The lack of volume in a high-dimensional cube

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at:, or follow me on Twitter.

I wish this felt more intuitive to me:

This is a starting point for intuition:

To provide some intuition consider the situation in two dimensions, as shown in Figure 10. For a point on the circle to be close to the equator, its
y-coordinate must be small.

So, this is sort of an expression of the Curse Of Dimensionality?

But perhaps the most amazing thing is that we can not figure out how many spheres might surround a central sphere, in such a way that they just barely touch (Kissing Spheres). In 5 dimensions, no one has been able to figure out the answer. I’m surprised by this. There is no way to calculate this? There are only 5 possible answers:






Given the small range of possible answers, I’d have assumed that someone could write a script to simply brute-force calculate the answer. It’s possible no one has done this because mathematicians don’t regard brute-force answers as proofs, and what they look for is proofs.

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