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Evenly distributing points on a sphere (extremelearning.com.au)
88 points by Signez on Aug 25, 2018 | hide | past | favorite | 20 comments

Author here. So amazed to find that my blog post got featured on Hacker news. Happy to answer any questions that I can! :)

I've seen the phylotaxis thing before but was unaware of the rather lovely Fibonacci Lattice. My maths is only Civ Eng grad level from >25 years ago but even I can appreciate this stuff. Your writing style is very approachable and well illustrated.

Thank you.

Thanks for the kind words. I’m glad you found it interesting and useful.

I think one of the advantages of writing blog posts rather than academic articles is that they are often more readable to a wider audience as the authors can be a little less formal in tone, expand on things (including copious illustrations), without worrying about space constraints.

For me, your style of writing provides a very decent balance. I'm not a scientist, nor engineer, mathematician or similar but I am a common (or garden) variety of nerd!

Quite often I will plough through papers and some of the more challenging blog posts that are linked here. A post like yours is challenging but only for the right reasons. You avoid a too "chatty" and "pally" style and present facts concisely but with a bit of context - enough to point amateurs in the right direction.

Your post shows values of d=2, 3.09, 3.29 and 3.31 with different latices. There must be some theoretical maximum d possible, right? Some upper bound that "d cannot be greater than X"? What do we know about the bounds of that maximum value?

The values for the tetrahedron, cube, octahedron, dodecahedron and icosahedron are:3.27, 3.27, 3.46,3.19 and 3.64, resp.

These produce the largest d for N=4,8,6,12 and 20 resp. Thus, it is presumed by almost everyone that d=3.64 is the global upper bound.

Unfortunately, I believe it is still an open problem to to prove this or to describe a general upper bound for specific N not equal to any of these five values.

Hey I'm a math undergrad and this is a question I've been curious about before. Can you link me to some literature on this? Would this fall under discrete and computational geometry? I know there are some well known books in that area

Hi I included about a dozen references at the end of my post for overall material on this topic. I specifically chose these ones as they are very readable. My favourite paper is defintiely the first one, "A comparison of popular point configurations on S2". It is very comprehensive in breadth as well as historical developments and includes copious references.

One of my other references is "Distributing many points on a sphere" as it is written by E.B. Saff who is basically a legend in this field. Hope that helps! Martin

How does the last method (modified Fibonacci sequence) so on the d* measure ?

Interesting question... I just calculated that for the modified Fibonacci sequence (last method): 2.95 < d_N < 3.12.

Technically d* does not exist, because as N-> infinity, d* alternates between 3.03 and 3.07 (depending on if k is odd or even).

Compare this to the canonical fibonacci sequence gives a value of d_N = 3.07 for all values of N, and so d* = 3.07

my question is can you get similar results by doing a physical simulation of charges repelling each other

One of the reasons why this field is still so active is because different measures and methods often result in slightly different optimal configurations. In the case of this post, I focused on direct construction methods; and measures relating to minimum distance or convex hulls.

For an excellent commentary on the latest for optimal Riesz energy (which includes Coulomb potentials) configurations can be found in the first paper that I reference: "A comparison of popular point configurations on S2", by Hardin, Michaels and Saff.

my other question is has anyone ever tried to find this with rational points only

If you are interested in spherical math, you should check out google's S2 library [0] which uses hilbert curves to classify areas on a sphere. Here is an overview of the cell hierarchy [1]

[0]: http://s2geometry.io/

[1]: http://s2geometry.io/devguide/s2cell_hierarchy

For anyone who was scratching thier head about how a plane filling curve gets mapped into a sphere; what S2 really does is project the sphere onto the six sides of a cube _and then_ fill each face with a Hilbert Curve index.

Lol and Holy shit! That totally explains why the s2 cell lengths vary based on where they are located in one of the six faces (aka, sides of a cube) of which the documentation does not explain clearly. I have spent many hours using this library and your comment has explained so much of what I was confused about it.

Thank you

I'm glad I could help! I just picked up that fun fact from recently.

You're going to get a kick out of: https://news.ycombinator.com/item?id=17765388

A somewhat related article recently on the HN front page - "Generating random points inside a sphere": https://karthikkaranth.me/blog/generating-random-points-in-a... ; https://news.ycombinator.com/item?id=17688599

Couldn’t this be used to model qubit states? Ie calculating function values on the resulting pairs.

In 2D, this is the porno theater problem.

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