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Comment by pdonis | Hacker News Reader

> The point is that when you're on the surface of a high-dimensional sphere, what you're standing on has characteristics that we, with our 3D intuition, would normally associate with spikes.

Can you explain why the "spikiness" should be associated with the hyperspheres instead of the hypercubes?

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Sure, although it's not an "instead". Let me explain.

No, it would take too long. Let me summarize.

If you look at high-dimensional cubes it's pretty obvious that the corners get further and further away from the center as the dimension goes up. People (quite rightly) feel that this can be visualized as taking an ordinary 3D cube and tugging on the corners, stretching them out and thus making them "spikey".

With spheres, though, there are no corners, so people think they remain "smooth". Which they do, in a sense, but that leads to the wrong intuition. Lopping off a high-dimensional spherical cap gives you virtually no volume at all, and in our 3D world the shape that is as symmetrical as you can make it, but which when you chop it off, has almost no volume, is a spike.

Add to that these comments: http://news.ycombinator.com/item?id=3997551 and http://news.ycombinator.com/item?id=3997681

These allude to the fact that when you've on the surface of a high-dimensional sphere, almost every step takes you out. Getting into the interior is like trying to get into the interior of a spike from its tip.

Does that help?

A nearly flat cap from the 3D sphere with big radius will also have small volume. I think the word "spikey" is bad for getting your idea across.

But in the 3D sphere the volume of the cap grows rapidly with the height. That's not true in higher dimensions, which is why in higher dimensions it behaves more like a spike than a sphere. It's the apparent contradiction of that image that helps prevent the usual mistake in visualization.

That's why I use the term, for that shock value.

> when you've on the surface of a high-dimensional sphere, almost every step takes you out. Getting into the interior is like trying to get into the interior of a spike from its tip.

This does, yes. Starting with how this aspect changes from a circle (1-sphere) to a 2-sphere might help to get across the "spikiness" you are talking about. It seems to be basically saying the same thing as the "lopping off the cap" description, but the latter was very hard for me to visualize. Saying that a smaller and smaller fraction of the total "hypersolid angle" intersects the hypersphere's interior as you increase the dimension makes it much easier (for me) to see what's going on.

Cool - thanks.

> almost every step takes you out

This doesn't seem to be true.

Imagine a sphere in n-dimensions, S = {Sum(x_i^2)<1}. Top of the sphere is p=(1,0,0,...,0). Now for any random direction r, if you take a very small step from p towards r, you have exactly 50% chance to be inside the sphere.

To be mathematically precise, for any r (unit vector) chosen at random, probability that there is e>0 such that p + e*r is within S is 1/2.

So half the time the steps take you out, half the time it takes you in - considering the step is small enough compared to the radius.

But for a given e, that probability goes to zero as n goes to infinity, or as n goes to infinity, the e required to be within a given delta of 50% gets smaller. In other words, choose your step size in an optimisation. As n gets larger, the chances of it working get smaller. In even moderately large dimensions the step sixe required is so small, you don't make any progress.

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