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I thought it does have curvature (like a non-flat torus), but that curvature is in effect compressed into 'singularities' at the corners.
by scrumper 9y ago
I thought it does have curvature (like a non-flat torus), but that curvature is in effect compressed into 'singularities' at the corners.
- tgb 9y agoIt's possible to hide curvature in singularities like that, but that's not the classical "grid with wrap-around on the top and bottom" you see in video games. In particular, there's nothing "special" about the corners - you can still go up/down/left/right at them. In essence, you could recenter the grid around any point without changing anything so the corners only look special but aren't. The torus actually needs total curvature 0 (the beautiful Gauss-Bonnet theorem says this and is true even for a torus that cannot fit in 3 dimensional space), which means you'd need some "singularities" with negative curvature and some with positive curvature to cancel out. But clearly all the corners in the torus are the same! So there's no way to hide the right curvature in them.
- scrumper 9y agoAha yes I see. I needed to think more carefully about total curvature. I have been doing a bit more reading on this, really a fascinating area. Thanks for the pointers.