[remcob]

The distance between two uniform random points on an n-sphere clusters around the equator. The article shows a histogram of the distribution in fig. 11. While it looks Gaussian, it is more closely related to the Beta distribution. I derived it in my notes, as (surprisingly) I could not find it easily in literature:

https://xn--2-umb.com/21/n-sphere

[zombot]

> The distance between two uniform random points on an n-sphere clusters around the equator.

This sentence makes no sense to me.

[p1esk]

He means it clusters around the distance from a pole to the equator.

[isoprophlex]

Pick an equator on an n-sphere. It is a hyperplane of dimensions (n-1) through the center, composed of all but one dimensions of your sphere. The xy plane for a unit sphere in xyz, for example.

Uniformly distribute points on the sphere. For high n, all points will be very near the equator you chose.

Obviously, in ofder for a point to be not close to this chosen equator, it projects close to 0 on all dimensions spanning the equatorial hyperplane, and not close to 0 on the dimension making up the pole-to-pole axis.

[akdor1154]

"clusters" is acting as a verb here, not a noun.

[7fYZ7mJh3RNKNaG]

beautiful visualizations, how did you make them?

[remcob]

The first one IIRC with Geogebra, all the rest with Matplotlib. The design goal was to maximize on 'data-ink ratio'.