Exploring the 3D Analog of the Proof of the 2D Isostatic Theorem for Sticky Disks

Abstract

The 2D Isostatic Theorem reveals that a packing of n disks with generic radii can have at most 2n − 3 disk pairs in contact. The motive of this thesis is to analyze the 3D analog of the 2D Isostatic Theorem proof and determine the truth of its dependencies within R^3. Through theoretical and computational methods, we have discovered that in R^3, Sg isn’t necessarily a smooth sub-manifold of dimension 4n − m (disproving the first dependency) and that (G, p) can have nontrivial edge-length stress equilibrium vectors (disproving the second dependency). Finally, in our search through the Miranda dataset of all possible packings where n ≤ 14, we determined that there were no additional examples where edge-length stress equilibrium vectors exist.