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The linked paper shows how the klein bottle maps onto the space. Basically, grids which have an orientation (horizontal or vertical stripes) are more common and
by jamii 15y ago
The linked paper shows how the klein bottle maps onto the space. Basically, grids which have an orientation (horizontal or vertical stripes) are more common and join together to form a klein bottle. That tendency towards orientation might be an artefact of the camera or might be because cameras are usually held horizontal or vertical relative to the sky.
- Drbble 15y agoThe paper notes that the vertical lines are gravity aligned, not camera aligned. The paper desperately needs a accompanying translation from extreme mathematician to mortal PhD level.
- jamii 15y agoThe second (expository) paper is pretty readable. I have only the barest grasp of algebraic topology but I believe I have a rough high-level understanding of what it is doing. It would be interesting to implement their barcode program independently to test whether or not I really understand.
- Tloewald 15y agoThe comment is interesting. He? argues that any 2 manifold with no edges would be likely to end up as a Klein bottle. The question is then why a 2 manifold. Well, the thing we're asking about is a 2d picture, right? Maybe that has something to do with it. Let's suppose we tried the same trick using 3x3x3 pixel representations of STL models — might that turn out to be form a simple edge less 3 manifold embedded in 27 dimensional space? (perhaps for simplicity we could use 2x2x2 cubes and look at 8 dimensional space... In essence we're talking about a population of 2d data arbitrarily but consistently mapped into higher dimensional space and we discover it maps to a 2 manifold.