# 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?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: https://stackoverflow.com/questions/9600801/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 , 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 are?

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!

• 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 to distribute points equally on a sphere? Thanks! Jul 12, 2015 at 15:48
• I should note that I originally asked the question here: stackoverflow.com/questions/31363178/… Also, I have emailed some experts, and I will let everyone know what I find out! :) Jul 12, 2015 at 17:45
• More cool information found! I came across an epic page here comparing some methods visually: 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? Jul 12, 2015 at 18:49
• (bendwavy.org c'est moi.) The golden sector spiral also has poles, though they're perhaps less obvious to the eye. Nov 3, 2015 at 9:20
• The N=2,3 are correct, but obvious ones. The reasons he forgot to mention them, is that they don't form platonic solids. They don't form solids at all. The Triangular bipyramid lacks the symmetry we seek, since the three middle points are 120 degrees apart but the pole ones are 90 degrees apart from nearest neighbors. Hence the poles are not the same as the middle ones. Jul 19, 2019 at 8:19