How to Find Conveniently Spaced Unit Vectors in N-Dimensions?

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SUMMARY

The discussion focuses on methods to generate a set of unit vectors in N dimensions, ensuring that the angle between neighboring vectors is approximately fixed. Participants suggest using points from an N-dimensional cube and normalizing them as a straightforward approach. Additionally, the idea of utilizing a cross-polytope is mentioned, although the parametrization remains unclear. The conversation also draws parallels to sphere-packing problems, emphasizing the challenge of covering surface area effectively while maintaining angular constraints.

PREREQUISITES
  • Understanding of N-dimensional geometry
  • Familiarity with unit vectors and normalization techniques
  • Basic knowledge of sphere-packing problems
  • Concepts of platonic solids and their properties
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  • Research methods for parametrizing cross-polytopes in N dimensions
  • Explore algorithms for sphere-packing in higher dimensions
  • Investigate the properties of platonic solids and their applications in vector generation
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Mathematicians, computer scientists, and researchers in fields requiring high-dimensional vector analysis, particularly those interested in geometric optimization and numerical methods.

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What is the most convenient way to get a set of unit vectors in N dimensions, such that the angle between two neighboring vectors is about a fixed value?

To be more precise I'd like to find a small number of vectors so that all possible other unit vectors are at most at a certain angle from a vector of this set.

It doesn't have to be precise, which probably would work for platonic solids like cubes only. Some rough method is to take all points on a N-dimensional cube and normalize the vectors. A whacky idea would be to do the same with a cross-polytope, but I don't know how to parametrize that.
But maybe one can do better?
 
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Sounds like a sphere-packing problem, where you're packing (N-2)-spheres into an (N-1)-sphere. What I mean is, in 3 dimensions, your normalized vectors live on S^2, and their angular neighborhoods are 1-spheres (circles).
 
Yes, it is. However a full sphere packing problem is highly non-trivial, but I would be satisfied with a more approximate solution. The requirement is to find a reasonable solution which covers all the "surface area".
 

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