Spreading out points on a sphere

There is an apocryphal story that someone from the Manhattan Project asked a mathematician how to uniformly distribute 100 points on a sphere. The mathematician replied that it couldn’t be done, and the project leader thought the mathematician was being uncooperative.

If this story is true, the mathematician’s response was correct but unhelpful. He took no initiative but strictly answering the question that was posed to him. Maybe he wasn’t trying to be uncooperative, but he was showing a lack of imagination, a lack of empathy.

Applied mathematicians are continually asked to do things that are impossible, strictly speaking. But rather than state what can and cannot be done exactly and with certainty, we reply by saying what can be done approximately or with high probability.

Here’s a more helpful response:

You can only evenly distribute points on a sphere if the points are the vertices of a regular solid. So you can exactly evenly space 4, 6, 8, 12, or 20 points on a sphere. Oh, also there’s the degenerate case of 2 antipodal points. If you want to put more than 20 points on a sphere, they can’t be exactly evenly spaced—some points will necessarily be closer to their neighbors than others—but there are ways to approximately position points on a sphere.

There are several ways to quantify how evenly points are distributed on a sphere. If you can tell me more about your problem, I can determine which metric best maps to your need. I can try a few ideas and let you know how well they do according to that metric, then you can let me know whether any of the solutions are good enough for your purposes.

To read more about how you might actually distribute points on a sphere as evenly as you can, see a recent article that describes the Fibonacci lattice method, and a variation called the offset Fibonacci lattice.

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9 thoughts on “Spreading out points on a sphere

  1. Hi John,
    Thanks for the shout-out. ;)
    It’s awesome to see that I can provide high value ideas and content to highly-regarded Applied mathematicians / Data consultants such as yourself.
    Regards, Martin

  2. This seems extremely strange to me.

    For counter-example, it seems clear that you can uniformly distribute 3 points around a sphere… in an equilateral triangle.

    It also seems like there should be “energy minimization” methods that should be successful. For example, if we considered the points to be point charges constrained to the surface of the sphere and performed a gradient decent on the potential energy, surely it would head towards a minimum. Are we just saying that at that minimal potential configuration, the points are not all symmetric with regards to their local configuration? Or is it just saying that there’s no exact solution (but there can be a numeric solution)?

    Relatedly, I wonder how this “availability of even spacings” property scales with dimension… Clearly, we can easily evenly distribute any number of points on the 1-d boundary of a circle. You’re saying there are restrictions for the 2-d surface of a sphere. What happens when we consider distributing points in a 3-d space? In a 4-d space? Etc?

  3. I wonder if we can say what ‘evenly spaced’ actually means? For example imagine 100 points placed at equal distances around a great circle on the sphere. Certainly we could increase the space between neighbouring points, but it still seems pretty ‘even’ in the sense that the symmetry group is transitive on the points.

  4. Another question worth asking might be “Does it have to be exactly 100 points?”

    It certainly wouldn’t surprise me if some other number of points (say, 92) was more optimal for spacing them approximately evenly, and was close enough for whatever application was planned.

  5. I agree with others that some definition of terms may be needed. It is not clear to me what counts as a “neighboring point”. If we just want to say that the distance from any point to its closest neighbor has to be the same for all points, then there are many other distributions possible than the ones determined by the vertices of a Platonic solid, though many of them may not be what we would commonly think of as “evenly spaced.”

  6. “Applied mathematicians are continually asked to do things that are impossible, strictly speaking. But rather than state what can and cannot be done exactly and with certainty, we reply by saying what can be done approximately or with high probability.” – brilliant, although I thought it’s a feature of engineering skill.

  7. In our infancy we viewed the sphere as mathematical perfection. Pick up a highly polished ball-bearing and one’s belief in the sphere is confirmed. It’s confirmed once again when we read the definition of a sphere; “a sphere is the set of points equidistant from a point termed as the spheres center.” Or something like that. Things start to collapse when we realize that there is no way to intrinsically coordinate/metricize a sphere without having at least one singularity. In order to regain our composure and the pride of our former youth we come up with an idea to describe a sphere as a set of evenly spaced points which by means of some process can be increased ad infinitum. Nope. 20 points and that’s it. You can’t go any further. Spheres are not perfect. Neither are musical scales. Sorry to be the one to shatter your world, lol.

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