[HN Gopher] Fibonacci Sphere ___________________________________________________________________ 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. ___________________________________________________________________ (page generated 2021-08-30 23:00 UTC)