How Many Edges in a Circle?

Date: 06/15/99 at 18:12:22
From: Dennis Mayhew
Subject: How many edges in a circle

Dear Doctor,

A group of teachers here in Edmonton, Alberta have been debating over 
the number of edges in a circle. Three answers have been presented: 
one, none, and infinite. What do you think? We want to know which one 
we should be teaching.

D Mayhew

Date: 06/18/99 at 07:23:23
From: Doctor Pete
Subject: Re: How many edges in a circle

Hello,

Here is what I think. I do not believe there is an "infinite" number 
of edges in a circle, by any common or logical definition of the word 
"edge." An edge can be thought of as a boundary - in which case a 
circle, being a single closed figure, does not have an infinite number 
of boundaries.

However, does a square, having in a sense one continuous boundary, 
therefore only have one edge? Certainly we do not wish the term "edge" 
to be so imprecise as to fail to distinguish between a square and, 
say, a pentagon. So, one might say that an edge is a continuous 
straight line on the boundary of a closed figure. But the circle, 
having no such lines, has *no* edges under this definition. This I 
find somewhat unsatisfactory, but less so than "infinite."

More pleasing to my sense of the word, suppose an "edge" is an open, 
nonempty subset of the boundary of a simple closed curve that is 
continuously differentiable. Under this definition a circle has one 
edge - the entire boundary. A semicircle would have two edges: the 
curved portion, and the diameter.  

This seems sensible because we wish to distinguish the circle and 
semicircle by an important property - a semicircle can be said to have 
two "vertices."  Clearly we would like to say that a vertex is a 
"point," no matter how blunt, so to speak. A regular 1000000-gon may 
be close to a circle, but it is still not yet a circle. Precisely 
speaking, a "vertex" is a point at which the boundary fails to be 
differentiable. A smooth curve, or a straight line, does not have a 
vertex.

But this last thought brings us back to the beginning. Think of a 
regular polygon of n sides. As n gets larger and larger, this polygon 
becomes more and more circular; that is, it gets closer and closer to 
approximating a circle. One might say that as n approaches infinity, 
we obtain a circle. And hence, it may be argued that a circle is a 
regular polygon with infinitely many edges. Well, perhaps so. But 
there are two reasons why this thinking is not satisfactory.

First, how long is an edge in such an infinite-sided polygon? It must 
have zero length, and therefore is not a line but a point. So can we 
truly call this zero-length segment an edge? The definition of "edge" 
I gave does not permit an edge to be a single point - it must contain 
more than one point.

Second, this limiting process, when applied to curves in general, 
implies that any curved figure has an infinite number of edges - for 
example, the semicircle would also have an infinite number of edges.  
But for the reasons previously given, we would like to distinguish 
between a circle and semicircle.

So, in short, it would seem that a circle has one "edge." But in fact 
the definition that permits this to be so is not a commonly perceived 
definition, as "edge" in common usage implies "straight line."  Thus, 
I feel that the matter is made more precise by simply not using the 
word "edge" to describe a circle at all.

In closing, here is a question: What kind of simple closed figure 
would have an infinite number of edges, under the definition I 
provided?  

The answer, I think, will amuse your teachers.

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