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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?

StayOnTarget
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  • A polygon is more than a bag of points and segments. You’ve elided the important part of that definition: “... 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.” – amd Dec 13 '17 at 23:57
  • @amd I included the portion which seemed directly relevant to the question – StayOnTarget Dec 14 '17 at 12:28
  • And in doing so left out the meat of the definition, which would answer your question. – amd Dec 14 '17 at 21:10
  • @amd maybe you are a mathematician and know the answer; I am not and do not. If you are sincerely trying to be helpful, then an answer to the question would be the way to go, rather than comments which are ambiguous to me (and any future readers). – StayOnTarget Dec 15 '17 at 13:17

1 Answers1

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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.

amd
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