NUMERICAL TETRAHEDRON

Abstract

Each triangle corresponds to an ordered triplet of non-negative real numbers – numeri­cal triangles that represent the lengths of its sides. A numerical triangle in which the numbers are arranged in a non-increasing order is called a proper triangle. This article examines the description of the set of all numerical triangles in three-dimensional Euclidean space. It is shown that the geometric location of this set is a "proper infinite tetrahedron" with a vertex at the origin and lateral edges coinciding with three rays emanating from this vertex and passing through the points of the three-dimensional unit cube (0, 1, 1), (1, 0, 1) and (1, 1, 0) respectively. In this tetrahedron, the lateral faces are represented by "infinite triangles" formed by two edges with angles between them equal to 60º. In the article, such a tetrahedron is referred to as a numerical tetrahedron. It is noted, that only the points on the surface of the numerical tetrahedron correspond to degenerate triangles. Triangular cross-sections of a numerical tetrahedron formed by planes intersecting all edges are considered. Among these edge cross-sections, the primary ones are distinguished, which are obtained by planes perpendicular to the height of the tetrahedron, and are represented by equilateral triangles. These equilateral triangles, in turn, are divided by their heights into six standard pairwise equal right triangles. One of them is equilateral, and represents all proper numerical triangles. A ray from the vertex of the numerical tetrahedron is considered, piercing the equilateral standard triangle at a specific point similar to the numerical triangle of this equilateral standard triangle. This point is called the primary black hole of the standard triangle, and the ray passing through it is called the primary black ray. It is shown that the numerical tetrahedron has a bundle of six primary black rays. Since any ray of the numerical tetrahedron pierces all possible edge cross-sections, it is noted that the numerical tetrahedron can have an infinite continuous set of such bundles.