Delaunay simplexes were used for structure analysis of hard sphere packings of different density. A special attention was paid to the simplexes of tetrahedral shape. Clusters of the tetrahedra adjacent by faces represent relatively dense aggregates of spheres atypical for crystalline structures. Such "polytetrahedral" aggregates reveal a characteristic feature of the dense disordered packings. The tetrahedra in point are not completely perfect. They coincide with the class of "quasi-regular tetrahedra" introduced by Hales in his proof of the Kepler conjecture. In this work we discuss a meaning of these tetrahedra and their role in formation of the dense disordered packings. Polytetrahedral principle of a spatial organization seems to be preferable from the statistical viewpoint, as in this case a variety of dense local configurations can be realized. However it has its limit and further increase in density can be provided by formation of crystalline nuclei.