Einsteinʹs special relativity: Unleashing the power of its hyperbolic geometry

We add physical appeal to Einstein velocity addition law of relativistically admissible velocities, thereby gaining new analogies with classical mechanics and invoking new insights into the special theory of relativity. We place Einstein velocity addition in the foundations of both special relativity and its underlying hyperbolic geometry, enabling us to present special relativity in full three space dimensions rather than the usual one-dimensional space, using three-geometry instead of four-geometry. Doing so we uncover unexpected analogies with classical results, enabling readers to understand the modern and unfamiliar in terms of the classical and familiar. In particular, we show that while the relativistic mass does not mesh up with the four-geometry, it meshes extraordinarily well with the three-geometry, providing unexpected insights that are not easy to come by, by other means.