New video early view
Added 2024-11-02 22:32:46 +0000 UTCThe next video includes a few truly delightful geometry puzzles, and I wanted to share an early cut with you now. There are some blank spots for animations, including the entirety of the last three minutes or so, but the core content is there.
Let me know what you think, and I'll factor it in during final edits in the next few days.
Comments
Thank you very much. I enjoy the video. I agree with your words; Anaogy is dangerous. The reason is that an analogy does not expand my world. Analogy always uses what I already know, but some ideas are entirely outside of my knowledge or experiences. In that case, I misunderstood the outside of my world by projecting to what I knew. In this case, I didn't acquire new knowledge. (But I feel something I know now. This is a danger to me.) I sometimes think I need to go outside of my comfort world and see the real unknown world by expanding my world (which usually confuses me a lot.) I have a hunch of intuition that high-dimensional vectors are almost perpendicular. I imagine a sphere and a random vector are directed in a specific direction. If the dot product of them is close enough, these vectors are on a small non-zero area. The surface area of a sphere is larger and larger in higher dimensions. Then, the chance of being in the neighboring area is smaller if I assume the random means that the vector and the sphere intersection with the epsilon (>0) hiper-area are well distributed. But this is my intuition, and I don't know whether that is a good way to think or not. Thanks a lot!
Hitoshi Yamauchi
2024-11-13 17:33:10 +0000 UTCnow I'd like to snap together oblate cubill magnets all day
Richard Kalhöfer
2024-11-09 12:21:08 +0000 UTCThis is the solution I also thought of. I remembered learning about this vanishing line/line of infinity from an art class I took in middle school long ago. From that I have the intuition that it should be a straight line in 2D
Stanley Sisson
2024-11-08 06:03:24 +0000 UTCMinor detail—maybe too minor to worry about, but: You said that the only restriction on Monge's theorem is that the circles can't be the same size, but also: one can't lie entirely within another, since then they won't even *have* common tangent lines.
Jason Taff
2024-11-08 04:30:12 +0000 UTCIt might be sad that we are unlikely to have a direct intuition about 4+ dimensions. But, for many folks (like me), the lower-dimension insights your videos offer would be nearly out of reach too without 3B1B. So thanks for a really incredible video! My neurons feel significantly rewired after watching!
Ron Goodman
2024-11-05 17:28:13 +0000 UTCPotentially, though more honestly I really do think that the level of intuition we have available to us for 3 dimensions is just not on the table for, say, 24 dimensions. And this is sad! One item on the topic list is a video to make strange-sounding facts about higher dimensional balls (like most volume being near the equator, or near the surface), feel less weird. Perhaps that can provide the counterbalanced note of optimism.
3blue1brown
2024-11-04 17:09:58 +0000 UTCNice, that is a fun generalization.
3blue1brown
2024-11-04 17:06:46 +0000 UTCFor details, see Page 17 of this pdf: https://www.weizmann.ac.il/math/klartag/sites/math.klartag/files/uploads/High_dim_lecture1.pdf The rabbit hole goes deeper as Klartag asks us to prove in Exercise 5 here: https://www.weizmann.ac.il/math/klartag/sites/math.klartag/files/uploads/hd_2020_ex1.pdf
Akash Kumar
2024-11-04 12:38:07 +0000 UTCA great video, as usual! A small comment. The proof by Archimedes which you presented -- which shows Area(sphere) = Area(An appropriate Cylinder) kicks off a really nice story. Namely, there are more theorems which share the same conceptual space and one of these theorems which Boaz Klartag attributes to (guess who) Archimedes asserts the following: Imagine taking a uniformly random point from the unit sphere S^{n-1} \subseteq R^n (which is the boundary of the unit ball). Call this point (X_1, X_2, \ldots X_n). Consider any n-2 tuple you obtain by dropping two of your favorite points. For instance, consider (X_1, X_2, \ldots, X_{n-2}). Klartag claims that this is a *uniformly random point* in B_{n-2} -- the unit ball in n-2 dimensions.
