A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space

We prove a plasticity principle of closed hexahedra in the three dimensional Euclidean space which states that: Suppose that the closed hexahedron A1A2⋯A5 has an interior weighted Fermat-Torricelli point A0 with respects to the weights Bi and let αi0j=∠AiA0Aj. Then these 10 angles are determined completely by 7 of them and considering these five prescribed rays which meet at the weighted Fermat-Torricelli point A0, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. By applying the plasticity principle of closed hexahedra to the n-inverse weighted Fermat-Torricelli problem, we derive some new evolutionary structures of closed polyhedra for n>5. Finally, we derive some evolutionary structures of pentagons in the two dimensional Euclidean space from the plasticity of weighted hexahedra as a limiting case.