On Polygons, Polygons within Polygons, and Definitions

Date: 07/22/2010 at 20:24:24
From: Mark
Subject: Is this a polygon

Suppose we take a rectangle and cut out a smaller quadrilateral within
it, as shown below:

    ____________________________
   |                            |
   |   _________________        |
   |  |_________________|       |
   |                            |
   |____________________________|

Would the resulting figure be a polygon? If it is, would it be a concave
octagon?

According to what I'm reading in this geometry textbook, a polygon is a
plane figure that is formed by three or more segments called sides such 
that

1. Each side intersects exactly two other sides, once at each endpoint.
2. No two sides with a common endpoint are collinear.

This wasn't a question I found in the textbook or anything. It was just a thought that 
popped into my head. I find it confusing because it is a very unusual example.

My assumption is that yes, it is a polygon since the figure fits the
description. I know that if it is a polygon then it must be concave, since
a line containing a segment of the inner quadrilateral would go through
the interior of the figure.

Date: 07/22/2010 at 22:29:56
From: Doctor Peterson
Subject: Re: Is this a polygon

Hi, Mark.

The kind of thinking you are doing is very much the way a mathematician
thinks!

Textbooks, alas, often give inadequate definitions of terms like "polygon"
because they try to keep things simple. A proper definition is something
like this:

  A polygon is a simple closed plane curve consisting of finitely many
  straight line segments.

"Simple," along with "closed," imply your textbook's first condition; and
while your second condition clarifies what we mean by a vertex, it is not
strictly necessary. But your book's definition entirely leaves out the
fact that a polygon must be a single connected curve -- which gets at the
ambiguity of your shape: it's really two shapes!

Note also that your idea of "cutting out a rectangle" suggests that you
are thinking not of the polygon itself (the segments), but of its
INTERIOR. The interior of your figure is connected (though not what we
call "simply connected"); but the boundary is two separate polygons, not
one.

As I mentioned earlier, thinking up "unusual examples" like the one that
"popped into your head" is what a good mathematician does. With any
definition, try to stretch it in various ways, to see if it either can be
extended to cover more than you originally had in mind, or needs to be
refined to eliminate things that you don't want it to include. For a
discussion of these ideas in connection with polyhedra, see

  Definitions as a Tool of Mathematics
    http://mathforum.org/library/drmath/view/62384.html 

  Polyhedra: Solids or Surfaces?
    http://mathforum.org/library/drmath/view/63135.html 

Ultimately, these discussions point out that definitions are flexible as
well as slippery! It can be hard to come up with a water-tight definition;
and when you do, it may not fit someone else's needs -- so he may prefer a
slightly different definition. Experimenting with the implications of a
definition, as you are doing, is a good way to develop a feel for math.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
  

Date: 07/23/2010 at 14:55:05
From: Mark
Subject: Thank you (Is this a polygon)

Well, thanks for the help.

In your reply, you wrote: "Your definition entirely leaves out the fact
that a polygon must be a single connected curve."

I take this to mean that the outer and inner rectangles by themselves are
polygons, but the region inside the large rectangle and outside the
smaller one is not a polygon.  Is this correct?

Date: 07/23/2010 at 19:39:19
From: Doctor Peterson
Subject: Re: Thank you (Is this a polygon)

Hi, Mark.

A "region" can't be a polygon at all; a polygon is a CURVE (something you
can draw with a pencil), not a REGION (part of a plane that can be shaded,
for example). When we talk about polygons, we aren't talking about
regions, unless we specify the interior of a polygon.

So yes: taken individually, the outer and inner rectangles (the boundaries
of your region) are polygons -- namely, rectangles. But their union (all
eight edges together) is not a polygon, because it consists of two
unconnected parts. And the region between them is not a polygon interior.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/24/2010 at 01:42:33
From: Mark
Subject: Thank you (Is this a polygon)

OK. I got it. Thanks.