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<head><title>isPolytopal -- checks if a Fan is polytopal</title>
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<div><h1>isPolytopal -- checks if a Fan is polytopal</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>b = isPolytopal F</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>F</tt>, <span>an object of class <a href="___Fan.html" title="the class of all fans">Fan</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <a href="../../Macaulay2Doc/html/_true.html" title="">true</a> if the <a href="___Fan.html" title="the class of all fans">Fan</a> is polytopal, <a href="../../Macaulay2Doc/html/_false.html" title="">false</a> otherwise</span></li>
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<div class="single"><h2>Description</h2>
<div><p/>
If <tt>F</tt> is projective, then there exists a polyhedron <tt>P</tt> such that <tt>F</tt>
is the normalFan of <tt>P</tt>. This means every codimension 1 cone of the Fan corresponds exactly to
an edge of the polytope. So consider <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> to the number of all edges. This can be considered as the
space of all edge lengths. If we take arbitrary lengths now for every edge we do not get a polytope. But
every codimension 2 cone of the fan corresponds to a 2 dimensional face of the polytope and if the edges
belonging to this face add up to 0 zero, they form in fact a 2 dimensional face. This gives linear
equations on the space of edge lengths and if we intersect these equations with the positive orthant in
the space of edge lengths we get a Cone. Thus, there exists such a polytope if and only if there is a
vector in this cone with strictly positive entries, since every edge has to appear in the polytope.<p/>
IF <tt>F</tt> is polytopal, the function <a href="_polytope.html" title="returns a polytope of which the fan is the normal fan if it is polytopal">polytope</a> returns a polytope of which <tt>F</tt> is
the normalFan.<p/>
Note that the function first checks if the fan is complete.<table class="examples"><tr><td><pre>i1 : C1 = posHull matrix {{1,0},{0,1}};</pre>
</td></tr>
<tr><td><pre>i2 : C2 = posHull matrix {{1,-1},{0,-2}};</pre>
</td></tr>
<tr><td><pre>i3 : C3 = posHull matrix {{0,-2},{1,-1}};</pre>
</td></tr>
<tr><td><pre>i4 : C4 = posHull matrix {{-1,-2},{-2,-1}};</pre>
</td></tr>
<tr><td><pre>i5 : F = fan{C1,C2,C3,C4}
o5 = {ambient dimension => 2 }
number of generating cones => 4
number of rays => 4
top dimension of the cones => 2
o5 : Fan</pre>
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<tr><td><pre>i6 : isPolytopal F
o6 = true</pre>
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<div class="waystouse"><h2>Ways to use <tt>isPolytopal</tt> :</h2>
<ul><li>isPolytopal(Fan)</li>
</ul>
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