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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.
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.
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.
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.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.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.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.Nevertheless, there are many aspects of symmetry which are embedded in the golden ratio and which are instructive for students to study.(Space limitation allow us to only mention a few.) The two points C and D are said to divide the line segment AB into its mean and extreme ratio We now have the simplest continued fraction and root expansion known to man; each of which can be continued on ad infinitum in a systematic, patterned way.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.