FrediFizzx wrote:minkwe wrote:Hi Fred,
The Cython (Cython is Python compiler with an extension to the Python language) code you posted is a declaration. It doesn't actually show how the Function Object is instanciated and used. In code declares a template for generating functions based on the provided base_point array. So that slightly different functions can be generated by simply changing base_point when generating an instance of fib_param2.
In your translation to Mathematica, you are selecting the elements of base_point as three random real numbers between -1 and 1. Why?
Now I haven't been paying close attention to everything on this thread but note that the selected randomization scheme appears to be critical in the distribution of points you obtain. See
https://en.wikipedia.org/wiki/Bertrand_ ... robability)#Jaynes's_solution_using_the_%22maximum_ignorance%22_principle.
Yes, here is a link to a text file of the rest of the Sage code that uses that function. It is quite long.
EPRsims/hopf_animate.sageNow that I understand what base_point is, I can maybe understand this code better. I just guessed at using random real numbers between -1 and 1 for a, b and c. It did give me some output.
And I have ph random between 0 and 2pi. If I set it to pi, I get half a ball. To pi/2, a 1/4 ball. Pretty wild and crazy!
What I am trying to do is to see if there is another way of getting parallelized 3-sphere points to feed the other simulations.
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Fred, another way of getting parallelized 3-sphere points to feed the other simulations: generate random unparallelized 3-sphere points in whatever way you like, and compute their coordinates wrt to a parallelization! (Note - there are many parallelizations, but all are isomorphic one another. Obtained from one another by rotations of the 3-sphere.
Regarding another question (about the picture of the cloud of points in the unit ball):
If you generate uniform random points (x, y, z, t) on the 3-sphere and make a 3d plot of the first 3 coordinates, the result will be hardly distinguishable, by eye, from uniform random points (x, y, z) in the 3-ball.
See a difference? Which is which?
However, if you just make a histogram of the x coordinate, you'll see a big difference. In the case of the 3-sphere, the *second* of the two 3-d plots, you'll see the semicircle law. In the case of the 3-ball you'll see something markedly different. It isn't difficult to figure out the probability density of x in that case (one needs a formula for the volume of the intersection of the ball and a half-space, and then differentiate with respect the distance of the hyperplane defining the half-space to the origin. The result is proportional, of course, to the area of the disk which is the intersection of ball and hyperplane!) The answer is 3/4 (1 - u^2), a parabola.
Notice that the 3d points from the 3-sphere come up closer to the boundary of the ball. At the boundary, the density is zero, but the slope of the density there is infinite. 3d points from the 3-ball kind of stay away from the boundary: at the boundary, the density is zero too, but its slope at the boundary is finite.
Conversely, the density of points near the centre of the ball is smaller for the 3-sphere than for the 3-ball case.
Immediately, the mean values of the squares of the x co-ordinate in the two cases are 1/4 (3-sphere) and 1/3 (3-ball). This suggests that the 3-sphere points are closer to the origin than the 3-ball points. But that is a wrong suggestion. Curious.