Building on his books since 1988, Ungar (North Dakota U.) explains the current status of analytic hyperbolic geometry, which emerged from Einstein's addition of relativistically admissible velocities. He surveys some of its recent triumphs--such as dissolving the dichotomy between Einsteinian and Minkowskian relativity--and emphasizes the interdisciplinary collaborations required to further develop the mathematical innovation and its applications. Readers are assumed to be familiar with Euclidean geometry from the perspective of vectors, and occasionally with differential calculus and functions of a complex variable.