Reflections on symmetry and proof

Abstract

This article uses notions of symmetry to approach the solutions to a broad range of mathematical problems. It responds to Krutetskii's criteria for mathematical ability as well as the outcomes which guide the Extension 1 & 2 Mathematics courses of the Board of Studies NSW. The concept of symmetry is fundamental, indeed foundational, to mathematics. Arguments and proofs based on symmetry are often aesthetically pleasing because they are subtle and succinct and non-standard. By symmetry, the person in the street usually means the exact correspondence in size and position of opposite parts, seen as an equal distribution of parts across a dividing line or about a centre. It is considered to be an attribute either of the whole or of the parts composing it. A more subtle appreciation of the term symmetry takes into account a harmony of parts with each other and with the whole, seen as a mutual relation of parts, as a fitting, regular or balanced arrangement and relation of parts or elements. (Onions, 1978).