Number theory and geometry, with a millenia-long history, together constitute the original mathematical disciplines. It is fascinating that the regularities and connections once discovered by mathematics retain their validity; more fascinating still is the fact that new relations are continually being discovered. Since mathematics is also a logical structure in which one step follows from another, it is by now quite a challenge to come to be at the cutting edge; the mathematical disciplines have evolved to the point that it is a lengthy trek to reach their frontiers. This is why we rarely get to know 20th-century mathematics through the school math curricula, and as a result one has the impression that mathematics is complete and finished. This book seeks to show that throughout human history new dimensions have continuously been established in geometry, and that its evolution is far from complete.