Joy Christian wrote:FrediFizzx wrote:I take it back. That is the coordinates of the points of a unit 3-sphere. But which ones are the x, y and z coordinates? The first set of three or the last set of three?

.

Any three of the four coordinates can be taken as x, y, and z. Normalized vectors in R^4 can certainly give a 3-sphere, but that would not be a quaternionic (or parallelizable) 3-sphere. Therefore the correlation among its points will not be as strong as -a.b. It can only give a linear correlation. What is needed are uniformly distributed points on a quaternionic 3-sphere.

Maybe I can help? What exactly do you mean by "quaternionic 3-sphere"? Then we can discuss what we should mean by "uniform distribution".

This should mean that you define a continuous group of transformations of the sphere. The 3-sphere is compact, so there exists a unique finite Haar measure which can (therefore) be normalised to a unique invariant probability measure. Given any parametrization of the sphere we can now calculate how to pick a random uniform point on it (we just figure out the joint probability distribution of the parameters).

So the starting point is to define the space of quaternions, then we need a metric on the space. Now we can define the sphere of radius 1. We had a metric, so we have a topology on the space of quaternions, and this gives us a topology on the sphere. Now we need a set of transformations of the sphere. The rest is calculus. Or differential geometry, if you prefer to use that language. We need to verify that the set of transformations has the required properties. We need a parametrization of the sphere. Then we are ready to calculate.

In fact, it is well known that the quaternions form a four-dimensional real vector space, and there is a very sensible way to define a (real) norm on the space making it a normed vector space of dimension 4. This Euclidean norm is our metric. The set of transformations could be just the right multiplications by a quaternion. The uniform distribution is normalized right Haar measure. It turns out that the quaternionic 3-sphere is just the ordinary sphere of radius 1 in 4-dimensional (real) Euclidean space, our old friend

, and the transformations are the ordinary "rotations". I think we already know how to pick uniformly distributed points on

.

See for instance the question and answers at:

https://stackoverflow.com/questions/15880367/python-uniform-distribution-of-points-on-4-dimensional-sphereThe questioner is muddled about terminology. The repliers sort him out and give some solutions.

See also

https://en.wikipedia.org/wiki/N-sphere#Uniformly_at_random_on_the_(n_%E2%88%92_1)-spherehttps://en.wikipedia.org/wiki/3-sphere