[HN Gopher] Fibonacci Sphere
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       Fibonacci Sphere
        
       Author : isaac21259
       Score  : 60 points
       Date   : 2021-08-30 07:09 UTC (15 hours ago)
        
 (HTM) web link (extremelearning.com.au)
 (TXT) w3m dump (extremelearning.com.au)
        
       | 10000truths wrote:
       | 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.
       | 
       | 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!
        
         | zarang wrote:
         | this is a really cool idea! Do you have any links to
         | posts/videos that further describe, analyse, etc this trick?
        
       | [deleted]
        
       | Datagenerator wrote:
       | Has e = 0.36 been named as constant or relations with other
       | optimal packing algorithms? It's approximately 1/4 Phi?
        
         | zarang wrote:
         | Not that I know of...
        
       | inasio wrote:
       | 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].
       | 
       | 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.
       | 
       | [0] http://neilsloane.com/packings/
        
         | zarang wrote:
         | Yes, He is legendary which is why i reference this page despite
         | it being rarely updated.
        
       | extremelearning wrote:
       | Author here. Happy to try to answer any questions! ;)
        
         | avmich wrote:
         | This link - http://neilsloane.com/packings/index.html#I - has
         | dead URLs. Like this -
         | http://www.teleport.com/~tpgettys/dodeca.gif . I specifically
         | wanted to check where the dodecahedron comes short.
         | 
         | Good article, but it'll take some time to understand it. %1 is
         | interesting, I used to use {..} for taking fractional part, %1
         | is intuitively easy, though not looking particularly good...
        
           | zarang wrote:
           | yeah. I think his website is extremely old and hasn't been
           | updated in the last decade or so. Despite this I linked to it
           | because he is a legend in this field and so i think this is
           | still the definitive reference.
           | 
           | As far as i understand, part of the story as to why
           | dodecahedron and the cube fall short is due their non-
           | triangular faces.
        
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