# Is the set of all unique (convex) polygons countable? If so, by what bijection to the natural numbers?

Polygons are, in this question, defined as non-unique if they similar to another (by rotation, reflection, translation, or scaling).

Would this answer be any different if similar but non-identical polygons were allowed? And if only if rotated/translated by rational coefficients?

Would this answer be any different if we constrained the length and internal angles of all polygons to rational numbers?

Assume the number of sides is finite but unbounded, and greater than two.

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This is part of a general fact: if $A$ is a countable set, then the collection of ordered $n$-tuples of elements of $A$ for all $n$ is still countable.
Justin: No, because the set of all infinite sequences over $Q$ is of the same cardinality as $R$. In fact, the set of all sequences over $\{0,1\}$ is of the same cardinality as $R$---this is by the binary encoding. If you are allowing polygons with infinitely many sides (finite but unbounded is OK), then I'll edit my answer - could you please clarify? – Akhil Mathew Jul 23 '10 at 2:40