I think one of the most interesting results in Elementary Group Theory is the so-called "[Burnside's Lemma](http://en.wikipedia.org/wiki/Burnside%27s_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](http://users.wpi.edu/~bservat/strippat.pdf)" by T. Pisanski, but it sounds to me rather a combinatorial result.



Burnside's Lemma and Geometry
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.


