Bonus Dimensional Puzzles
Added 2024-11-07 23:12:52 +0000 UTCI just uploaded a finalized version of the latest video, "Why 4d geometry makes me sad". There were a couple of other puzzles on my list that could have followed this theme, and I thought I'd throw together a quick and casual bonus video showing two more for everyone here, as a thanks for the support. Enjoy!
Comments
I just remembered a problem with an interesting dimensionality trick - To find a Delaunay triangulation of a set of points, map each point to (x, y, x^2+z^2) and find the lower 3D convex hull of these points. Those triangles are precisely the ones in the Delaunay triangulation! The proof is pretty mechanical and not too insightful imo, but an interesting trick nonetheless
Aman Karunakaran
2024-11-14 15:02:52 +0000 UTCAh, I see. I like that section, it's lovely. ๐ I still think there are ways that we could in principle build intuitions (perhaps less visual and more proprioceptional) about higher dimensional spaces if we interact with many simulations โ like video games โ that involve efficiently navigating them. Then folk can learn how to build world maps in their mind and know where things are relative to one another and how to navigate from one point in that space to another on an intuitional level. Games like 4D Golf, 4D Miner, Hyperbolica, Hyperrogue, Manifold Garden, et al are good starts for building intuitions for higher dimensional and non-Euclidean spaces.
Jesse Thompson
2024-11-08 01:30:46 +0000 UTCIt's 25:53 onward; the idea is that although the human mind's capabilities for visualization diminish drastically above 3 dimensions, the potential for spatial insights via higher dimensional perspectives does not necessarily stop above 3 dimensions. Personally I think the real tragedy is that it seems for many purposes, 5 dimensions and up is pretty uninteresting, but 4 dimensions seems to be incredibly rich with nontrivial problems. We were so close!
Aman Karunakaran
2024-11-08 01:08:52 +0000 UTCNice puzzles! 0:40 provided they are not collinear. I don't know if there's a slick obvious argument that they are not all collinear, but you basically get it for free with your proof anyway (or at the least you show that if all three are collinear then at least one is the same as one of the others) I love the fact that the pink/yellow/green lines in the second problem shrink in the distance, already foreshadowing the 3d perspective you're about to reveal EDIT: deleted some other comments bc I didn't realize this was a patron-only video at first. Thanks for the extra puzzles!
Aman Karunakaran
2024-11-08 01:00:11 +0000 UTCI saw the previous preview for this video, and the unlisted video presently has title "Why 4D geometry makes me sad". I don't recall anywhere that you mentioned it making you sad though, is there a timestamp to that topic in the finished video?
Jesse Thompson
2024-11-07 23:48:28 +0000 UTC
