New video! A trick to visualizing higher dimensions
Added 2017-08-11 22:50:26 +0000 UTCHello all,
This is a fun one on trying to grapple with higher dimensional spheres. The puzzle that it centers on is a very famous example of where our intuition for higher dimensions breaks down, but the goal I set for myself here was to present it in such a way that the final 10-dimensional example doesn't actually feel that weird. I'm curious to hear what you think!
Perhaps it's worth saying a few words on why those interested in math care about higher dimensions. Why care about 20-dimensional space when the world we live in is not 20-dimensional? Broadly speaking, spatial reasoning is not just useful for describing the physical world, it's a sort of problem-solving tactic for any scenario where you want to think about collections of things (like lists of numbers) as individual points of some object.
For example, viewers of the "Who cares about topology" video will remember how nice it was to associate pairs of points on a loop with individual points on a Mobius strip. But this is only associating collections of two things with individual points. As the relevant collection gets larger, the corresponding object is harder to hold in one's mind, since it lives in a higher dimensional space. For example, there is something analogous to a Mobius strip that represents unordered quintuplets of points on a loop, but seeing it would require being able to visualize 6 dimensions. That doesn't stop mathematicians from working with such objects, though, it's just that the work becomes a bit more abstruse.
The tactic I provide in this video is not some magic make-you-feel-like-neo-from-the-matrix visual that lets you hold a full high-dimensional object in your head. Instead, its main function is to emphasize that what we really mean by "a point in n dimensions" is "a list of n real numbers", and moreover that it can still be fruitful to think visually about that idea even when you can't conceptual the list of numbers as an individual point in space as we can for 1, 2 and 3 dimensions.
-Grant
Comments
Solving one of Brilliant question: “What is true of all odd numbers?... And one of answers “They look strange” 😂🤣😂🤣
2018-08-04 01:08:16 +0000 UTCYou Know, Professor! Your videos lectures are so imaginative! Seeing the numbers taking real forms of things and with this song... It is amazing! Have you ever thought to translate in others languages? So many countries all around finding a mathematician professors like you. In my opinion, it would be a fantastic tool for others professors. ----- "Good ideas need to be spread." ✌️😉
2017-08-27 02:16:03 +0000 UTCMy intention was for the box example to be a playground for thinking of higher dimensional spheres this way. If all you want is to know the radius of the inner sphere, I'll agree that it is much more immediate to simply write sqrt(N)-1. For me at least, and maybe this doesn't apply to everyone, there was always something unsatisfying about that. The conclusion that the inner sphere grows without bound was right there in the numbers, but it felt like such an alien result. What made it click for me was to actually think about all the numbers in x_1^2 + x_2^2 +...+x_n^2 = 1, and how most of them need to be very near zero, and hence unit spheres are very small relative to unit cubes in higher dimensions. The sliders are, of course, just a way too see and play with those numbers more readily. Maybe it seems more convoluted for some. And I can see how if the impression I gave was that they are meant just for this one example, it would seem like more of a diversion than it's worth. But I had fun thinking about things this way, and wanted to share that :)
3blue1brown
2017-08-18 22:11:50 +0000 UTCSorry, but it seems like an extremely roundabout way of explaining something immediately obvious from the formula. And I don't see how the sliders make any use of our wonderful visual cortex.
Alexey Badalov
2017-08-17 13:30:44 +0000 UTCIt's <a href="https://brilliant.org/3b1b." rel="nofollow noopener" target="_blank">https://brilliant.org/3b1b.</a> This should be in the video description and on the screen. I hope you like them!
3blue1brown
2017-08-17 06:27:52 +0000 UTCHi, I can't find the link to 'brilliant' that lets them know I came from your site, Grant.
2017-08-16 22:58:52 +0000 UTCCongrats
Illuminati Games
2017-08-14 00:52:09 +0000 UTCGrant--your program is terrific. I also wish to thank you for discussing Brilliant!!! I have great fun with it at all levels and I contiually learn that which I already thought I knew!1 One surprise after another.
John C. Vesey
2017-08-12 17:52:46 +0000 UTCYou are doing really good.
Illuminati Games
2017-08-12 17:46:57 +0000 UTCAt 22:41 your cute li'l confession doesn't really amount to anything more than "yay, perturbation theory activate!" I mean, were you planning to disclaim your Millenium Falcon of math visualization software doing the same trick half a dozen times, and then the video codec doing the same trick a half dozen times, and then my browser's HTML5 rendering engine doing.. AND THEN my video card.. my LED monitor.. my retina.. my occipital lobe.. my memory recall of the event.. look, I'm still fairly certain there was a graph there, yeah? :D
Jesse Thompson
2017-08-12 05:24:54 +0000 UTCBill Russell
2017-08-12 00:24:27 +0000 UTCI just finished watching the clip. It was outstanding as usual. Do you see the link between the Riemann zeta function, or am I just dumb. Also the space or real-estate is a concave equal to the space convex.
Bill Russell
2017-08-12 00:23:05 +0000 UTCAt that point, x is 0.5, but x^2 is 0.25.
3blue1brown
2017-08-12 00:22:02 +0000 UTC@7:30, did you mean to say 0.25 instead of 0.5?
PseudsPie
2017-08-11 23:12:32 +0000 UTCWow. I'm working at 4 dimensions and 1 time path.
Bill Russell
2017-08-11 22:53:15 +0000 UTC
