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<title>GAMGI Tutorials: Voronoi tesselation</title>
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<h1>Voronoi tesselation</h1>
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<ul>
<li><a href="voronoi_page1.html">Page 1</a></li>
<li><span>Page 2</span></li>
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Then we import a simple cubic structure and analyse the data
produced by the tesselation, to ilustrate the various properties
reported, first with <b>Variancy</b> set to 0.0 and then with
<b>Variancy</b> set to 1.0, in both cases with periodic boundary
conditions. The two polyhedra obtained can be seen in the figure
at <a href="http://www.gamgi.org/images/voronoi2.png">
http://www.gamgi.org/images/voronoi2.png</a>.
<h3>Cubic structure</h3>
<ol>
<li>Select <b>Gamgi->Reset</b> and press <b>Ok</b>, to remove the various
layers and restart Gamgi. Press <b>File->Import</b> and import the file
<b>dat/cluster/cs111.xml</b>, containing a cluster with eight atoms of
different elements, in a cubic arrangement.</li>
<li>Select <b>Cluster->Measure</b>, set <b>Method</b> to <b>Voronoi</b>,
<b>Boundaries</b> to <b>Periodic</b>, <b>Offset</b> to <b>1.5</b> and
click on the cluster. A cubic polyhedra is shown, in a new layer, formed
by eight octant cubes, one for each atom.</li>
<li>Try to understand the topological data reported. For example,
<b>Polyhedron Faces</b>, <b>Polyhedron Edges</b> and <b>Polyhedron
Vertices</b> are 6, 12 and 8, respectively, and <b>Face Vertices</b>
is 4.
<p/>
<b>Edge Polyhedrons</b> can be 1 (edges on the edges of the total
polyhedron), 2 (edges on the faces of the total polyhedron) or 4
(edges inside the total polyhedron). <b>Edge Faces</b> can be 2
(edges on the outside of the total polyhedron) or 4 (edges on the
inside of the total polyhedron).
<p/>
<b>Vertex Polyhedrons</b> can be 1 (vertices on the corners
of the total polyhedron), 2 (vertices on the edges of the
total polyhedron), 4 (vertices on the faces of the total
polyhedron) or 8 (the vertex inside the total polyhedron).
<b>Vertex Faces</b> can be 3 (vertices on the corners
of the total polyhedron), 5 (vertices on the edges of the
total polyhedron), 8 (vertices on the faces of the total
polyhedron) or 12 (the vertex inside the total polyhedron).
<b>Vertex Edges</b> can be 3 (vertices on the corners
of the total polyhedron), 4 (vertices on the edges of the
total polyhedron), 5 (vertices on the faces of the total
polyhedron) or 6 (the vertex inside the total polyhedron).</li>
<li>Try to understand the geometrical data reported. The atoms are
separated by 2.0 and the <b>Offset</b> is 1.5 so their cubes have
a length of 2.0 / 2 + 1.5 / 2 = 1.75. Therefore <b>Edge Lengths</b>
is always 1.75 and <b>Edge Lengths</b> is always 90.0.
<p/>
<b>Face Areas</b> is 1.75 x 1.75 = 3.0625, <b>Face Lengths</b>
(the perimeter) is 1.75 x 4 = 7.0 and <b>Face Angles</b> is 90.0.
<b>Polyhedron Volumes</b> is 1.75 x 1.75 x 1.75 = 5.359375, <b>Polyhedron
Areas</b> is 1.75 x 1.75 x 6 = 18.375 and <b>Polyhedron Lengths</b> is
1.75 x 12 = 21.0.
The <b>Total Volume</b> is 1.75 x 1.75 x 1.75 x 8 = 42.875.</li>
<li>Move the cluster to the left side of the graphic area, and select
again the first layer, containing the cubic atomic arrangement. Select
<b>Cluster->Measure</b>, set <b>Method</b> to <b>Voronoi</b>, <b>Boundaries</b>
to <b>Periodic</b> but this time change <b>Variancy</b> to <b>1.0</b>
(<b>Voronoi</b> page), before clicking on the cluster. An error is produced
because this weighted tesselation is more restrictive than the normal Voronoi
tesselation and the small H atom is too close to the virtual large Na atom.</li>
<li>Repeat the same task but this time increase <b>Offset</b> to 1.5 before
clicking on the cluster. Eight complex polyhedra are produced, with larger
ones for larger atoms, as Na, and smaller ones for smaller atoms, as H.
Move the total cluster to the right side and select again the first
layer, to see the two polyhedra together.</li>
</ol>
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