# Uniqueness of a polygon

Suppose I have two $n$-sided polygons A and B. Is there a non-trivial upper bound on the number of parameters (eg. area, perimeter, etc) of the two polygons, that need to be the same, for A and B to be identical? (For example, if we consider a simpler case to the problem when say A is constrained to lie inside B, then if area(A) = area(B), then A and B are identical. Hence area is the only non-trivial parameter in this case)

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I guess you are looking for a nice answer, but here is a stupid one. On the other hand I'm sure that there is no "nice answer".

A polygon can is uniquely determined by length of sides $\ell_i$ and angles $\alpha_i$. Thus we have to find a complete set of invariants for sequence $(\alpha_1,\ell_1,\alpha_2,\ell_2,\dots\alpha_n,\ell_n)$ which survive after even cyclic shifts and reversing order.

Then you prepare symmetric polynomials for your group. Say take all monomials of degree at most one in each $\alpha_i$ and $\ell_i$ and take its mean value it along the group. You obtain a big collection of polynomial expressions in $\alpha_i$ and $\ell_i$ which gives complete invariant (perimeter will be one of them).

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I liked the first version of your answer better --- somehow symmetric polynomials of complex coordinates of vertices seem more natural to me. –  t3suji Jan 21 '10 at 5:21
@t3suji. Well, it was not quite correct... –  Anton Petrunin Jan 21 '10 at 15:42

The moduli space of $n$-gons up to orientation preserving similarity can be identified with $\mathbb{C}P^{n-2}$, so I would say $2n-3$.

See On the moduli space of polygons in the Euclidean plane by Kapovich and Millson for more about this space.

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In what sense do two parameters suffice to pick out a triangle? Naively one needs three (say, the lengths of the sides). –  Michael Lugo Jan 21 '10 at 3:37
Oh, sorry, it's the moduli space up to orientation preserving similarity. So two angles suffice for a triangle. –  Richard Kent Jan 21 '10 at 4:03
Edited to fix this. –  Richard Kent Jan 21 '10 at 4:08

There are many structures that mathematicians study because of their intrinsic interest, simplicity and being quick starting. Examples that come to mind are plane polygons and graphs.

Plane polygons have a variety of properties that one can look at: area, perimeter, minimum number of vertex guards, number of reflex angles, number of right angles, etc. An individual graph can have a variety of "invariants" that can be studied: coloring number, clique number, being eulerian, or hamiltonian, etc. For two graphs there is no list of invariants that guarantees that the two graphs are isomorphic. The complexity of checking when two graphs are isomorphic is still a dynamic area to investigate. I do not know any list of properties that guarantee that two polygons are congruent going beyond specifying the lengths of the sides and the measure of the angles. Finding new combinatorial/geometric properties of polygons seems to continue to be very worthwhile.

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