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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.

flag	asked Sep 20 2010 at 18:32 Michele Triestino 678●6●15

	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 2010 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 2012 at 19:01
	@Marty Thank you for your comment, that's why I wrote "so-called" – Michele Triestino Feb 2 2012 at 17:39

Benoît Kloeckner · Accepted Answer · 2010-09-20 19:07:28Z UTC

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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.

answered Sep 20 2010 at 19:07

Benoît Kloeckner
5,489●18●29

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Which, of course, can then be used to classify the Platonic solids. – Qiaochu Yuan Sep 20 2010 at 19:38

Burnside’s Lemma and Geometry

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Burnside’s Lemma and Geometry

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