CONSTRUCTION OF SECTIONS OF POLYHEDRON METHOD OF TRACES

In this article it is proved that in solving problems on the construction of sections of polyhedra, students not only perform constructions, they also apply axioms, properties of planimetry and stereometry, but also learn algorithmic thinking, the ability to reason logically, make correct arguments and conclusions. It is established that the solution of problems on the construction of sections of polyhedra occupies a special place in the process of forming a spatial representation and in the development of mathematical thinking, both students and schoolchildren. Based on the definition of the trace of the secant plane, the rules for constructing sections of the polyhedron by the traces method are formulated. Problems on construction of sections of polyhedra are developed in the case when:: the section of the prism is given by the trace l , which is located on the plane of the base of the prism and does not have common points with the base of this prism and by point K , belonging to some side rib; the secant plane is defined by the trace l and some point M , belonging to the side rib of the pyramid; the section of the pyramid is determined by points M, N, K two of them are located on different ribs, and the third is the internal point of the face of this pyramid.

On the inner crystalline structure of ferrite and cementite in pearlite

In a previous paper on “The Inner Structure of the Pearlite Grain” read to the Iron and Steel Institute in 1922 the author advanced views on the stereometry of the pearlite grain. It was shown that the angle of inclination of a secant plane ω could be computed from the equation cos ω = Δ 0 /Δ ω , (1) where Δ 0 is the actual distance between two consecutive cementite or ferrite lamellæ and Δ ω is the trace of that distance on the secant plane. It follows from (1) that Δ ω = Δ 0 sec ω. (2) As the diameter d c of a cementite lamella or the diameter d f of a ferrite lamella is a constant fraction of Δ, it may be assumed that d ω = d 0 sec ω, (3) where d ω is the trace of d 0 on the secant plane.