A Generalized Fermat-Torricelli Tree in the Three Dimensional Euclidean Space

We obtain a generalized Fermat-Torricelli tree for a boundary pentagonal pyramid (closed hexahedron) and a boundary closed enniaedron in the three dimensional Euclidean Space by placing two mobile vertices with positive weights at the interior of their convex hull and by introducing a method of differentiation of the objective function with respect to two variable dihedral angles and four variable length of linear segments which connect each mobile vertex with two boundary vertices. The generalized Fermat-Torricelli tree with degree at most four with respect to a closed hexahedron or a closed enniaedron is an Euclidean minimal tree structure which may be considered as an intermediate class of a tree structure between a weighted Fermat-Torricelli (tree) structure and a weighted Steiner minimal tree structure with respect to a closed hexahedron or closed enniaedron.