Of Mobius Strips and the Shape of Things

[E]ven though Earth is (roughly) spherical, the directions you travel can be broken down into two directions: North-South (longitude lines) and East-West (latitude lines). However, there are two places where the latitude-longitude coordinate system breaks down: the North and South Poles, where a single latitude corresponds to all the longitudes simultaneously. In fact, no coordinate system can describe the surface of a sphere without a breakdown at at least one point! It’s not for lack of trying, it’s just a mathematical fact. (There are coordinate systems that break down at only one point, but they’re less useful for cartography.)

Arguments about Möbius are always so one-sided.