Proof or law of the number of sides of a polygon

Question

The definition of "polygon" seems to include the fact that any polygon will have the same number of sides as points.

For instance:

A polygon can be defined ... as a geometric object "consisting of a number of points (called vertices) and an equal number of line segments (called sides)... (Coxeter and Greitzer 1967, p. 51)."

(from Wolfram Mathworld)

Is the fact that # of sides = # of points just an assertion or is there an actual proof of this? Or is it just taken to be axiomatic in some sense?

Answer 1

A polygon isn’t just a bag full of points and segments. There’s structure to it that’s described in the next part of the definition, which you didn’t quote:

...namely a cyclically ordered set of points in a plane, with no three successive points collinear, together with the line segments joining consecutive pairs of the points.

The property you’re asking about is really just a part of this definition, or a basic consequence of it, depending on one’s point of view.

“Cyclically ordered...” Write down a list of the points in order, adding another copy of the first point at the end of the list: $$P_1\;P_2\;P_3\;\dots P_n\;P_1$$

“together with the line segments joining consecutive pairs of the points.” Draw a line segment in each of the spaces between consecutive points in the list: $$P_1—P_2—P_3—\dots—P_n—P_1$$

How many line segments did you draw? In other words, how many spaces are there in the augmented list of points?

The non-colinearity part of the definition is there to ensure that adjacent line segments aren’t part of a longer line segment, so that you haven’t subdivided what should be a single edge with an extra point.