Graph or lattice topology discretization

  • #1
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Main Question or Discussion Point

Mathematicians, I summon thee to help me identify which field deals with this stuff. I come here not as a physicist but as a sunday programmer trying to solve some numerical problems.

I set out to model a lattice version of a smooth space. A discretization procedure not uncommon in physics, but there's a catch: every connected point has the same distance. Which essentially breaks everything I knew about lattices. So I'm not putting points on a plane (where each point would have its own coordinates) but making a plane out of points. So each point only has has information about its connectivity with its surroundings. This is closer to graph theory, I believe.

This smells a lot like topology, that's why I came here. Is there a way to retrieve information, like number of dimensions, or even geometrical aspects like curvature, in some limit where this graph/lattice tends to a continuum space? This limit would be an increase in number of points or something like that.
 

Answers and Replies

  • #2
84
35
Not an expert, but I don’t think graphs are appropriate structures for most topological or geometric problems. Graphs have only vertices and edges, whereas a triangulation of a topological or geometric space has k-simplices, where k<= the dimension of the space being considered.

I get the distinct impression that you are interested in geometric data and not topological data (that would be invariants like Betti numbers and homology groups and such). If you have a representation of a space as a simplicial complex, you could consider the discrete exterior calculus, but to extract meaningful geometric information I assume you’d some more geometric data, perhaps a metric which assigns edges to their respective distances.
 
  • #3
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Thanks for pointing to (discrete) exterior algebra. I will have to do some reading to see if that is the way but it's better than what I had before! (nothing)
 
  • #4
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OK, I did some more reading. Exterior algebra is promising but I suspect the solution might be simpler. I found some more fancy words to describe what I was talking about. I learned the concept of taxicab geometry, where distance is defined as an absolute value from point to point. So maybe if I can tesselate a surface in a specific way where the taxicab distance in this lattice approximates the continuum version, that might work.

Anyway, just wanted to do a update/closure to this topic.
 

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