Chasles’ mystic tetrahedron

Abstract

The results of the experimental study and testing of the theorem by M. Chasles are given. The theorem shows the peculiarities of the intersection of an arbitrary tetrahedron with an arbitrary quadratic surface (second-order surface). The multiple variants of the intersection stipulated in the theorem have been considered. It has been experimentally proven that for the majority of these variants the four intersecting lines constructed according to the algorithm of this theorem belong to the surface of a single one-sheeted hyperboloid. For two of the variants, the results that differ from the theorem have been received. The experiments implied building and studying 3D computer models obtained using the AutoCAD and SolidWorks packages. All the variants of the quadratic surface at different mutual position with respect to the tetrahedron have been considered. The methods of holding the experiments have been considered in detail. The proof of the theorem for one of its variants has been obtained and presented.