Do Circles Have Corners?

Date: 08/06/99 at 22:28:50
From: Anderson Kazuo Tanaka
Subject: Corners in Circles?

My question is whether circles have corners. I believe that circles 
could have corners because a circle is a continuous line, and what 
stops it from becoming a straight line is an angle. To have an angle 
there needs to be a corner - even if the corner is microscopic, it's 
there. I would like to know your answer. Thank you very much.

Sincerely,
Anderson Kazuo Tanaka

Date: 08/07/99 at 10:14:31
From: Doctor Peterson
Subject: Re: Corners in Circles?

Hi, Anderson.

There is a sense in which you could say that a circle is "all corner," 
if you just think of a corner as a turning point, since the circle is 
always turning. But technically a corner, or 'vertex' in more 
mathematical terminology, is a point where two straight edges of a 
figure meet, and a circle has no straight edges, so it can have no 
corners.

A circle can be seen as the limit of a set of polygons with increasing 
numbers of edges. If we approximate the circle by, say, a "googolgon" 
with a huge number of edges, then we will have many microscopic 
corners. But when you go from polygons to the actual circle, you no 
longer have any edges or corners, only a smooth curve. Without edges, 
you have no angles or corners.

If you have not yet learned about limits, and about calculus, which 
lets us work with continuously changing quantities and curves, you can 
look forward to some ideas I think you will enjoy. Your thoughts about 
tiny angles will tie into this, if you can wait.

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