Projective Geometry — Where Parallel Lines Meet

In ordinary geometry, parallel lines never meet. In projective geometry, they meet at a 'point at infinity' — one ideal point for each direction. This is not mystical: stand on a long straight road and the two edges converge at the horizon. Projective geometry has profound symmetry: every theorem about points and lines has a 'dual' theorem obtained by swapping the two words. The railway tracks that seem to converge, the perspective in every Renaissance painting, the mathematics of every camera — all are projective geometry.