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Thinking outside the 10-dimensional box
3Blue1Brown
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All the comments and stories posted to Hacker News that reference this video.As you increase the dimensionality of a sphere, an ever-increasing proportion of the sphere is within epsilon of the surface.This 3Blue1Brown video addresses enough related stuff that the rest should become reasonably comprehensible: https://www.youtube.com/watch?v=zwAD6dRSVyI This particular result is tossed in as a side note around 23:25, so it's not addressed directly, but it'll help.
I like this video on the topic: https://youtu.be/zwAD6dRSVyI
Yeah, I see now. “times i” is discrete 90 degree rotation itself, not just ‘i’, nor ‘b’. Thanks everyone for making that clear.This though shows that explanations via analogies or non-strict wording may confuse one rather than enlighten. I’m not good at math, but once understood to not search analogies or geometry in things. Instead it is better to “shut up and calculate”. Not sure if imagining something is required to manage it. It’s only our brain’s faulty quirk.
Don't forget 3Blue1Brown's incredible video about visualizing higher dimensions[1]. He's getting to xkcd levels of "Of course there's a relevant video"
Great 3blue1brown video to accompany this – Thinking Visually about Higher Dimensions:
Great article! I've been thinking about this on and off recently, as I wonder if we might be having intuitive issues when it comes to gradient descent optimization.3 Blue,1 Brown had a video recently that kicked off my head scratching and is a great complement to your article: https://www.youtube.com/watch?v=zwAD6dRSVyI
⬐ marckhouryThank you! It's funny you should mention that because I've been thinking about continuous optimization a lot lately.That's a great video, I really like his slider method for understanding the coordinates. Thanks for linking it!
I would reiterate that simply because there are problems with naive visualization, we shouldn't discredit visual thinking.There are several key elements to effective visual thinking. The primary importance is to keep it grounded in proofs and theorems, so you know exactly what are your limitations. Often you can use a geometric argument on top of a few theorems and you get a very strong result intuitively, and then use this intuition with a tiny amount of algebra to prove it (which might take you forever to arrive from a purely algebraic perspective). Another key is that there are several ways of visualizing things. You can almost always transform a problem into an equivalent one that is easy to visualize (just need a little bit of care with the transformation, etc).
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For example, you can show functions form a vector space, visualizing some interesting algebraic properties about them, even if it constitutes an infinite-dimensional space.
You can show several operators (such as d/dx) are linear, you can give it a norm, internal product, etc. This trick lets you use visual tools (and linear algebra tools) with arbitrary functions. You can visualize projection of a function into a subspace, or into some non-canonical basis -- yielding useful applications -- such as Fourier analysis.
Fourier analysis itself is a fertile ground for visual thinking. You'll be finding trivial arguments for seemingly difficult decisions such as "Does this linear system have a bounded output for any bounded input?". There isn't one right way of thinking about anything.
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On the other hand, it can't be stressed enough the importance of keeping track of formal assumptions, axioms, definitions, theorems to construct valid, correct proofs. That way you minimize the risk of fooling yourself, and can safely use your intuition.
This 3B1B video exemplifies many of those elements:
A trick for visualizing higher dimensions: https://www.youtube.com/watch?v=zwAD6dRSVyIOnly 383k subscribers. Hmm. Remember to subscribe, comment, and smash that like button.
Real Engineering is another great channel. And of course Veritasium and Numberphile.
⬐ dahartThat's a cool video, I'm subscribing. It's surprising that the embedded sphere has an unbounded radius. He didn't mention in the video that this problem is due to, and is a sort of dual or inverse to the fact that the volume of an N-dimensional sphere goes to zero as N goes to infinity. That hurt my head a little the first time I learned about it!
⬐ rocquaI really like 1blue3brown, but this one didn't quite resonate with me. To me, it felt like a long winded way to make it intuitive that the point (1, 1 ... 1) has distance sqrt(n) from the origin, where n is the number of dimensions. I think there'd be move value in explaining things like 'most volume is near the edge' and 'most points are far apart'. Granted, that is less about building intuition, and more about specifically dispelling heuristics from 2 and 3 dimensions.I'd still wholeheartedly recommend his other videos though. Especially [1] where he gives a very nice topological result regarding inscribed rectangles in closed loops. In the same vein is [2] proving the borsuk-ulam theorem.
