Fibonacci Sphere

One neat trick I’ve learned is that you can use the points on a Fibonacci sphere to optimally compress unit vectors, for things like normal textures. For example, if you have an array of 1024 points representing a Fibonacci sphere, you can compress unit vectors into lg(1024)=10 bits with a nearest neighbor search and decompress with an O(1) table lookup.<p>In fact, the general strategy works for higher dimensions as well. Spread some points on the hypersurface of a unit 3-sphere with some kind of energy minimalization simulation, and the resulting array of 4D unit vectors can can be used to compress quaternions!

I ran into this problem working on differential equations that model pattern formation (reaction-diffusion equations, originally postulated by Turing in the 1950s). The equations are highly nonlinear, but some solutions can be found when solving the problem on a sphere. You get spot solutions that dynamically move essentially to the minimum energy configuration (Fekete points I believe are called). BTW, Neil Sloane, of OEIS fame, has a list of the best packings, up to n=100 I believe [0].<p>Things get interesting when you also allow the sphere to grow, the spots start to split (and sometimes annihilate), understanding how the spots move on the sphere is itself a very interesting problem.<p>[0] <a href="http:&#x2F;&#x2F;neilsloane.com&#x2F;packings&#x2F;" rel="nofollow">http:&#x2F;&#x2F;neilsloane.com&#x2F;packings&#x2F;</a>

Quite cool how the author picks a topic, explains it well, and on top of that presents some novel results. He did this before with his article on minimum-discrepancy sequences, which is one of my favourite results in math.

Author here. Happy to try to answer any questions! ;)

In past weeks I&#x27;ve done some work on procedural meshes and filling out the space with various layout strategies. The posted article was a good introduction, and then I found this very approachable paper with great overview of different algorithms: &quot;Point Picking and Distributing on the Disc and Sphere&quot;.<p><a href="https:&#x2F;&#x2F;apps.dtic.mil&#x2F;dtic&#x2F;tr&#x2F;fulltext&#x2F;u2&#x2F;a626479.pdf" rel="nofollow">https:&#x2F;&#x2F;apps.dtic.mil&#x2F;dtic&#x2F;tr&#x2F;fulltext&#x2F;u2&#x2F;a626479.pdf</a>

Has ε = 0.36 been named as constant or relations with other optimal packing algorithms? It&#x27;s approximately 1&#x2F;4 Phi?