It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.3K
- Chat 499
- Study Groups 18
- Petri Nets 9
- Epidemiology 3
- Leaf Modeling 1
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- MIT 2019: Applied Category Theory 339
- MIT 2019: Lectures 79
- MIT 2019: Exercises 149
- MIT 2019: Chat 50
- UCR ACT Seminar 4
- General 67
- Azimuth Code Project 110
- Statistical methods 3
- Drafts 2
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 147
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 708

Options

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:

## Comments

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

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

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!

`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!`

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

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

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

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