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In mathematics, certain basic concepts, such as symmetry and infinity, are so pervasive and adaptable that they can become elusive to the student. Understanding these concepts and the tools for studying them is often a long process that extends over many years in a student’s career. Students first see infinity appearing as the potential infinite inherent in the positional number system, then implicit in plane geometry, and eventually underlying all of calculus and analysis. This paper will describe how the concept of symmetry in a way gives them a comprehensive understanding of the mathematical approach to symmetry.
Students are fascinated by concrete examples of symmetry in nature and in art. One can simply locate the symmetries of designs and patterns, or one use symmetry groups as a comprehensible way to introduce students to the abstract approach of modern mathematics. Furthermore, the ideas used by mathematicians in studying symmetry are not unique to mathematics and can be found in other areas of human thought. Students can see the interconnectedness of mathematics with other branches of knowledge.
In the Bhagavad-Gita, Lord Krishna lays out the complete knowledge of life to his pupil Arjuna, just as a great battle is about to begin. This work has long been appreciated for the great wisdom that is expounded in just a few short chapters. A verse that seems to me to capture the essence of the mathematical study of symmetry is part of Krishna’s explanation of the field of action (Chapter 4, verse 18, translated by Maharishi Mahesh Yogi):
He who in action sees inaction
and in inaction sees action is
wise among men. He is united, he
has accomplished all action.
How is this related to symmetry? A geometric figure that we wish to study is usually given as a set of points existing in some ambient space. For example, a tiling pattern may be given as a collection of line segments in the plane. A symmetry transformation can be regarded as “action” and invariants can be regarded as “inaction.” We begin with a non-dynamic situation (the set of points of the tiling pattern sitting in the plane) and then find some dynamism (the symmetry transformation). Thus, in inaction, we see action. But a symmetry transformation is not just any action; it must leave the pattern (as a set of points) invariant. Thus, what is important to us is that in this action (the transformation), we are able to see inaction (the invariance of the set of points making up the pattern).
This is the seed of all that to know about symmetry: action and inaction, a transformation and its invariants, what changes and what stays the same.
With this, the students gain a unifying perspective on the concept of symmetry that can help them understand it initially and that can later help them simplify and unify all the occurrences of this concept as they are met and eventually understand symmetry groups, invariants, and so on.
Mathematics is part of life; mathematicians doing mathematics are subject to the same natural laws that govern all of life. A deep understanding of the whole of life should give us the kind of insight that will help us understand the parts of life, including some very specific aspect of mathematics. In the Bhagavad-Gita, expression of knowledge about the nature of life can be used to go deeply into the mathematical study of symmetry and, hopefully, acts as a suggestion that this bringing together of mathematics and life as a whole can be done in other ways.