The Art and Craft of Problem Solving

3.1 SYMMETRY 63

Why is symmetry important? Because it gives you "free" information. If you

know that something is, say, symmetric with respect to 90-degree rotation about some

point, then you only need to look at one-quarter of the object. And you also know that

the center of rotation is a "special" point, worthy of close investigation. You will see

these ideas in action below, but before we begin, let us mention two things to keep in

mind as you ponder symmetry:

• The strategic principles of peripheral vision and rule-breaking tell us to look
  for symmetry in unlikely places, and not to worry if something is almost, but
  not quite symmetrical. In these cases, it is wise to proceed as if symmetry is
  present, since we will probably learn something useful.


• An informal alternate definition of symmetry is "harmony." This is even vaguer
  than our "formal" definition, but it is not without value. Look for harmony,
  and beauty, whenever you investigate a problem. If you can do something that
  makes things more harmonious or more beautiful, even if you have no idea how
  to define these two terms, then you are often on the right track.

Geometric Symmetry

Most geometric investigations profit by a look at symmetry. Ask these questions about

symmetry:

• Is it present?


• If not, can it be imposed?


• How can it then be exploited?

Here is a simple but striking example.

Example 3.1. 4 A square is inscribed in a circle that is inscribed in a square. Find the

ratio of the areas of the two squares.

Solution: The problem can certainly be solved algebraically (let x equal the length
of the small square, then use the Pythagorean theorem, etc.), but there is a nicer ap­
proach. The original diagram is full of symmetries. We are free to rotate and/or reflect
many shapes and still preserve the areas of the two squares. How do we choose from
all these possibilities? We need to use the hypothesis that the objects are inscribed
in one another. If we rotate the small square by 45 degrees, its vertices now line up
with the points of tangency between the circle and the large square, and instantly the
solution emerges.