I wrote another little post, "Exploring Spaces 2: Twisted Taping". I made a twisted paper strip this time, and made some more interactive spaces you can try moving around in https://vezwork.github.io/polylab/dist/demo/articles/exploring_spaces_2/. I really enjoyed the feedback and adjacent thoughts on the last one from @Joshua Horowitz and @Konrad Hinsen! Once again, I'd love to hear feedback, criticism, and adjacent thoughts 🙂

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!)

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 https://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.