
Random Points on a Sphere - johndcook
https://johncarlosbaez.wordpress.com/2018/07/10/random-points-on-a-sphere-part-1/
======
unholiness
High dimensional spheres are very counterintuitive! The volume of a
n-dimensional unit sphere goes to zero as n increases, almost all the volume
is contained near the surface.

Some interesting musings in this realm:
[https://marckhoury.github.io/counterintuitive-properties-
of-...](https://marckhoury.github.io/counterintuitive-properties-of-high-
dimensional-space/)

~~~
TorKlingberg
Related: In 4D and higher space, the concept of rotating around an axis does
not make sense.

In 3D there are three ways to turn: yaw, pitch and roll. In 4D, there are not
4 but 6 ways.

~~~
JadeNB
> Related: In 4D and higher space, the concept of rotating around an axis does
> not make sense.

I think that this is like saying "rotating in 3D doesn't make sense"; it's not
so much that it doesn't make sense as that it's not uniquely specified.

(Then there's the fact, which maybe is what you meant (since you referenced
the dimension of SO(4)), that a 4D Euclidean rotation need not fix an axis at
all!)

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unwind
Awesome.

Also it seems Greg Egan is the SF author[1] which for me makes it extra cool.

[1]
[https://en.wikipedia.org/wiki/Greg_Egan](https://en.wikipedia.org/wiki/Greg_Egan)

~~~
pmiri
It was a real treat seeing his name at the start of this.

~~~
starshadowx2
Definitely. I had no idea he was involved going in, I thought it would just be
some neat math I didn't understand (which it still was). Greg Egan has been my
favourite author for a few years now, ever since reading Diaspora.

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max_likelihood
Wow, mind blown. Unfortunately, even with an undergraduate degree in
mathematics, I was lost after this paragraph:

`This made me eager to find a proof that all the even moments of the
probability distribution of distances between points on the unit sphere in
\mathbb{R}^d are integers when \mathbb{R}^d is an associatve normed division
algebra.`

Nonetheless, very interesting!

~~~
crasshopper
read his post on octonions
[http://math.ucr.edu/home/baez/octonions/](http://math.ucr.edu/home/baez/octonions/)

The 1, 2, 4, 8 phenomenon surprised 20th-century mathematicians, and derives
from weird facts about how S7, S3, and S1 fiber. (fibration is lining one
shape with other shapes)

~~~
JadeNB
> read his post on octonions
> [http://math.ucr.edu/home/baez/octonions/](http://math.ucr.edu/home/baez/octonions/)

Not for a discussion on _associative_ normed algebras ….

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crasshopper
Persi Diaconis [https://www.ams.org/notices/200511/what-
is.pdf](https://www.ams.org/notices/200511/what-is.pdf)

Niles Johnson on Hopf/Milnor fibrations
[https://nilesjohnson.net/hopf.html](https://nilesjohnson.net/hopf.html)

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pathsjs
The obvious question is actually the converse: assume all even moments of the
distance are integers - does it follow that S^(n-1) is a group (hence n=1, 2
or 4)?

~~~
qmalzp
N.b. the article uses d for the dimension, and n for the moment index. I
haven't cranked out the details, but I believe that this is already true just
assuming the _fourth_ moments of the distance are integers. That is, for n !=
1, 2, or 4, the fourth moment is never an integer. Idea of the "brute force"
proof:

Take the formula in the article for the 4th moment of the d-dimensional
sphere, which is always a rational number. Basically, for n=2^k, the
denominator should be divisible by a larger power of two than the numerator
(specifically, if I crunched the numbers right, the gap should be k-2). When n
is not a power of two, then for any odd prime p dividing n, I _believe_ the
denominator should be divisible by a larger power of p than the numerator.
This requires calculating exactly how many powers of p divide various
factorial expressions, but you get the idea.

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ttoinou
Wait, there are a lot of ways in computer science to choose points randomly,
and points randomly with constraints (here belong to unit sphere) and that
could change the result, no ?

~~~
imh
>We’ll be ‘randomly choosing’ lots of points on spheres of various dimensions.
Whenever we do this, I mean that they’re chosen independently, and uniformly
with respect to the unique rotation-invariant Borel measure that’s a
probability measure on the sphere. In other words: nothing sneaky, just the
most obvious symmetrical thing!

There are an infinite number of probability distributions over most objects,
yes, but there's also often a good default that is the "uniform" distribution.
That's what they're talking about here.

~~~
OscarCunningham
As an aside, there's an interesting way to generate a uniform distribution on
a sphere. It uses the fact that the joint distribution of several independent
standard Gaussians has rotational symmetry. So to generate a random point on
the surface of the n-sphere, you can sample n+1 numbers independently from the
standard Gaussian and then normalise them so that their squares sum to 1 (if
they're all 0 then you have to reroll).

