Cute! This reminds me of the kind of app github.com/ccorcos/tuple-database was designed for

Cool. I need to look into that still

Very cool

I think I saw tuple DB via the local first discord

localfirstweb.dev

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.

Thank you Konrad Hinsen! Its so interesting to hear that molecular simulations are done in S1^3! I have an idea for illustrating this space as well, but I was avoiding getting into 3D rendering for this post.

Why do people do molecular simulations in that space? It would be strange to me if a molecule was connected as a circle across the entire space since I don't know what would correspond to in real life.

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.

I've been playing around with having a "clone" dot that makes it look like the dot is peeking around from the other side to address the visual jump. You can see the clone in the circular interactive at the bottom of this new post: vezwork.github.io/polylab/dist/demo/articles/exploring_spaces_2

I'm wondering if the clone dot is better or if the jump is better.

Konrad Hinsen I found this video on youtube visualizing water molecules turning into ice youtube.com/watch?v=zRUFzJrDtq0 (It looks really cool 😉:ice_cube:). Is this visually representative of the sort of simulation you do? Do your simulations usually have visualizations attached? Or do you get out raw data mostly?

That's a nice one! Not the kind I do because I work on biomolecules. Mine are more like this one: youtube.com/watch?v=7AhQ19m2ok4 (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.

Clone dot is dope I love it. It’s more durably physical than jump dot – like it treats the dot as a real object with spatial extent sitting in the space in question. (I would enjoy pulling clone dot into the corner of the torus square and seeing it cut into four quarters!)

And circular RP2 interactive is super interesting! I can drag the dot to a place on the screen and go in little circles in a way which makes the dot circulate around a much bigger circle. This is cuz you’re not mapping from screen space to the circle with a function, but instead you’re mapping a path on the screen to a path in the circle in a path-dependent way. (Lmk if that doesn’t makes sense.)

Edit: This is kinda hysteresis? Kinda? (joshuahhh.com/projects/hysteresis)

I know you’re wary of getting into 3D rendering right now, but I do think that parallel rep would be very valuable for people who haven’t seen this stuff before.

And a wacky suggestion for a puzzle I’d love to see explored interactively: What’s the configuration space of lines in the plane?

Wow, I love these comments.

re: clone dot comment: I hadn't thought about this torus example! I would need 3 clones, interesting! I like this point you're making.

I was on the fence about the clone dot because it feels kind-of magic-y and less like "there's just a point moving around" from a programming perspective.

re: circular RP2 interactive comment: I also played around with this circulation around the edge of the circle lol. Yea, I kind of know what you mean about the path dependence. If it wasn't path dependent, then I would have a continuous map f: Screen Plane -> Disk, probably such that f(Disk) = Disk, is there a map like that?

And whoa!! This post about hysterisis is super cool. I love how I can see that the space of these points joined by slack is parameterized by a rhombus. Also how the space is like two lines (vertical sides of the rhombus) connected by a line at each corresponding point and how the connecting lines don't connect to eachother.

re: 3D: you're right. I will do it. I'm gonna continue on my 2D journey for a bit more tho, because if I do 3D I will want to add point picking and interaction and I think I want to do raytracing to avoid triangle artifacts.

re: wacky puzzle: had to think about this a bit. Is this space equivalent to an infinitely tall cylinder? Let theta be the angle of a point on the cylinder and h be its height; the corresponding line is perpendicular to the vector, in polar coords (h, 2*theta).

edit: no, this is really wrong lol. I was thinking about how from each line through the origin there's a real line worth of lines parallel to it.

edit2: found this math.stackexchange.com/questions/1848739/a-topology-on-the-set-of-lines/1848770#1848770.

and I see that I was on the right track that locally it looks like each point is on a circle and a line, but I was missing the global twist in the space.

Very nice!

Cool! Looked up miller columns, neat usecase here!

Nice!

On my computer, dragging those lines & circles selects number text, which is distracting. I think some user-select: none should fix that.

so fun to play with