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17 January 2022

Hollow Spheres in High Dimensions

by Rylan Schaeffer

Stan just wrote a post about the counter-intuitive properties of high dimensions, focusing on a sphere touching surrounding spheres packed inside surrounding squares:

He then shows as the dimensionality increase, the inner sphere becomes this spiky object, popping out from between the surrounding spheres:

It’s a common example, but one that (in my opinion) is too complicated to explain simply, especially when there’s a much simpler example available: a single sphere. Let’s ask a simple question: how much of a sphere is close to its boundary in high dimensions?

Dimensions = 2

Let’s start with $$D=2$$ dimensions and ask what fraction of a radius-$$r$$ sphere (i.e. a circle) is within $$\epsilon << r$$ of its surface. We know that the total volume is $$\pi r^2$$ and the inner volume is $$\pi (r - \epsilon)^2$$, so the fraction not near the periphery is:

$\frac{\pi (r - \epsilon)^2}{\pi r^2} = \Big(\frac{r - \epsilon}{r}\Big)^2$

For a small $$\epsilon$$ e.g. 0.01r, a high fraction of the volume is in the middle e.g. 98%. and so most of the circle is not near the surface.

Dimensions = 3

In $$D=3$$ dimensions, we again ask: what fraction of a radius-$$r$$ sphere is within $$\epsilon < r$$ of its surface. We know that the total volume is $$\frac{4}{3}\pi r^3$$ and the inner volume is $$\frac{4}{3} \pi (r - \epsilon)^3$$, so the fraction not near the surface is:

$\frac{\frac{4}{3} \pi (r - \epsilon)^3}{\frac{4}{3} \pi (r)^3} = \Big(\frac{r - \epsilon}{r}\Big)^3$

Again, most of the volume is in the middle, and so most of the sphere is not near the boundary.

Dimensions = D

What happens as the number of dimensions grow? We find that the volume in the center of the n-sphere is given by

$\Big( \frac{r-\epsilon}{r} \Big)^D$

As $$D$$ gets larger, that fraction converges to 0 since the term being exponentiated, $$(r-\epsilon) / r$$ is less than $$1$$. Consequently, the fraction of the sphere in the middle goes to zero and the sphere becomes hollow! In 2D, almost all of a circle is more than $$\epsilon$$ away from its shell, but in high dimensions, almost all of an n-sphere is within $$\epsilon$$ from its boundary.

To summarize: spheres in high dimensions become hollow!!

So much simpler than boxes, spheres and spikes!!

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