One of the funny implications is that many 2D video games implicitly have toroidal worlds.
This is reasonable for something like Pacman, but many games with overworld maps (SNES RPGs like Chrono Trigger, for example) implicitly have toroidal planets due to the navigation.
Pretty cool: describes a way of creating a 3D figure with the same properties as a torus, but avoiding a certain stretching/distorting you get with plain tori (simplification for one sentence summary...). Their approach seems related to the idea that if you have a closed curve bounded by another curve (a square, let's say), as you make the inner curve more wrinkly it gets correspondingly longer--though it remains bounded by the same square. Here, the three dimensional embedding is more 'wrinkly' along the axes that need to be extended in order for the emedding to be isometric.
I wonder if you only need infinite layering of corrugations if the 'stretching factor' is irrational. Probably not that simple :)
The infinite layering of corrugations is required regardless, because the "true" surface has undefined curvature. They are doing an approximation to that of arbitrary precision.
This is a really delightful peice of work. The existence of surfaces with the curvature undefined everywhere is an easily described concept that comes up fairly frequently in topology, but I'd always assumed it was just impossible to visualize them.
The way I was thinking about was that the curvature is named undefined because it has discontinuities in it, which result from the fact that the object is constructed by sticking pieces together, and the joints aren't entirely smooth. Is there something deeper to its 'undefined curvature'? Or I guess that's just more important than it seems to me: like it being amenable to classification by curvature type is the essential thing here. And I guess it must be something deeper than not being smooth at the joints, since there are probably applicable smoothing algorithms.
I'd like to see this answered by someone knowledgeable too.
From my limited knowledge I would guess the function being C2 gives the curvature a "global" domain; while if the function is only C1 the curvatures will have limited influence, and you can use that to enforce other curvatures "prohibited" under C2. Anyway, I might be completely wrong, just the idea I got from the article :)
I was just wondering the other day how to visualize a square flat torus, and I didn't realize that this was such a compelling mathematical question.