Emanuel (Menachem) Lazar

Bar-Ilan University

Voronoi Topology for Structure Analysis
A common challenge in studying many physical systems is describing the "structure" of a set of points in space. Group theory provides powerful tools to study perfectly ordered systems, and probability theory provides tools to study "perfectly disordered" systems. Real systems, however, tend to be neither perfectly ordered not perfectly disordered. A meaningful way to describe this structure, in an elegant and practical manner, motivates the development of the Voronoi topology method, implemented in VoroTop.
Statistics of Three-Dimensional Poisson-Voronoi Structures
Voronoi tessellations of Poisson point processes are widely used for modeling many physical and biological systems. In this paper (preprint), we analyze simulated Poisson-Voronoi structures containing a total of 250,000,000 cells to provide topological and geometrical statistics of this important class of networks. To date, this is the largest data set for this type of system (by at least two orders of magnitude); moreover, many topological features of this system have not been considered until now.

Please be in touch with any questions; we are especially eager to provide additional statistics. Additional raw data for these structures is available here.

If you use this data in published work, please consider citing one of the papers below.

Filename Description
all_pvectors.dataDistribution of all 375,410 observed p-vectors
top_10000_wvectors.dataDistribution of 10,000 most common Weinberg vectors
edges_dist_areas_perims.dataDistribution of edges, and average area and perimeter
faces_dist_vols_areas.dataDistribution of faces, and average volume and surface area
p_vectors_and_types.dataCounts of possible and observed p-vectors and topological types
symmetries.dataDistribution of symmetry orders
areas_distribution.dataDistribution of areas of all faces
faces_areas_distributions.dataDistribution of areas of all faces with fixed number of edges
perims_distribution.dataDistribution of perimeters of all faces
faces_perim_distributions.dataDistribution of perimeters of all faces with fixed number of edges
volumes_distribution.dataDistribution of volumes of all cells
faces_volumes_distributions.dataDistribution of volumes of all cells with fixed number of faces
surface_areas_distribution.dataDistribution of surface areas of all cells
faces_surface_areas_distributions.dataDistribution of surface areas of all cells with fixed number of faces
weaire-aboav.dataExpected number of neighbors m(F) of cells with F faces
kentropy.datak-entropy for 0 <= k <= 8
vertices.dataVertices, characterized by total faces of four neighbors (*)
face_round.dataRoundness of faces with n edges
cell_round.dataRoundness of cells with F faces
Addition fun data
The 250,000,000 Poisson-Voronoi cells contained: Click here for Schlegel diagrams of the 35 most common grain topologies in the Poisson-Voronoi microstructure.
Random fun fact
You might know that the `Kelvin cell', a very symmetric 14-faced polyhedron, has 6 quadrilateral faces and 8 hexagonal faces. Kelvin conjectured that this shape tiles three-dimensional space with a minimal surface area among all space-filling polyhedra. What you might not know is that the Kelvin cell is not the only polyhedron with this set of faces. Below are Schlegel diagrams of the Kelvin cell and of two other polyhedra with 6 quadrilateral faces and 8 hexagonal faces which are most certainly not the Kelvin cell; these are the only other such polyhedra.

Kelvin cell
non-Kelvin cell 1
non-Kelvin cell 2
Schlegel diagrams of three polyhedra with 6 square and 8 hexagonal faces.
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