Event Details

Fall 2003 Keck Foundation Student Research Day
Host:
Chapman University
Location:
Orange, CA
Date(s):
November 19, 2003
Abstract/Description:
Bioscapes: Patterns in Nature Through the Eyes of Artists, Mathematicians, and Biologists
What do epithelial cell boundaries, fish boundaries on sandy lake bottoms, and cross-sections of leaves as well as two-dimensional projections of three dimensional patterns such as the packing of side chains in polypeptides, bird territories, and forest canopies have in common? The famous Dutch artist, Escher, dealt with symmetrical tessellations and his art has greatly intrigued numerous biologists. The techniques that I will discuss handle asymmetric as well as symmetric patterns. These techniques range from elementary school mathematics to contemporary research in modern mathematics; however, the esthetic motivation is the primary one and I promise to show lots of pictures and biological applications. Pedagogically, multiple ways of knowing and isomorphism of various approaches will be emphasized. 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 science 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. I will demonstrate six different techniques for studying these planar patterns: diffusion, constructing perpendicular bisectors, origami paper folding, calculating circumcenters, projective and computational geometry, and Delaunay duals with graph theory.
BioQUEST Staff Attending:
John Jungck

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