Akash Kumar
2024-11-04 12:34:08 +0000 UTCVery fun! My first impression of the rotations of the hexagons in puzzle #1, was that since you're going only part way around, there must be several possible orientations. I had to go back later to realize that rotating twice in the same direction got you back to the same figure you started with, so there are only two possibilities. It might be useful to point that out.
Neal McBurnett
2024-11-03 22:24:04 +0000 UTCFor problem #3, I think you can prove it with perspective. The 3 circles are identical balls (having the same radius) at different distances from the viewer. The balls are connected by 3 infinite cylinders. The cylinders form a triangle, and so lie on the same plane. Under perspective, the plane has a vanishing line on which each cylinder's vanishing point must lie.
Sam Sartor
2024-11-03 17:05:20 +0000 UTCThis was a juicy mixtape of puzzles, thank you! Re: the "sad ending", if you'd like to end on a bittersweet note of hope, & make a reference to an old video, you could mention your 10D Sphere Puzzle video! In that video, you come up with (afaik) novel "hybrid" visualization between the pure-geometric & pure-analytic approach, visualizing pt's on a 10D sphere as 10 sliders. More generally: maybe, like how complex analysis was really hard until someone came up with a clever visualization, maybe some punk mathematician will come up with a way to visualize objects in arbitrary N dimensions. That could be a note of hope (& possible research direction) to end on!
Nicky Case
2024-11-03 14:36:49 +0000 UTCI can puzzle around with it. Maybe smoothly transitioning to a more realistic field of view could do the trick
3blue1brown
2024-11-03 14:14:33 +0000 UTCImagine not loving projective geometry! :b Also, not sure if it's just Patreon, but that section with the spheres and cones had too much going on for the bitrate to keep up :( But the video was super cool as usual, many thanks Grant!
_ericBG
2024-11-03 12:54:46 +0000 UTCOhhh I see, that does make sense. I rewatched, and FWIW (data set size of 1 here) the emphasis on *number of MOVES* and *one of these to another* is what caught me up. So it sounded like maximizing the steps of _any two configurations_ rather than minimizing the steps of _the two farthest_. 🤷♂️ > what is the maximum number of MOVES that it might take to get from one of these to another
Colin Gray
2024-11-03 12:12:49 +0000 UTCI noticed that when you were showing the 3-dimensional views for part #1, I was having a lot of trouble with the images flipping in my mind between pointing to the back at times and forward at others. I'm not sure how to change that, but it's probably a well-known issue with isometric projections and fixable. I'm not sure whether it would help to have shadows? Is there a better way?
William Smith
2024-11-03 05:06:39 +0000 UTCI guess the phrasing is confusing. What two states are the "farthest" apart, in that the minimum number of steps required is as high as possible, and how high is that?
3blue1brown
2024-11-03 00:26:30 +0000 UTCGolay code and Leech lattice !!! I hope you make a video at some point! Some random thoughts, not in any order: a little more of a lead time to pause and ponder would be good, some of the solutions crept up on me before hitting pause. Puzzle #3 was my favorite. #5 reminds of me of the 4-d (and higher) rubiks cubes. Though the video theme is "dimensional crossover", I wonder if there is a separate theme that can tie the puzzles together? Thank you for calling out the math olympiad, I hope more of your viewers learn about the world of competitive math! In fact, I wonder how much of your views are from people who might not otherwise know about the world of math competitions and research. Thanks for another great video :)
Michael Kokosenski
2024-11-03 00:25:32 +0000 UTCIn problem #1 there's a question "what's the maximum number of steps needed?" But should that instead be "minimum steps"?
Colin Gray
2024-11-02 23:10:14 +0000 UTCHave you read up on the new soft cell shapes? I'd love to see how you would present those ideas
Steve Chantry-Taylor
2024-11-02 23:03:34 +0000 UTCAnd I thot #3 was "Mongo only pawn in game of life's" Theorem.
Jon Adams
2024-11-02 22:51:25 +0000 UTC
