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Comment by rocqua | Hacker News Reader

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

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

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> 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'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...

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).

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.

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.

funny no one's been noticing that it's 3blue1brown

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