Tetrahedral geometric network and its development along five points

Abstract

There is nothing more natural than to generalize the conclusions of the previous note relating to three-digit numbers or a triangular geometric network to numbers of higher significance and, above all, to four-digit numbers, and the result is a tetrahedral network. Such a network of numbers has found its application for the chemical tetrahedron in petrography. Reflecting on the mathematical foundations of the construction of a triangular network, we will find that the main theorems remain valid for this network with a corresponding complication of the constructions themselves. This complication lies in the fact that the total number of points associated with one elementary tetrahedron (more precisely, a sphenoid) of any period no longer 7(3 + 3 + 1), but 15: four at the vertices, six midpoints of the edges, four midpoints of the faces and one midpoint of the tetrahedron itself.