An evolutionary design of weighted minimum networks for four points in the three-dimensional Euclidean space

We find the equations of the two interior nodes (weighted Fermat–Torricelli points) with respect to the weighted Steiner problem for four points determining a tetrahedron in R 3 \mathbb{R}^{3} . Furthermore, by applying the solution with respect to the weighted Steiner problem for a boundary tetrahedron, we calculate the twist angle between the two weighted Steiner planes formed by one edge and the line defined by the two weighted Fermat–Torricelli points and a non-neighboring edge and the line defined by the two weighted Fermat–Torricelli points. By applying the plasticity principle of quadrilaterals starting from a weighted Fermat–Torricelli tree for a boundary triangle (monad) in the sense of Leibniz, established in [A. N. Zachos, A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature, Acta Appl. Math. 129 (2014), 81–134], we develop an evolutionary scheme of a weighted network for a boundary tetrahedron in R 3 \mathbb{R}^{3} . By introducing the inverse weighted Steiner network with two interior nodes built by two different quantities of the subconscious (remaining weights) for boundary tetrahedra, we describe the evolution of a weighted network with two nodes that have inherited a subconscious. The cancellation of the dynamic plasticity of these weighted networks may be applied to the creation of evolutionary scenarios, in order to prevent the development of a quadrilateral or tetrahedral virus (a monad that has got a subconscious) and the cancerogenesis of quadrilateral cells. We continue by giving the plasticity equations for a generalized weighted minimum network with two nodes that have got a subconscious whose vertices are replaced by Euclidean spheres. This evolutionary approach may be applied to the determination of the bond strengths of molecular structures between atoms in the sense of Pauling. We obtain the analytical solutions of the weighted Fermat–Torricelli problem for the case of pairs of equal weights or one pair of equal weights. We consider as a DNA-like chain a sequence of tetrahedra whose vertices possess some symmetrical weights. By calculating the twist angles of each sequence and by applying the weighted Fermat–Torricelli tree structures with symmetrical weights or weighted Steiner tree structures, we may approximate the curve axis of a DNA-like tree chain. Finally, we construct a minimum tree, which is not a minimal Steiner tree for some boundary symmetric tetrahedra in R 3 \mathbb{R}^{3} , which has two interior nodes with equal weights having the property that the common perpendicular of some two opposite edges passes through their midpoints. We prove that the length of this minimum tree may have length less than the length of the full Steiner tree for the same boundary symmetric tetrahedra, under some angular conditions.

Calibrations for minimal networks in a covering space setting

In this paper, we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover, we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.