Euclid would hate this one simple trick: draw straight lines connecting the northernmost and southernmost points of the contiguous U.S., and the easternmost and westernmost points, and the two lines never intersect. Here’s why it happens: on a flat surface, if one straight line connects the top and bottom of a shape while another connects the left and right, you’d expect the two lines to intersect. The Earth, however, is not flat. It’s roughly spherical, which means the shortest path between two points is a great-circle route—the same type of route long-distance aircraft often follow. When those routes are projected onto a flat map, they can appear curved and no longer behave like straight lines in ordinary Euclidean geometry. As a result, the great-circle route connecting the northernmost and southernmost points of the contiguous United States and the one connecting its easternmost and westernmost points never actually intersect. It seems to defy Euclid’s intuition, but it’s exactly what spherical geometry predicts.(Historic Vids)