ABSTRACT

In the fifteenth century artists discovered perspective, which revolutionized the drawing of three-dimensional scenes. This chapter discusses parallels from the standpoint of measurement: they are lines with the same slope, or lines a constant distance apart. The passages remind mathematicians that parallels are quite different from the standpoint of vision: if the plane is spread out horizontally in front of them, parallels appear to meet on the horizon. By choosing to keep "horizontal" lines horizontal, mathematicians force the "verticals" to meet on the horizon. And since the intersections of a diagonal with the"horizontals" are also its intersections with the "verticals," the latter intersections give mathematicians the positions of the horizontals. The chapter discusses that there is a long history of projective configuration theorems that explain coincidences. Their proofs lie outside projective geometry so if mathematicians want to explain coincidences within projective geometry they must take certain configuration theorems as axioms.