Thanks for that response - I can see that you've missed the point, so I need to go back and consider how to re-work it to make the point clearer.

I'm trying to explain that every point on the surface of a high-dimensional sphere has characteristics that we, with our 3-dimensional intuition, would more usually associate with a spike.

When I get a bit of time I'll go back and re-read it with your comment in mind.

I think the square getting spikier is the salient issue at hand, not the sphere getting spikier. The inner sphere has an ever increasing radius based on the distance from the origin of a unit n-sphere placed at (1, 1, 1, ...). The inner sphere isn't spiky, that point just keeps getting farther away because n-squares get spiky.

You're focussing here on something other than the point I'm trying to get to . Yes, the corners of the cube get further from the surface of the sphere, but that's not the intended point. As I said in the comment to which you're replying, the idea is that points on the surface of the high-dimensional sphere have characteristics we normally associate with a spike. As so often happens with analogies, people (and you're not the only one, so the problem is with the writing, not the audience) are concentrating or fixating on aspects other than the one the author had in mind.

Thanks for your comment.

Maybe you could elaborate on what you mean by "spikey". I associate spikes with some kind of discontinuity, having little to do with volume.

Two things in particular.

Firstly, if you stand on the surface and chop off a cap, it has almost no volume. Secondly, if you step in a random direction, you'll be outside, not inside, the sphere. These are almost equivalent if consider a ball around your current location - almost none of it intersects the sphere.

In our 3D world this are characteristics of a spike, not of a sphere, so thinking of high-dimensional spheres as spikey helps stop you from making natural mistakes driven by otherwise perfectly good visualization abilities.

That second thing is perhaps your best example so far.

The cube example suffers from construction. It's simple to see that the bounding spheres are fixed to the cube's corners, which escape with dimension. While very interesting, this doesn't do much to constrain the ball's shape.

Volume is also bad measure to use here, since it's already agreed that the volume of the entire sphere rapidly decreases. This is mostly due to the immense expansion of the reference cube. The ratio of the cap to the sphere is better, but you left that as an exercise to the reader, ignoring things like what height relative volume starts materializing in (a reasonable person might expect cos π/4).

But picking a random direction from the surface is illuminating. It's hard to get rid of that effect without switching to spherical coordinates.

Is there some quantitative measure of curvature that works for the (n-1) dimensional surface of an n-dimensional solid for any n? If such exists, it would allow you to quantify this.

After some thought, I think I see what you were trying to get at. Let’s say you’ve got a 3-sphere, and you lop off a thin slice. That slice has a fair amount of volume, whereas with an n-sphere for increasingly high n, the volume is less and less—and thus intuitively more and more spike-like. Right?

Yes, that's one part of it. The other part is the fact that from the surface, whichever direction you go is not into the sphere.