

Hevea Project – Flat tori finally visualized - DiabloD3
http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html

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bglazer
That's one of the best math articles I've ever read frankly.

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.

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TTPrograms
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.

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bglazer
Someone (smarter than me) should program a Pacman clone that takes place on
one of these tori.

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throwaway_bob
pacman already takes place on one of these tori

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bglazer
Yes I understand that, I meant on the 3d embedding of it presented in this
article

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westoncb
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 :)

Edit: added a summary to the beginning.

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tjradcliffe
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.

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westoncb
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.

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darkmighty
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 :)

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myhf
Greg Egan produced a good visualization of flattening a torus:
[https://vimeo.com/71479159](https://vimeo.com/71479159)

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trurl
Not to be confused with the other Hevea:
[http://hevea.inria.fr/](http://hevea.inria.fr/)

