I have been trying to figure this problem out for a while, and while I believe someone must have figured it out hundreds of years ago, I still can't quite get it.
Suppose we have a 3-dimensional sphere and three distinct planes that pass through the center of the sphere. Each pair of two planes will cut a lune from the surface of the sphere. Consider the intersection of the three lunes, which is a triangle on the surface of the sphere. Now, suppose you can reflect the triangle across any of the three planes, take the union of both the triangle and its reflection and repeat the process with this new surface (reflect the 4-sided polygon on the sphere across any of the three planes, union the two surfaces together, etc.) The question is, is it possible to cover the entire sphere using this method? If not, is it possible to at least reach the diametrically opposed triangle (the reflection of the original triangle across the center of the sphere)? Is it possible to do so in a finite number of steps?
The motivation for this question can be thought of in terms of optics: if the three planes were mirrors, and we shine a laser pointer infinitely close to their intersection, would the light bounce back after a finite number of reflections? My intuition says yes, but alas I can't prove it.