# Voronoi tesselation

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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 Variancy set to 0.0 and then with Variancy set to 1.0, in both cases with periodic boundary conditions. The two polyhedra obtained can be seen in the figure at http://www.gamgi.org/images/voronoi2.png.

### Cubic structure

1. Select Gamgi->Reset and press Ok, to remove the various layers and restart Gamgi. Press File->Import and import the file dat/cluster/cs111.xml, containing a cluster with eight atoms of different elements, in a cubic arrangement.
2. Select Cluster->Measure, set Method to Voronoi, Boundaries to Periodic, Offset to 1.5 and click on the cluster. A cubic polyhedra is shown, in a new layer, formed by eight octant cubes, one for each atom.
3. Try to understand the topological data reported. For example, Polyhedron Faces, Polyhedron Edges and Polyhedron Vertices are 6, 12 and 8, respectively, and Face Vertices is 4.

   Edge Polyhedrons 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). Edge Faces can be 2 (edges on the outside of the total polyhedron) or 4 (edges on the inside of the total polyhedron).

   Vertex Polyhedrons 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). Vertex Faces 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). Vertex Edges 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).
4. Try to understand the geometrical data reported. The atoms are separated by 2.0 and the Offset is 1.5 so their cubes have a length of 2.0 / 2 + 1.5 / 2 = 1.75. Therefore Edge Lengths is always 1.75 and Edge Lengths is always 90.0.

   Face Areas is 1.75 x 1.75 = 3.0625, Face Lengths (the perimeter) is 1.75 x 4 = 7.0 and Face Angles is 90.0. Polyhedron Volumes is 1.75 x 1.75 x 1.75 = 5.359375, Polyhedron Areas is 1.75 x 1.75 x 6 = 18.375 and Polyhedron Lengths is 1.75 x 12 = 21.0. The Total Volume is 1.75 x 1.75 x 1.75 x 8 = 42.875.
5. Move the cluster to the left side of the graphic area, and select again the first layer, containing the cubic atomic arrangement. Select Cluster->Measure, set Method to Voronoi, Boundaries to Periodic but this time change Variancy to 1.0 (Voronoi 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.
6. Repeat the same task but this time increase Offset 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.

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