Circle and Polygons: Lines of Symmetry

Date: 04/14/97 at 15:08:57
From: Anonymous
Subject: Lines of symmetry

Hello!

How many lines of symmetry are there in a circle? This has been an on-
going conversation in our class. We've asked many teachers and we
have come up with 3 answers: 180, 360, and infinitely many.  

Thank you for your time

Date: 04/14/97 at 22:28:18
From: Doctor Steven
Subject: Re: Lines of symmetry

It seems you are asking the question: "How many lines can a circle be 
reflected about and still be self-coincident (i.e., fall back onto 
itself)?"

The answer is infinitely many.  Take any diameter of the circle and 
reflect the circle about that diameter and it will be self-coincident.  
There are an infinite number of diameters of a circle, so there is an 
infinite number of such lines.

Notice that the circle is also self-coincident under any rotation.  So 
there are an infinite number of symmetry rotations of the circle.

A more difficult question would be to ask how many lines and rotations 
of symmetry a polygon has.  Number the corners of a square like so:

    1  _______  2
      |       |
      |       |
      |       |
      |_______| 
    3           4

When we flip the square about a line of symmetry or rotate the square, 
we will call this a rigid motion, because the square maintains its 
shape (i.e., it doesn't get squashed or anything).  A square has 4 
lines of symmetry: the horizontal line, the vertical line, and the two 
diagonals.  It also has 4 rotations: the 90 degree turn, the 180 
degree turn, the 270 degree turn and the 360 degree turn.  

The horizontal line flip switches 1 and 3, and switches 2 and 4.
The vertical line flip switches 1 and 2, and switches 3 and 4.
The 1,4-diagonal line flip switches 3 and 2 and leaves both 1 and 4           
   fixed.
The 3,2-diagonal line flip switches 1 and 4 and leaves both 3 and 2 
   fixed.
The 360 degree (or 0 degree, however you look at it) rotation leaves     
   everything fixed.
The 90 degree rotation moves 1 to 2, 2 to 4, 4 to 3, and 3 to 1.
The 180 degree rotation moves 1 to 4, and 2 to 3.
The 270 degree rotation moves 1 to 3, 3 to 4, 4 to 2, and 2 to 1.

Note that following any one of the rigid motions by another rigid 
motion gives us a different rigid motion. For example: The horizontal 
flip, followed by the 90 degree rotation switches 3 and 2 and leaves 1 
and 4 fixed, which is the same as the 1,4-diagonal flip.

I leave it to you to figure out the symmetries of a pentagon, and 
polygons in general.

Hope this helps.

-Doctor Steven,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/