*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.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.

*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.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.*

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.

It is almost as though the notion of symmetry is built into us as a standard against which we measure aesthetic appeal to assess both mental and physical constructs.

Hargittai and Hargittai in their text Symmetry: A Unifying Concept illustrate how deeply seated and ubiquitous symmetrical relationships are through hundreds of photographs of man-made objects, from examples in architectural symmetry, to those found in nature, as exemplified by the markings on the wings of a butterfly.

Everywhere we turn we can see symmetrical relationships.

They are both visual and audio, and they are so pervasive in our daily lives that one is led naturally to wonder if the notion of symmetry is innate in human beings.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.We have now found the two desired points for the golden ratio. The aesthetic appeal of the golden ratio and its ties to the Fibonacci sequence, as well as its far reaching connections to nature and science are well documented in the literature (Huntley, 1970; Herz-Fischler, 1998; Dunlap, 1998).But the connection of this number to the human psyche, in the spirit of Fechner's work with investigating our attractions toward it, is an open question.It has been our experience that most students cannot solve these problems, because they do not use symmetry as a heuristic tool.(Partial answers are presented at the end of this paper.) integers into the cells of an n×n square so that the sums obtained by adding the numbers in each column, each row and each diagonal are equal.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.It is well known that there seems to be a small set of real numbers which appeal to our psyche more than other numbers.E.g., more than a hundred years ago the psychologist Gustav Fechner made literally thousands of measurements of rectangles commonly seen in everyday life; playing cards, window frames, writing papers, book covers, etc., and he noticed that the ratio of the length to the width seemed to approach the golden ratio 5)/2.

## Comments Symmetry Problem Solving

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