4 ms·

A trick to visualizing higher dimensions [video]

76 points by sillysaurus3 9y ago

rocqua 9y ago
I 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. [1] https://www.youtube.com/watch?v=AmgkSdhK4K8 https://www.youtube.com/watch?v=AmgkSdhK4K8 [2] https://www.youtube.com/watch?v=FhSFkLhDANA https://www.youtube.com/watch?v=FhSFkLhDANA
- fizixer 9y ago
  I 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 9y ago
    The 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.
    - fizixer 9y ago
      I 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.
- yorwba 9y ago
  > 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.
- karmakaze 9y ago
  funny no one's been noticing that it's 3blue1brown
- darkmighty 9y ago
  I'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/Astroid.svg/220px-Astroid.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/As...
emerged 9y ago
Beautifully animated as always, but it didn't help me visualize higher dimensions at all.
npgatech 9y ago
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).
- sillysaurus3 9y ago
  How 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.
- FullMtlAlcoholc 9y ago
  > 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.
jpeanuts 9y ago
A 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" [1] http://www.math.psu.edu/tseng/mathjoke1.html http://www.math.psu.edu/tseng/mathjoke1.html
zengargoyle 9y ago
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.

76 points by sillysaurus3 9y ago

I 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. [1] https://www.youtube.com/watch?v=AmgkSdhK4K8 https://www.youtube.com/watch?v=AmgkSdhK4K8 [2] https://www.youtube.com/watch?v=FhSFkLhDANA https://www.youtube.com/watch?v=FhSFkLhDANA

fizixer 9y ago
I 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 9y ago
  The 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.
  - fizixer 9y ago
    I 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.
yorwba 9y ago
> 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.
karmakaze 9y ago
funny no one's been noticing that it's 3blue1brown
darkmighty 9y ago
I'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/Astroid.svg/220px-Astroid.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/As...

emerged 9y ago

Beautifully animated as always, but it didn't help me visualize higher dimensions at all.

npgatech 9y ago

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).

sillysaurus3 9y ago
How 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.
FullMtlAlcoholc 9y ago
> 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.

jpeanuts 9y ago

A 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" [1] http://www.math.psu.edu/tseng/mathjoke1.html http://www.math.psu.edu/tseng/mathjoke1.html

zengargoyle 9y ago

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.