[Math] The Disco Ball Problem

mg.metric-geometrysphere packing

Let me first give some of a background as to where I got this problem. I had a math teacher ask me a few months ago: "How many 1 unit by 1 unit squares could one fit on a sphere with a radius of 32 units?" The only problem was that it was a high school math course, and that I, as a high school student, had no way of answering the question in its true sense, as the teacher simply wanted a division of surface area.

But the problem remains unanswered, and it has been bothering me for some time now. How many mirrors can you fit on a disco ball? This isn't a question for the construction of a disco ball simply the construction of a theoretical "perfect" disco ball given the parts. My teacher didn't realize that the problem answer she was expecting was for an entirely different problem relating to how many areas of 1 unit squared you could create from the surface. Given my severely limited math knowledge, I am simply unable to answer a problem as complex as this.

Here is the problem in its entirety:

If you have a sphere of radius $r$ and squares of length $l$, then how many squares can you fit on it in tangent with the sphere with only the centers of each square touching the sphere, and with no square overlap?

This is a disco ball so the parameters get a bit tight:

  1. The squares can only be attached by their centers

  2. The squares can't overlap

  3. The squares can't go into the sphere

  4. The squares can't be broken or bent (don't want bad luck)

  5. The squares go on the outside of the sphere

I don't know what math this would take to solve this, but I know that it has a solution lurking in the voids of some imagination if so gifted to me.

What this question might mean to a mathematician:

In this given situation the square mirrors could be interchanged with spheres without any adverse effects to the problem. So where this problem might find a solution could be in the already well-defined field of sphere packing, the only caveat being the fact that it's packing with set radii ratios and where non-touching spheres are discarded.

Best Answer

This is not a direct answer, but it's close. The following paper partitions a sphere into equal-area, near-squares:

"The system has several degrees of freedom that can be fixed, for instance, by enforcing the cells aspect ratios. Therefore, the cells can have very comparable forms, i.e. close to the square."

Beckers, Benoit, and Pierre Beckers. "A general rule for disk and hemisphere partition into equal-area cells." Computational Geometry 45, no. 7 (2012): 275-283. (Elsevier link.)

          Fig11
          Figure 11. Dome, equal areas.

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