In spherical geometry, the shortest-length curve between two points on the surface of the sphere turns out to be part of a Great Circle – an equator-line circle that cuts the sphere in half. So lines are circles, which is fun to share with philosophers. (Note – taxicab geometry provides that same amusement, where circles are squares.)
So a natural question, where “natural” means I never actually thought of it but wish I had, is What is the longest line along the surface of the earth that goes entirely through water? This would be the longest possible straight-line sailing distance, if you ignored all the physical aspects of sailing like wind and water currents. Fortunately, before I even thought of the question, someone had answered it. Behold!
Longest straight-line distance through the water on earth.
This gif appears to be from a youtube video by Patrick Anderson of 2012 (here) which has the advantage of being a little slower.
So that raises the question of the longest straight-line distance through land. And here’s a guess at it: http://i.imgur.com/nbNfl.jpg and then another one https://sites.google.com/site/guybruneau/fun-stuff/longest-distance-on-land, although that second one it doesn’t quite look like part of a Great Circle so possibly the projection imposed a different geometry. Or possibly I have trouble visualizing projections of Great Circles, which is also possible because they are weird. (The cool kind of weird, of course.)
Thanks CJ for sending me that gif, although now that I’m finding myself asking questions like “What line passes through the most countries?” I can tell that it’s going to keep me from my grading for longer than it should.
360 
