An Evolutionary Structure of Pyramids in the Three-Dimensional Euclidean Space

We solve the problem of evolution for some classes of pentahedra (pyramids) in the three dimensional Euclidean space by applying the inverse weighted Fermat-Torricelli problem of 5 rays that meet at the weighted Fermat-Torricelli point A(0) and the invariance property of the weighted Fermat-Torricelli point. The main result is the three dimensional property of plasticity which states that: If we decrease the weights that correspond to the first, third and fourth ray which passes from the apex of the pyramid, then the weights that correspond to the second and fifth ray increase. Finally, we introduce the notion of the generalized plasticity for weighted pyramids via a specific discretization of the five weights along the five given prescribed rays.