Elie Cartan and pan-geometry of multispatial hyperspace

Abstract

Elie Cartan has proved that highest dimensionality of any simple geometric space is three and that an exterior differentiation of a 3D+ geometric object gives bivector, which may correspond to some two 2D surfaces as if the 3D+ geometric object comprised two 3D objects. Since one cannot increase the dimensionality of a 3D space even though more than four independently varying physical magnitudes do exist, then an expansion of dimensionality requires a multispatial hyperspace that contains many simple geometric 3D spaces. Presence of such a hyperspace prompts for an entirely new concept of vectors with an isometric operation of vector multiplication of traditional vectors (3-tuples). This new operation on 3-vectors implies presence of a 3D mass-based linear vector space and consequently thus a 9D geometric hyperspace for classical mechanics alone. Also an outline of entirely new, synthetic approach to physics and mathematics is introduced. This synthetic approach can be used to design a computer-aided knowledge extracting system, which could generate entirely new scientific knowledge.