o8 : HashTable</pre>
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To avoid repeated computation, the matrix of <tt>i</tt>-faces is cached at <tt>D.cache.faces#i</tt>. This function will use this value if it has already been computed.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="simplicial complexes">SimplicialComplexes</a> -- simplicial complexes</span></li>
<li><span><a href="_facets.html" title="the facets of a simplicial complex">facets</a> -- the facets of a simplicial complex</span></li>
<li><span><a href="_boundary.html" title="boundary operator">boundary</a> -- boundary operator</span></li>
<li><span><a href="_f__Vector.html" title="the f-vector of a simplicial complex">fVector</a> -- the f-vector of a simplicial complex</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>faces</tt> :</h2>
<ul><li>faces(ZZ,SimplicialComplex)</li>
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