The plasticity of some mass transportation networks in the three dimensional Euclidean Space

Abstract

We obtain an important generalization of the inverse weighted Fermat-Torricelli problem for tetrahedra in R^3 by assigning at the corresponding weighted Fermat-Torricelli point a remaining positive number (residual weight). As a consequence, we derive a new plasticity principle of weighted Fermat-Torricellitrees of degree five for boundary closed hexahedra in R^3 by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in R^3. We also derive a complete solution for an important generalization of the inverse weighted Fermat-Torricelli problem for three non-collinear points and a new plasticity principle of mass networks of degree four for boundary convex quadrilaterals in R^2. The plasticity of mass transportation networks provides some first evidence in a creation of a new field that we may call in the future Mathematical Botany.