# Voronoi tesselation

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Voronoi tesselation is used in a wide range of scientific areas, from medicine to economics, as a technique to split some intensive quantity among a given set of objects. When these are atoms, the Voronoi tesselation provides a way to relate the electronic distribution of the atomic structure with its topologigal and geometric properties.

First we create a water molecule and calculate its Voronoi tesselation, changing the boundary offset, the boundary conditions, and the atomic weights. The four polyhedra obtained can be seen in the figure at http://www.gamgi.org/images/voronoi1.png.

### Water

1. Press Atom->Create, set Element to O, set X, Y, Z to 0, 0, 0, and press Ok, to create the O atom.
2. Set Element to H, set X, Y, Z to 0.6, 0.7749, 0, and press Ok, to create the first H atom. To create the second, set Element to H, and X, Y, Z to 0.6, -0.7749, 0, before pressing Ok.
3. Select Molecule->Create, write a name for the molecule, for example water, and press Ok. The new molecule has no contents yet, so the dialog Molecule->Link replaces automatically the previous one. Press the mouse over one of the atoms and press Ok, to link it to the molecule. Repeat for the other two atoms.
4. Select Bond->Create and click over the O and a H atom to create a bond. Repeat with the other H atom. To see the object hierarchy, select for example Molecule->Select. To save the molecule, press File->Export and write the file name, for example water.xml, before pressing Ok.
5. Select Molecule->Measure, set Method to Voronoi, and click on the molecule. A polyhedra cluster object is shown, in a new layer (with Visibility to the Outside set to None), describing the tesselation of the water molecule. A long report is also produced, with geometrical and topological properties. To see what these properties mean press Help->Current. Close the report window, rotate the cluster and move it to the top right corner of the graphic area.
6. Select Layer->Select (or press Layer on the top menu with the mouse middle button) and click the mouse on the screen, to list the available layers and select the first one, that contains the water molecule. The water and the cluster are now both visible (because this layer has Visibility to the Outside set to All).
7. Select again Molecule->Measure, set Method to Voronoi, but this time change Offset to 0.1 (Voronoi page), before clicking on the molecule. The 6 outside planes are now much closer to the atoms, so the cluster produced is much smaller and its shape is also different. Rotate the cluster and move it to the bottom right corner of the graphic area.
8. Select again the layer containing the water molecule. Select again Molecule->Measure, set Method to Voronoi but this time change Boundaries to Periodic (Voronoi page), before clicking on the molecule. Instead of defining 6 outer planes, the water molecule was now surrounded by 26 virtual copies of its own, and the cluster obtained reflects these changes.
9. Select Cluster->Copy, change Number to 2, click on the cluster and press Ok, to produce two new copies of the original cluster. Select Cluster->Select and confirm that there are now three clusters exactly equal. Then click on the visible cluster to select it and move it to the right until it just touches the central one. Click again on the visible cluster to select it and move it to the left until it just touches the central one. The three clusters together define a continuous pattern, showing that periodic boundary conditions were applied. Select Cluster->Remove and click on the left and right clusters to remove them. Move the remaining cluster to the top left corner of the graphic area.
10. Select again the layer containing the water molecule. Select again Molecule->Measure, set Method to Voronoi, Boundaries to Periodic but this time change Variancy to 1.0 (Voronoi page), before clicking on the molecule. This time larger atoms have larger polyhedra so these are much smaller for H than for O. Move the cluster to the bottom left corner of the graphic area and select again the layer containing the water molecule, to see the four polyhedra obtained so far.

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