Symmetry Problem Solving

Algebra, geometry, trigonometry and calculus are four main domains of school and collegiate mathematics.And in each of these domains, students are introduced to generalizations on the notions of symmetry; e.g., in algebra they are introduced to symmetric functions, symmetric determinants, symmetric groups, symmetric systems of equations and to symmetric forms, as in the symmetric form of the equatation of a line; in geometry they meet the notions of point and line symmetry, and n-fold symmetry.

In trigonometry the students meet basic symmetric relations like cos), and at a more advanced level, the notions of symmetric spherical triangles.

In elementary calculus they again see that symmetry plays a pivotal role in applying integration techniques and in working with differential forms.

And as soon as one thinks of patterns, one thinks of symmetry.

So here we have a systematic (symmetric) way to represent an irrational number, that in and of itself is almost a contradiction of terms, for irrational numbers do not have systematic decimal representations; yet with the notions of continued fractions and infinite square roots there is a certain symmetry to them.

Constructing the points C and D for given line segment AB is an instructive exercise which is closely associated with the Appolonian circle for fixed points A and B; the Appolonian circle is the locus of a point P such that AP/PB is a constant k APB internally and externally.

Let the internal bisector meet line segment AB at C; and let the external bisector meet AB extended at D.

The notion of symmetry is itself a mathematician's dream, for point and line symmetries which have been extensively studied in their own right, have been generalized and applied to almost every area of mathematics, even school mathematics.

Moreover, entire domains of mathematics, such as group theory have arisen out of the study of symmetry.

One of these domains is problem solving, where symmetry must be seen or imposed on a problem to effect its solution.

Another domain is in concept formation, where it is often advantageous to think of basic mathematical notions in terms of symmetrical properties which surround them.