  The formal structure of the early Einstein's Special Relativity follows the
axiomatic deductive method of Euclidean geometry. In this paper we show the
deep-rooted relation between Euclidean and space-time geometries that are both
linked to a two-dimensional number system: the complex and hyperbolic numbers,
respectively. By studying the properties of these numbers together,
pseudo-Euclidean trigonometry has been formalized with an axiomatic deductive
method and this allows us to give a complete quantitative formalization of the
twin paradox in a familiar "Euclidean" way for uniform motions as well as for
accelerated ones.
