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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.

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There may be something in the book by Paul B Yale, Geometry and Symmetry, published in 1968 (2nd edition, Dover, 1988). Well, it has the word "Geometry" in the title, and Burnside's Lemma is discussed in the 1st chapter. – Gerry Myerson Sep 21 '10 at 6:30
"Burnside's Lemma" is not due to Burnside according to a 1979 paper of P. Neumann called (appropriately) "A lemma that is not Burnside's". Neumann attributes the result to Cauchy and Frobenius. – Marty Isaacs Jan 31 '12 at 19:01
@Marty Thank you for your comment, that's why I wrote "so-called" – Michele Triestino Feb 2 '12 at 17:39
up vote 7 down vote accepted

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.

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Which, of course, can then be used to classify the Platonic solids. – Qiaochu Yuan Sep 20 '10 at 19:38

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