Tetrahedral geometric network and its development based on five points
Abstract
Nothing is more natural than to generalize the conclusions of the previous note relating to three-digit numbers or the triangular geometric network, to numbers of higher digit counts and, first, 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 fundamental 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 an elementary tetrahedron (more precisely, a sphenoid) of any given period is no longer 7 (3 + 3 + 1), but 15: four at the vertices, six midpoints of the edges, four centroids of the faces, and one at the centroid of the tetrahedron itself.
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