  The evolution from Mobius to gyrogroups began in 1988, and is still ongoing
in [14, 15]. Gyrogroups, a natural generalization of groups, lay a fruitful
bridge between nonassociative algebra and hyperbolic geometry, just as groups
lay a fruitful bridge between associative algebra and Euclidean geometry. More
than 150 years have passed since the German mathematician August Ferdinand
Mobius first studied the transformations that now bear his name. Yet, the rich
structure he thereby exposed is still far from being exhausted, as the
evolution from Mobius to gyrogroups demonstrates.