⬐ darkmighty⬐ npgatechI've been thinking of ways of making some sort of rigorous analogue (in the sense of obeying a small set of properties) to higher dimensional manifolds, like spheres and cubes.For example, an unit sphere inscribed inside a high dimensional unit cube looks a lot like an astroid[1] inscribed in a square in 2D with regards to metric properties (using an L^2 metric):
- The astroid touches the cube, while it is very far to the cube in the direction of the corner
- We can see that the corners, being far from the unit-L^2 distance manifold, occupy most of the volume
- As the number of dimensions increase, the astroid becomes increasingly compressed near the origin
[1] https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/As...
⬐ fizixerI think a major qualifier to all the 3+ D vis. schemes is that, no matter how elaborate or clever the trick is, you can never experience such a space the way you experience 2D or 3D space. (all the tricks are only approximate visualizations).⬐ JoshTriplett⬐ karmakazeThe levels in Braid that have time moving forward and backwards as you move left and right are the closest thing I've seen to "experiencing" space-time where time is a dimension. As you think hard enough about the puzzles, you can imagine moving in time the way you move in space, treating it as just another dimension.⬐ fizixerI knew someone would invoke time. I'm aware of 4D space-time as a possible visualization scheme for 4 dimensions.By '2D or 3D space', I specifically meant being able to experience 3+ dimensions the way we experience 2 or 3 spatial dimensions.
funny no one's been noticing that it's 3blue1brown⬐ yorwba> I think there'd be move value in explaining things like 'most volume is near the edge' and 'most points are far apart'.You can do that in 2d already. Take a square. Double its side, yielding quadruple the area. The 75% of the area that were added are closer to the edge than to the center. That most points are far apart is a consequence, since you can always take one of the points to be the center.
The curse of dimensionality is that this gets worse as the number of dimensions increases. You can observe this in the step from 1d -> 2d: 50% near the edge vs. 75% near the edge.
To really drive home the point, consider 3d: doubling the side of a cube yields 8x the volume, 7/8 = 87.5% of which are near the edge.
For n dimensions, the volume near the edge is 1 - 2^-n. Already at n = 10, more than 99.9% are near the edge.
I totally applaud 1blue3brown's videos but for some reason, his approach leaves me more confused than ever. Long winded unnecessary visualizations, especially when animated, annoys me for a reason that I cannot explain.Don't get me wrong, visualizations are powerful but I think I prefer static visualizations. Animations overload the visual system with "visual bloat", if you will.
Also, this video did not help me visualize higher dimensions. I prefer a simpler approach - just project down to 3 dimensions (or 4 if you add time at the risk of animating).
⬐ FullMtlAlcoholc⬐ zengargoyle> I prefer static visualizations. Animations overload the visual system with "visual bloat", if you will.Then perhaps learning from a book or slides would be better for you. Animations are not visual bloat, they've helped me understand topics that had eluded me for years. Also, I can now visualize how a projection of a hypercube/tesseract rotates because of animations. Visualizing that process with static images would probably take me until the heat death of the universe to understand.
⬐ sillysaurus3How would you project a 10D sphere into 3 dimensions? Worse, the projection tricks you into believing you've gleaned some insight.The animation was critical for understanding the relationship between the values as they change.
I have found with 1blue3brown that I sometimes have to wait a while and ponder things and go back and watch the video again some time later. So this particular video was working for me up to a point and then I maybe just missed some important point that hopefully I'll catch on a re-watch next week or so.⬐ emergedBeautifully animated as always, but it didn't help me visualize higher dimensions at all.⬐ jpeanutsA Mathematician and an Engineer attend a lecture by a Physicist. The topic concerns Kulza-Klein theories involving physical processes that occur in spaces with dimensions of 9, 12 and even higher. The Mathematician is sitting, clearly enjoying the lecture, while the Engineer is frowning and looking generally confused and puzzled. By the end the Engineer has a terrible headache. At the end, the Mathematician comments about the wonderful lecture. The Engineer says "How do you understand this stuff?" Mathematician: "I just visualize the process" Engineer: "How can you POSSIBLY visualize something that occurs in 9-dimensional space?" Mathematician: "Easy, first visualize it in N-dimensional space, then let N go to 9"
