4
$\begingroup$

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.

  • 0
    Very interesting question!2016-11-16

1 Answers 1

7

There are uncountably many, because for example one can have rectangles with arbitrary side ratios. For your second question, if everything is constrained to be rational, there will be countably many, because a polygon is uniquely determined by its ordered collection of sides and angles.

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.

  • 0
    Then it's countable, because for each $n$ the set of all such polygons is countable, and taking the union over all $n$ gives a countable union of countable sets, which is (by a diagonal argument, countable - see http://en.wikipedia.org/wiki/Countable_set.2010-07-23