# Cluster Measure

- Type
- Symmetry
- Voronoi
- Tolerance
- View

Select here the main parameters controlling the Voronoi tesselation.

### Boundaries

GAMGI supports two methods to handle the tesselation in the boundaries, Planar and Periodic.

The Planar method defines six outer planes, forming a closed space that includes all the seeds. Conceptually, this is equivalent to create six virtual seeds around each real seed, positioned in such a way that they define six planes at half-distance, common for all seeds, forming a closed background.

The Periodic method applies periodic boundary conditions, using the extended image convention, which means to create 26 virtual systems around the central system (as 26 boxes around the central one, in a 3x3x3 arrangement). The minimum image convention, often used in Molecular Mechanics simulations, is not suitable in Voronoi tesselation, because it considers only the nearest neighbours, while Voronoi tesselation requires all the 26 nearest neighbours.

The Offset parameter defines the border width between: 1) the seeds and the outer planes, in the Planar method; 2) the central system and the other 26 around, in the Periodic method.

Offset cannot be 0 in the Planar method, because virtual seeds would coincide with real seeds, producing an error. Offset can be 0 in the Periodic method, if real and virtual seeds do not coincide, otherwise an error is flagged. An error is also produced when two real seeds coincide (current tolerance: 1.0E-4).

### Weights

By default, GAMGI calculates the normal Voronoi partition, so all atoms are given the same weight, corresponding to Variancy equal to zero. However GAMGI can partition the space giving different weights to different atoms (Radical Tesselation), according to their radius. In general, if d is the distance between seed 0 and neighbour 1, the plane between them passes at a distance d * fraction from the seed, where fraction is given by:

    r0 = radius0 * variancy
    r1 = radius1 * variancy
    fraction = 1/2 [1 + (r0**2 - r1**2)/d**2]

When Variancy is 0, fraction is 1/2 (normal Voronoi), whereas when Variancy is 1, unscaled atomic radius are used to determine fraction. Variancy can vary continuously from 0 to a given limit, GAMGI_PHYS_VORONOI_VARIANCY, currently 10.0.

When Variancy is 0, the tesselation can always be obtained (as long as there are no atoms with essentially the same coordinates). When Variancy increases, the tesselation becomes progressively more restrictive. The atoms must be far enough from each other, particularly when the radius are quite different, otherwise the tesselation cannot be determined and and error is shown.
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