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I wrote up a collection of answers to questions frequently asked in the math newsgroups concerning spheres: principally the question, "How can I space some points equally on the sphere?" but also some topics regarding parameterization, volumes, and so on.

So... Here's the FAQ.

A major purpose of the FAQ was to squelch repeated requests for information about placing points in a regular way around the sphere. The FAQ incorporates the high points, but some related files are discussed at the end of this page.

Just so we're clear here, this is not a research paper or a summary of state-of-the-art techniques. Rather, this article just clarifies some basic points:

• "uniformly distributed" has more than one meaning;
• for most n there is no answer which is particularly elegant;
• quick-and-dirty approximations are easy;
and so on.

Here now are some links to other areas of mathematics which cover topics somewhat related to themes in the FAQ.

Regular divisions of the sphere also arise from the action of the crystallographic point groups on 3-dimensional space. These groups are discussed in the page for three-dimensional geometry. More generally, this is a part of real geometry, which includes discussions of the volumes of (hyper)spheres.

The FAQ contains a short treatment of the most regular subdivisions of the sphere, those resulting from the Platonic solids. These are treated along with other (more irregular) polyhedra and their generalizations. On that page are files related to the decompositions of spheres into subsets (e.g. the Banach-Tarski paradox)

Packings of spheres are related to the distribution of points around spheres; these packings are considered in the page on convex geometry (52C: packing problems)

There is a page with a few short pieces concerning basic issues of spherical geometry, e.g. navigation.

Spheres may be viewed as topological spaces, metric spaces, manifolds, and so on. There are some classic questions regarding diffeomorphisms of spheres and balls treated with Manifolds. Also there are comments on the Grassmannians, which parameterize subspaces of the spheres rather than just points. Sometimes questions are treated with the tools of algebraic topology -- questions involving homotopy groups of spheres, K-groups (vector bundles), etc. A discussion of the possible vector fields on spheres (and which ones are parallelizable) is relevant to the classification of real division algebras, and so is part of the division-algebra FAQ.

Generation of random points on the sphere (with a uniform distribution) comes under the purview of probability theory and random processes (where you will find suggestions which are more carefully thought out than the one in the FAQ). Similarly the organization and analysis of statistical data taken from the surface of the sphere would be in Statistics. Spherical harmonics are listed among the special functions.

For applications to a particular sphere known as Earth you may want information on: Geodesy and Geophysics (in particular, topics concerning spherical navigation are there); Mechanics, which includes topics concerning celestial mechanics (orbits and so on); or Astronomy and astrophysics

I also mentioned map-making in the FAQ. Here are some pointers to map-making tools (esp. the Mercator projections)

With those topics moved to other pages, we have only some further files specifically related to the issue of placing points around a sphere.

(Also saved, for some reason, are the posted announcements that the FAQ was in the works and then done.)

Some other useful links:

Some other useful references:

• Conway, J. H., N. J. A. Sloane. "Sphere Packings, Lattices and Groups", Springer-Verlag.
• Saff, E.B., Kuijlaars, A.B.J., "Distributing many points on a sphere", Mathematical Intelligencer, v 19 #1 (1997) 5-11
• Croft, H.T., Falconer, K.J., Guy, R.K., "Unsolved Problems in Geometry", Springer-Verlag 1991 (see problems D7 and F17)
• Fejes Tóth, László. "The isepiphan problem for n-hedra", Amer. J. Math. 70 (1948) 174--180. (How much of the sphere can be covered with n spherical caps?)

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Last modified 2001/09/17 by Dave Rusin. Mail: