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.