Random Points on Spheres and Groups

John Baez

I can't work all the time... I need to goof off a bit to stay happy. I'm afraid this, together with working hard on some papers, has slowed down the production of lectures here.

This week I've been doing some recreational mathematics with some Greg Egan, Dan Piponi, John D. Cook and other people on Twitter. We've shown things like this:

• the average distance between \(xy\) and \(yx\) for unit quaternions \(x,y\) is

$$ \displaystyle{\frac{32}{9 \pi}} \approx 1.13176848421 \dots $$

• the average distance between \((xy)z\) and \(x(yz)\) for unit octonions \(x,y\) is

$$ \displaystyle{ \frac{2^{14}3^2}{5^3 7^3 \pi} \approx 1.0947335878 \dots } $$

• You can take the unit sphere in \(\mathbb{R}^n\), randomly choose two points on it, and compute their distance. This gives a random variable, whose moments you can calculate. When \(n = 1, 2\) or \(4\), and seemingly in no other cases, all the even moments are integers. However, for every even \(n\), if we compute the first \(k\) even moments, figure out the fraction of them that are integers, and take the limit as \(n \to \infty\), this fraction approaches 1.

You can see the whole story here:

• Random Points on a Sphere (Part 1).

• Random Points on a Sphere (Part 2).

• Random Points on a Group.

Keith E. Peterson

What happens if you put the \(n\)-sphere under a deformation?

John Baez

I'm not sure what you mean, but it probably destroys all the delicate beautiful stuff we're talking about in these posts! It's like asking what happens to trig functions if you take the unit circle and squash it. There's probably some answer, but it ain't as pretty as trig functions!

Christopher Upshaw

Though that raises an interesting question, can we construct a canonical probability distribution on the "unit hyperbola" by using sinh and cosh?

Pierre Prado

Is it possible to devise some important link of this results , on the n-sphere, to antipodals and the Borsuk Ulam theorem? best