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Two-Sided Polygon?

Date: 12/01/2003 at 16:44:51
From: Erik
Subject: Is it possible to form a polygon with two straight lines?

My 5th grade math teacher said that we had to draw a polygon using two
straight lines.  We are not allowed to use curved lines. 

I think this is not possible. I thought that a polygon had to have at
least 3 sides.  It had to have line segments that meet at two sides at
each vertex.
 
Am I wrong?


Date: 12/01/2003 at 18:53:39
From: Doctor Ian
Subject: Re: Is it possible to form a polygon with two straight lines?

Hi Erik,

It depends on how flexible you're willing to be in your definition of
'straight' and 'polygon'. 

For example, one definition of a 'straight' segment is that it
connects its endpoints with the shortest possible path.  Find a globe,
and look at one of the meridians of longitude (which run from pole to
pole).  Pick any two points on that meridian.  

Now, the shortest path ALONG THE SURFACE of the globe is that
meridian.  So while in three dimensions we think of it as a curve, ON
THE SURFACE of the globe, it's a straight line.  

Now, pick any two meridians--say 0 degrees and 30 degrees.  They
intersect at the north and south poles, and enclose an area between
them, right?  So now you have two straight line segements, which
intersect at vertices, and enclose an area.  That looks like a polygon
to me!

Or, you could try this.  Take a lined piece of paper, and draw two
intersecting lines across is, so that they leave the paper at the same
heights on each side:

    +---------------+
    |               |
    A               A'
    | .           . |
    |    .     .    |
    |       .       |
    |    .     .    |
    | .           . |
    B               B'
    |               |
    +---------------+

Now roll the paper into a cylinder, so that point A touches point A',
and point B touches point B'.  Once again, you have straight line
segments meeting to form vertices, and enclosing an area.  

However, if you restrict yourself to using flat spaces (like a piece
of paper that you can't pick up), then you're correct:  two lines can
form only one angle, so they can't enclose a polygon.  What curving
the space allows you to do is have the lines intersect more than once.   

Unless, of course, you're willing to draw a 'degenerate' polygon:

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

In that case, you could just draw a line segment from point A to point
B, and another one from point B to point A.  This is the limiting case
of a rectangle, as the lengths of one pair of opposite sides goes to
zero.  But not everyone would be willing to call it a polygon. 

Does this help?

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

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