Can a Circle be a Polygon?

Date: 5/22/96 at 0:42:48
From: Bill Smith
Subject: Can a Circle be a Polygon?

Could a circle be considered a polygon with an infinite number of 
sides?

Date: 5/22/96 at 9:19:47
From: Doctor Sarah
Subject: Re: Can a Circle be a Polygon?

Hi Bill -

If you've been reading our newsgroup geometry puzzles you'll have seen 
some of the following discussion, but I'm guessing you haven't, so 
here it is:

From: Mike Slack
Newsgroups: geometry.puzzles
Subject: Circles
Date: 20 May 1996 09:14:52 -0400

How many sides does a circle have?
Not as easy as it may seem...

It could have 1 curved side!

It could have 2 sides - inside and outside!

It could have infinite sides?  Each side being very small.

It could have no sides!  If a side is defined as a straight line, no
matter how small, then surely a circle has no sides!  Why?  Because
no matter how small a side is it can be halved, and the midpoint of
that side is no longer equidistant from the center of the circle
compared to each end!

Who has the definitive answer?
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From: Curtis - Stephen Dean
Newsgroups: geometry.puzzles
Subject: Re: Circles
Date: 21 May 1996 08:08:34 -0400

By definition a circle is the locus of all points equidistant from a 
central point.  If asked then the number of sides, it is my opinion 
that the question itself does not apply if "sides" is implied to have 
the same meaning as in a rectangle or square.  However, you could in a 
non-mathematical sense say that the circle has an inside and an 
outside, which is mathematically irrelevant.

S. Dean Curtis   
Baltimore County Public Schools
Sudbrook Magnet Middle School  
Baltimore, MD 21208
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From: Vince Devlin
Newsgroups: geometry.puzzles
Subject: Re: Circles
Date: 21 May 1996 09:22:14 -0400

Mike,

You refer to a "curved side."  You might want to post your definition 
of a "side."  That might make your puzzle easier to discuss.

Vince Devlin
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From: DougKuhlmann-PhillipsAcademy-Math
Newsgroups: geometry.puzzles
Subject: Re: Circles
Date: 21 May 1996 10:49:36 -0400

If you ever studied the Jordan Curve Theorem, you would know that the
question of whether a curve determines an inside or an outside is 
hardly mathematically irrelevant.

Doug Kuhlmann   
Phillips Academy
Andover, MA 01810
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From: Mike Slack
Newsgroups: geometry.puzzles
Subject: Re: Circles
Date: 22 May 1996 07:05:28 -0400

The definition of a "side" is all inclusive in the discussion.  But
for the sake of completeness here is a definition.....

side n. 1. any of the more or less flat outer surfaces of an object,
especially as distinct from the top or bottom, front and back.
2. either surface of a flat object.  3. any of the bounding lines
of a flat figure such as a triangle or square.  (taken from the
Oxford Dictionary!)
----------------------------------------------------------------------
From: Walter Whiteley
Newsgroups: geometry.puzzles
Subject: Re: inside of Circles
Date: 21 May 1996 10:30:36 -0400

The fact that a circle, in the plane or on a sphere, has an 'inside' 
and an 'outside' is mathematically very relevant.  In fact, this is a 
special case of the Jordon Curve Theorem, that any simple closed curve 
(image of a a continuous 1-1 map from the circle) divides the plane or 
sphere into two components.

This is, of course, not true of some circles on a torus.

Walter Whiteley
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