TITLE: Burnside's Lemma and Geometry
QUESTION [7 upvotes]: I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action.
I wonder if there is any (interesting) application in Elementary Geometry (I mean Euclidean, hyperbolic or elliptic geometry).
Searching on Google, I've found the article "Applying Burnside’s lemma to a one-dimensional
Escher problem" by T. Pisanski, but it sounds to me rather a combinatorial result.

REPLY [7 votes]: Burnside Lemma can be used as a first step to classify all finite subgroups of $\mathrm{SO}(3)$: it gives you that there are at most $3$ orbits in the action of any finite group $G$ on the set of intersections between axes of elements of $G$ and the unit sphere.