Symmetry Problem Solving

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.

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One of these points C is an interior point to the segment AB.

The other point D is an exterior point to the segment; i.e., the segment AB must be extended to reach it.

The centrality of symmetry as a notion in and of itself, not to mention its use as a heuristic in problem solving, is easily documented in general mathematics, and in school mathematics too.

But whether or not there is a natural, innate, gravitation towards symmetry is an open question, although many giants in mathematics and the physical sciences (Poincare, 1913; Einstein, 1935; Weyl, 1952; Polya, 1962 and Penrose, 1974) have addressed their own propensities for symmetry and aesthetics, individually saying that they believe it to be one of the driving forces behind their work.

Whether or not we have a subconscious gravitation towards symmetry, and special numbers such as the golden ratio, is admittedly a hazy area and perhaps best left to psychologists to investigate.

But like learning to appreciate art and music, where one learns what to look for with respect to a painting, and what to listen for with respect to a piece of music of a particular period, one must be taught how to look for symmetrical relationships; gravitation towards symmetry might happen naturally, but learning how to utilize symmetry must be taught (Dreyfus and Eisenberg, 1990).

Huntley's listing of Fechner's data is presented in Figure 1.

The golden ratio is a number closely tied to symmetry. Given the line segment AB there are two special points C and D on it that divide the line segment into the golden ratio.

At the most basic level it helps to look at texts like the one written by the Hargittais, where the ubiquity of point and line symmetries are vividly pointed out and awake our sensitivity to geometrical symmetry.

But the notion of symmetry enters many domains of school mathematics other than geometry.


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