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Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: Evenly distributing n points on a sphere.

But what I would like to know is: "Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best. Does anyone know of a better method?"

I have a Ph.D. in physics and may have an application for some of this research in physics.

I came across this wonderful paper:

http://arxiv.org/pdf/0912.4540.pdf "Measurement of areas on a sphere using Fibonacci and latitude–longitude lattices"

The paper states, "The Fibonacci lattice is a particularly appealing alternative [15, 16, 17, 23, 65, 42, 66, 67, 68, 76, 52, 28, 56, 55]. Being easy to construct, it can have any odd number of points [68], and these are evenly distributed (Fig. 1) with each point representing almost the same area. For the numerical integration of continuous functions on a sphere, it has distinct advantages over other lattices [28, 56]."

It the Fibonacci lattice the very best way to distribute N points on a sphere so that they are evenly distributed? Is there any way that is better?

As seen above, the paper states, "with each point representing almost the same area. "

Is it impossible, in principle (except for special rare cases of N such as 4, etc.), to exactly evenly distribute N points on a sphere so that each point/region has the exact same area?

So far it seems to me that the Fibonacci lattice the very best way to distribute N points on a sphere so that they are evenly distributed. Do you feel this to be correct?

Thanks so much!

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  • Regarding whether it's important or not to evenly distributed them with almost the same area per region, I could see one use of this in a computer graphics algorithm. Maybe something to do some stochastic sampling from a point into its surroundings, possibly to make a cube map? Each point on the sphere would represent a vector from the center of the sphere, shooting out into the surroundings to sample what is there - like maybe for incoming light values or something. – Alan Wolfe Jul 12 '15 at 03:38
  • As far as if there is anything better, I wonder how this compares to poisson disc sampling on the surface of the sphere? Maybe more computationally efficient? Maybe not... Dunno – Alan Wolfe Jul 12 '15 at 03:40
  • Sorry one more comment. There's a mathematics stack exchange. The people there can probably help you a lot more than here imo! – Alan Wolfe Jul 12 '15 at 04:34
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    Technically speaking, it is impossible to equidistribute points on a sphere unless they are the corners of a Platonic solid, so N=4,6,8,12,20. So, for any N's other than this, what is the best way to equidistribute N points on a sphere? So far it seems that the Fibonacci lattice the very best way to distribute N points on a sphere so that they are evenly distributed. Does anyone know of a better method? Thanks! – Physics Ph.D. Jul 12 '15 at 15:45
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    Thanks Alan Wolfe! I followed your advice and asked the question at the Math stack exchange! http://math.stackexchange.com/questions/1358046/is-the-fibonacci-lattice-the-very-best-way-to-evenly-distribute-n-points-on-a-sp I have also emailed some experts, and I will let everyone know what I find out! :) – Physics Ph.D. Jul 12 '15 at 17:43
  • More cool information found! I came across an epic page here comparing some methods visually: http://bendwavy.org/pack/pack.htm The middle column, representing "a golden section of the circle," seems to be the most symmetric under rotation? For instance Rusin's and Saff & Kuijlaars methods seem to have poles, so one would be able to note the rotation of the spheres. Having noted that, would it be logical to say that the center method utilizing the golden section (the Fibonacci lattice) provides the best way to symmetrically distribute N points on a sphere in an equidistant manner? – Physics Ph.D. Jul 12 '15 at 18:46
  • When you say "best" do you mean in terms of quality out? Or in terms of computational costs? Or maybe best quality for lowest computation? (sorry, i'm a programmer / engineer by trade hehe) – Alan Wolfe Jul 12 '15 at 19:09
  • Thanks Alan! I would mean in terms of "quality out," with quality meaning "as near perfect symmetry" as possible, as it seems that perfect symmetry is elusive but for N=4,6,8,12,20. "Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere in the most symmetric manner possible? So far it seems that it is the best. Does anyone know of a better method?" Hope that helps! – Physics Ph.D. Jul 12 '15 at 19:18
  • It seems to me that you can improve upon the Fibonacci lattice by doing a second pass with an electron repulsion simulation. – fluffybunny Jul 12 '15 at 22:20
  • Great. I also want to know whether Fibnacci lattice has the lowest discrepancy. – cmjdxy Jul 01 '20 at 01:46

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