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Counting Collinear Sides

Date: 04/12/2013 at 11:24:00
From: Beth
Subject: definition of octagon and polygon

Can a figure drawn as a rectangle with one point on each side be
classified as an octagon?

Generally speaking, can sides of a polygon be collinear?

Some argue that this fits the definition of both polygons and octagons. I
see their argument that the figure has 8 vertices and 8 sides. And we're
clearly dealing with a closed figure made of 3 or more line segments.

But I say no. I believe consecutive sides of a polygon cannot be
collinear. 



Date: 04/12/2013 at 17:23:45
From: Doctor Peterson
Subject: Re: definition of octagon and polygon

Hi, Beth.

Many statements of the definition of a polygon are not quite complete,
because we generally "know what a polygon is when we see one!" Some
definitions do explicitly include a restriction that adjacent sides must
not be collinear; others, even from fairly careful sources, do not,
perhaps because we don't think of the possibility, or else because we
don't want our definitions to be too complicated.

The fact is, definitions are more flexible than most people realize, even
in math. For some purposes, we don't want what might be called degenerate
vertices, as in your example, because it would mess up what we want to do;
for example, it would affect the number of diagonals in a convex polygon,
unless we disallow this as a convex polygon. For other purposes, such as
in topology, collinearity doesn't matter.

Wikipedia makes just this sort of statement:

    http://en.wikipedia.org/wiki/Polygon 

   In geometry a polygon is a flat shape consisting of straight lines
   that are joined to form a closed chain or circuit.

   The basic geometrical notion has been adapted in various ways to
   suit particular purposes. Mathematicians are often concerned only
   with the closed polygonal chain and with simple polygons which do
   not self-intersect, and may define a polygon accordingly.
   Geometrically two edges meeting at a corner are required to form
   an angle that is not straight (180 degrees); otherwise, the line
   segments will be considered parts of a single edge; however
   mathematically, such corners may sometimes be allowed.
 
Note that the definition itself does not happen to mention collinearity,
but the additional comment points out variations in whether
self-intersecting polygons are allowed (if they are, then we need the
qualifier "simple" to exclude them), and also your issue of whether
straight angles are allowed.

MathWorld does explicitly reject straight angles, by a restriction on
successive vertices:

    http://mathworld.wolfram.com/Polygon.html 

   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), 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. In other
   words, a polygon is a closed broken line lying in a plane" (Coxeter
   and Greitzer 1967, p. 51). 
 
   There is unfortunately substantial disagreement over the definition
   of a polygon. Other sources commonly define a polygon as a "closed
   plane figure with straight edges" (Gellert et al. 1989, p. 162), "a
   closed plane figure bounded by straight line segments as its sides"
   (Bronshtein et al. 2003, p. 137), or "a closed plane figure bounded
   by three or more line segments that terminate in pairs at the same
   number of vertices, and do not intersect other than at their
   vertices" (Borowski and Borwein 2005, p. 573). These definitions all
   imply that a polygon is a set of line segments plus the region they
   enclose, though they never define precisely what is meant by "closed
   plane figure" and universally depict polygons as a [set of] closed broken
   black lines with no shading of the interiors.

Here, too, we find a note on variations -- this time about whether the
polygon is just the outline or also the interior.

I've made some similar comments about variations in the definition, though
I didn't mention collinearity:

  On Polygons, Polygons within Polygons, and Definitions
    http://mathforum.org/library/drmath/view/76073.html 

The questioner above mentioned that his text requires non-collinearity,
though that was not an issue and I happen not to have mentioned it in my
own definition. (You may find the links in that page interesting, too.)

So I'd say you're correct with regard to common usage, but it is entirely
reasonable, for some purposes, to extend the definition to allow straight
angles.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Definitions
High School Triangles and Other Polygons

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