The bifurcation approach to hyperbolic geometry

The Thomas precession of relativity physics gives rise to important isometr ies in hyperbolic geometry that expose analogies with Euclidean geometry. T hese, in turn, suggest our bifurcation approach to hyperbolic geometry, acc ording to which Euclidean geometry bifurcates into two mutually dual branch es of hyperbolic geometry in its transition to non-Euclidean geometry. One of the two resulting branches turns out to be the standard hyperbolic geome try of Bolyai and Lobachevsky. The corresponding bifurcation of Newtonian m echanics in the transition to Einsteinian mechanics indicates that there ar e two, mutually dual, kinds of uniform accelerations. Furthermore, while cu rrent hyperbolic geometry "does not use the notion of vector at all" (I.M. Yaglom, Geometric Transformations III, p. 135, trans. by Abe Shenitzer, Ran dom House, New York, 1973), our bifurcation approach exposes the elusive hy perbolic vectors, that we call gyrovectors.