> It starts with a closed loop — any kind of curvy path that ends where it starts. The problem Greene and Lobb worked on predicts, basically, that every such path contains sets of four points that form the vertices of rectangles of any desired proportion.<p>> Their final proof — showing the predicted rectangles do indeed exist — transports the problem into an entirely new geometric setting. There, the stubborn question yields easily.<p>Except that the article then contradicts itself by saying how they haven't actually proved this. They proved it for smooth closed curves, not for any closed curves.<p>This is a really bizarre article. They seem like they want to describe the math. But they can't bring themselves to do it in a way that might be helpful. They're just waving words around.<p>And then there's this:<p>> it’s possible to rotate the Möbius strip in four-dimensional space so that you only change one of the coordinates in each point’s four-coordinate address — like changing the street numbers of all the houses on a block, but leaving the street name, city and state unchanged. (For a more geometric example, think about how holding a block in front of you and shifting it to the right only changes its x coordinates, not the y and z coordinates.)<p>You can certainly translate a space along an axis without affecting its coordinates along other axes. But that's not a rotation.