Lobachevsky’s Imaginary Geometry as Specular and Hyperdimensional Structure

This article proposes a reinterpretation of Lobachevsky’s so-called imaginary geometry as a hyperdimensional, specular structure emerging from the dynamic intersection of two three-dimensional Euclidean spaces. In this model, non-Euclidean parallelism arises not from intrinsic curvature but from phase-driven oscillations in spatial curvature. The resulting topological configuration consists of four interrelated subspaces—two transverse and two vertical—whose geometry depends on whether the source spaces evolve in synchronized or opposing phases. This framework offers a dynamic and dual interpretation of non-Euclidean structure, challenging traditional reductions of Lobachevskian geometry to hyperbolic models.