Manifold Mirrors: The vicissitudes of geometry

We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing . A. Einstein (1926) The period between the fifteenth and nineteenth centuries saw a mutual influence between painting and geometry, with the latter allowing for the dawn of perspective and the former for the dawn of projective geometry. An imperfect narration of this process occupied us in Chapter 9. We then proceeded in Chapter 11 to survey the many vicissitudes that accompanied perspective in its journey from prominence to abandonement. The words of Einstein opening this chapter suggest that during this period geometry was not free of its own vicissitudes and that the prominence of the Euclidean order was gradually eroded as well. The goal of this chapter, obeying a sense of symmetry, is to describe these vicissitudes. Along the way, we will gain a better understanding of projective geometry and face some perplexities seldom associated with mathematics. This chapters bears little on art. One may argue that the erosion of the prominence of Euclidean geometry began with the seed of projective geometry planted by Renaissance painters. But this is a weak justification for this chapter, which is otherwise denser than previous ones. Probably a stronger reason for its inclusion is the fact that it will be a basis for the additional material on symmetries in Chapter 13, as well as for the discussion on the shape of the universe which will serve us to return to our early speculations about the nature of space in Section 1.1.