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## FREQUENTLY-ASKED QUESTIONS ABOUT SPHERES |

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;

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.

- An example: how to put dimples in a ball?
- How to place mirrors on a sphere to create a disco mirror ball!
- Variations on the theme (hoping that anyone who asks the question can find the form(s) of the answer they like best...)
- Getting up to 120 points on a sphere in a symmetrical arrangement.
- Using alternative notions of best to decide on point placement.
- How do electrons distribute themselves?
- Distributing points on spheres with point-repulsion methods: codes, pointers.
- sphere.bas -- a hohum BASIC program showing how to implement a couple of the approximation procedures mentioned in the FAQ
- Lame UBASIC code to give a quick distribution of points
- Easy method for a fairly good point distribution [Saff/Kuijlaars]
- Placing points uniformly around other shapes
- A similar question: how many disks of radius r needed to cover the unit sphere in R^n ?
- How big can k non-overlapping (equal) disks be on the N-sphere?
- Another interpretation of well-distributed one might give.
- Using a generalized spiral to distribute points on spheres.
- How many equilibrium configurations for the placement of N points on the sphere?
- How many minimum energy configurations are there for N points on a sphere?
- Citations to Sloane's book. (The pointers are a little old, but new links to Sloane are below.)
- Application of automorphic forms(!) to the question at hand.
- Names of several people who work in this area
- Description of some optimal distributions of N points on a sphere for small N.
- Approximately uniform distribution of points on surfaces using creation/destruction of points
- Using even distributions of points on a sphere for interpolation
- [Offsite] Estimates of potential energy of point distributions, as functions of the number of points and the power (=exponent) of the assumed potential-energy function. Many references.

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

Some other useful links:

- The Geometry of the Sphere [John C. Polking, Rice University]
- Neil Sloane's home page; in particular,
- Information on minimal potential energy arrangements by Neil Sloane.
- Information on packings by Neil Sloane.

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?)

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org