Valuing Voronoi Visualization: Scale Independence in Irregular Biological Patterns and Its Importance to Structural Bioinformatics
John R. Jungck
Mead Chair of the Sciences and Professor of Biology
Beloit College
700 College Street
Beloit, WI 53511
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Allometry and fractals have captured the imagination of mathematical biologists as well as amateurs because both apply across ten orders of magnitude of biological phenomena and structures from the molecular to the ecological level. Voronoi polygons and polyhedra are less well known to both audiences, but scale equally well. Furthermore, Voronoi polygons and polyhedra are associated with additional mathematical methods that allow deeper insight into a variety of biological phenomena such as growth, diffusion, division, packing, docking of ligands, strength of materials, molecular folding, foraging behavior, predator avoidance, and crowding as well as to their utility in making measurements, modeling interactions, relationship of two- and three-dimensional tomographic structures, and visualization per se. By employing Voronoi polygons and polyhedra in mathematical biology education, we will illustrate the five fold approach of curricular reform in mathematics: (1) analytical (theorem/proof), (2) numerical, (3) symbolic, (4) visual/graphical, and (5) applied to relevant scientific and social problems. While various approaches to constructing Voronoi polygons and polyhedra, Delaunay triangulations, and minimal spanning trees may be formally isomorphic from a mathematical or computer science perspective, different techniques are much better than others in helping students relate a causal mechanical and material model of their biological phenomena of interest, simulating phenomena realistically, or in making appropriate measurements. Multiple methods for constructing Voronoi polygons and polyhedra, Delaunay triangulations, and minimal spanning trees will be applied to epithelial cell boundaries, fish boundaries on sandy lake bottoms, dragonfly wing veination, cross-sections of leaves, fiddler crab flocking behavior, drug design, packing of side chains in polypeptides, bird territories, and forest canopies to illustrate their commonalities and differences? Finally, statistical analyses of Voronoi polygons and L-mosaics will be compared to determine whether nearest neighbor or long-range interactions better apply to a given set of biological data.
Key words: Computational geometry, Voronoi polygons and polyhedra, Delaunay triangulations, minimal spanning trees, convex hulls, perpendicular bisectors, coordinate geometry, Voronoi fractals, simplex, alpha shapes, structural bioinformatics