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Ok, to start with, please go easy on me - I only have secondary school level maths, and that was 24 years ago.

I'm looking to work out how to evenly distribute points over a sphere.

Specifically 20 points and 24 points.

With the 20 points, I know that a regular icosahedron has 20 faces made of equilateral triangles. But it's not a sphere.

But with the faces all having a centre point equidistant from each corner, I presume that running a line through the centre points to their opposite faces would project out onto the sphere, but would they be equally spaced?

The 24 faces are a bit more problematic.

The 2 most equal designs are the trapezoidal or deltoidal icositetrahedron where each face is kite shaped, or the pentagonal icositetrahedron where each face is pentagonal, but not a regular polygon.

In both of these cases, the centre point of each face is not equidistant from the corners.

Without 3D modelling software, how can I work this out?

Escribblings
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  • Equidistribution of points on a sphere is, in general, a hard problem. Part of the difficulty is in defining what you mean by "evenly spaced", but even when you arrive at a precise definition you're comfortable with, it's still hard. – Gerry Myerson Apr 08 '19 at 23:56
  • See, e.g., https://en.wikipedia.org/wiki/Tammes_problem – Gerry Myerson Apr 09 '19 at 00:02
  • Some other resources: https://www.youtube.com/watch?v=rwV_dAlWWnw and https://www.maths.unsw.edu.au/about/distributing-points-sphere and https://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere and https://stackoverflow.com/questions/14805583/dispersing-n-points-uniformly-on-a-sphere and https://www.cmu.edu/biolphys/deserno/pdf/sphere_equi.pdf and https://math.stackexchange.com/questions/2355241/equally-spaced-points-on-a-sphere and https://math.stackexchange.com/questions/299981/generating-evenly-distributed-points-on-a-sphere – Gerry Myerson Apr 09 '19 at 12:45
  • Also https://math.stackexchange.com/questions/279544/return-an-array-of-evenly-distributed-points-on-a-sphere-give-radius-and-origin and https://www.sciencedirect.com/science/article/pii/S0955799713002592 and https://mathoverflow.net/questions/212550/distributing-points-evenly-on-a-sphere and http://extremelearning.com.au/evenly-distributing-points-on-a-sphere/ and https://www.physicsforums.com/threads/evenly-distributed-points-on-a-sphere.680031/ and probably a few hundred others. Have a look, and then report back to us! – Gerry Myerson Apr 09 '19 at 12:50
  • Have you had a look at those references? – Gerry Myerson Apr 10 '19 at 13:00
  • Yes, feeling a bit overwhelmed with our all at the moment. – Escribblings Apr 10 '19 at 14:11

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