Yesterday I taped together a paper torus, built some inputs that wrap around like a torus, and drew some lines on a torus. I wrote a little post about my exploration! https://vezwork.github.io/polylab/dist/demo/articles/exploring_spaces_1/ Would love to hear your feedback, criticism, adjacent thoughts, etc!

Feedbacks: It threw me off that moving one of the sliders moved the other slider! At first I thought it was some weird bug. Now I see that they’re both actually 2D sliders. That’s kinda confusing. It’s interesting! Maybe there’s something there. But: kinda confusing. It seems like you’re interested here in “parallel representations” – seeing the torus as a square (with identified opposite sides), as a 3D object, and as the configuration space of a pair of sliders. I’d really like to see more of that. For instance, in the “Input Spaces” section, could you show a point on a 3D torus alongside the two reps you have already?

Nice! Molecular simulations are mainly done in hypertorus geometries, meaning a 3D space resulting from the Cartesian product of three circles. Teaching this is quite challenging. I never thought of using a paper torus for illustration, but next time I will!

Thanks so much for the feedback @Joshua Horowitz! I hear you on the slider weirdness, fixed it! The 1D sliders should be actually 1D now. I'm excited to hear that you'd like to see more parallel representation stuff. I will come back to this (3D torus) once I start getting into 3D space stuff.

The fundamental problem in molecular simulation is that we can simulate only much smaller volumes than a realistic sample should have. Even a drop of water is too large to simulate. Moreover, simulating a complete drop would be of no interest (for reason that I can go into if you want, but that involves quite some details about modelling liquids). So we simulate a finite volume, and even a rather small one. What shape should it have? And what should be around it? There is no good answer to the shape question as the volume is much too small to be realistic anyway. And if we put walls around it, we make a system much too small to show certain forms of motion, such as slow diffusion. So it's best not to have walls at all. Conclusion: we use 3D hypersurfaces, i.e. the surfaces of some 4D volume. Finite volume, and yet without boundaries. Both hypertori and hypersheres have been used, but hypertori won because they cause fewer unwanted artifacts.

Wow cool. Since you said you can't simulate large volumes, does that mean that the simulations you are doing are quantum field simulations?

Oh also both sliders have zones off the edge where you can drag the dot and the never get it back.

Some people do quantum simulations (though not quantum field, that's for subatomic particles). Personally I don't. They are interesting mostly for simulating chemical reactions. But even for "classical" simulations, it's not reasonable to aim for realistic volumes. A small drop of water, say 0.05 ml, has 10^{21} water molecules. Most properties of water for which the molecular scale matters can be computed with 1000 molecules on a hypertorus. And if you are interested in the specificities of the droplet, for things like surface tension, a continuum mechanics simulation (fluid dynamics, as used for engineering) is both cheaper and provides more insight.

Thanks again @Joshua Horowitz! I originally intentionally added the non-visible zones the dot can go into. I wanted to hide the "jump" when the dot teleports from one side to the other. I don't like that you can get stuck though. I fixed it.

That's a nice one! Not the kind I do because I work on biomolecules. Mine are more like this one: (note that there should be water all around the protein, but it's not shown because it would hide the proteins). Visualizatons are crucial to make sense of such simulations, but there's always a second phase of numerical data analysis.