# Graph or lattice topology discretization

• A
• diegzumillo
In summary, the conversation revolved around the topic of using lattice models to represent smooth spaces and the challenges that arise with this approach. The person asking for help was unsure about which field of mathematics would be most applicable and mentioned graph theory and topology as possibilities. The experts suggested using discrete exterior calculus to extract geometric information from a simplicial complex and also mentioned the concept of taxicab geometry as a potential solution. The conversation ended with the person expressing their gratitude for the new information and sharing their idea for a potential solution.

#### diegzumillo

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.

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.

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)

• suremarc
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.