Fibonacci Sphere
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Fibonacci Sphere
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Re: Fibonacci Sphere
#2Author here. Happy to try to answer any questions! ;)
Re: Fibonacci Sphere
#3Author here. Happy to try to answer any questions! ;)
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...
Re: Fibonacci Sphere
#4Has ε = 0.36 been named as constant or relations with other optimal packing algorithms? It's approximately 1/4 Phi?
Re: Fibonacci Sphere
#5I 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.
Re: Fibonacci Sphere
#6Author here. Happy to try to answer any questions! ;)
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...
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.
Re: Fibonacci Sphere
#7I 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,…
Yes, He is legendary which is why i reference this page despite it being rarely updated.
Re: Fibonacci Sphere
#8Has ε = 0.36 been named as constant or relations with other optimal packing algorithms? It's approximately 1/4 Phi?
Not that I know of…
Re: Fibonacci Sphere
#9One 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!
Re: Fibonacci Sphere
#10One 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 we…
this is a really cool idea!
Do you have any links to posts/videos that further describe, analyse, etc this trick?