The Plasticity of some Mass Transportation Networks in the Three Dimensional Euclidean Space

We obtain an important generalization of the inverse weighted Fermat-Torricelli problem for tetrahedra in $\mathbb{R}^3$ by assigning to the corresponding weighted Fermat-Torricelli point a remaining positive number (residual weight). As a consequence, we derive a new plasticity principle of weighted Fermat-Torricelli trees of degree five for boundary closed hexahedra in $\mathbb{R}^3$ by applying a geometric plasticity principle which lead to the plasticity of mass transportation networks of degree five in $\mathbb{R}^3$. 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 $\mathbb{R}^2$. The plasticity of mass transportation networks provides some first evidence for a creation of a new field that we may call Mathematical Botany in the future.