Theoretical and Applied Discrete Geometry

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Research on this project involves applying the theory of rigid and non-rigid frameworks and tensegrities to problems in discrete geometry. Examples involve extending the principle investigator's results on the Kneser-Poulsen problem concerning the volume of the union or intersection of spherical disks under contractions of their centers, bellows problems involving the invariance of the volume bounded by flexible surfaces, the opening of robot arms, the cataloguing of certain stable highly symmetric tensegrity structures, and stability questions involving packings of granular materials.Consider granular materials, which are seen in avalanches, sand, rock piles, piles of M&M's, pills, grain, seeds, powders, and the regolith on asteroids and planets. A good example of an application of the theory above is when an external force is applied to a granular material. Understanding the way that force is transmitted throughout the material is important, and the mathematical theory of rigid structures can be used to justify, elucidate, and simplify the calculations of these forces.