Equation that Couldn't Be Solved (eBook)

Chapter One: Symmetry

An inkblot on a piece of paper is not particularly attractive to the eye, but if you fold the paper before the ink dries, you may get something that looks like figure 1 that is much more intriguing. In fact, the interpretation of similar inkblots forms the basis for the famous Rorschach test developed in the 1920s by the Swiss psychiatrist Hermann Rorschach. The declared purpose of the test is to somehow elicit the hidden fears, wild fantasies, and deeper thoughts of the viewers interpreting the ambiguous shapes. The actual value of the test as an 'x-ray of the mind' is vehemently debated in psychological circles. As Emory University psychologist Scott Lilienfeld once put it, 'Whose mind, that of the client or the examiner?' Nevertheless, there is no denial of the fact that images such as that in figure 1 convey some sort of attractive and fascinating impression. Why?

Is it because the human body, most animals, and so many human artifacts possess a similar bilateral symmetry? And why do all those zoological features and creations of the human imagination exhibit such a symmetry in the first place?

Most people perceive harmonious compositions such as Botticelli's Birth of Venus as symmetrical. Art historian Ernst H. Gombrich even notes that the 'liberties which Botticelli took with nature in order to achieve a graceful outline add to the beauty and harmony of the design.' Yet mathematicians will tell you that the arrangements of colors and forms in that painting are not symmetric at all in the mathematical sense. Conversely, most nonmathematical viewers do not perceive the pattern in figure 3 as symmetrical, even though it actually is symmetrical according to the formal mathematical definition. So what is symmetry really? What role, if any, does it play in perception? How is it related to our aesthetic sensibility? In the scientific realm, why has symmetry become such a pivotal concept in our ideas about the cosmos around us and in the fundamental theories attempting to explain it? Since symmetry spans such a wide range of disciplines, what 'language' and what 'grammar' do we use to describe and characterize symmetries and their attributes, and how was that universal language invented? On a lighter note, can symmetry provide an answer to the all-important question posed in the title of one of the songs of rock star Rod Stewart -- 'Do Ya Think I'm Sexy?'

I will try to provide at least partial answers to all of these questions and many more. Along the way, I hope that the story as a whole will depict both the humanistic side of mathematics and, even more importantly, the human side of mathematicians. As we shall see, symmetry is the paramount tool for bridging the gap between science and art, between psychology and mathematics. It permeates objects and concepts ranging from Persian carpets to the molecules of life, from the Sistine Chapel to the sought-after 'Theory of Everything.' Yet group theory, the mathematical language that describes the essence of symmetries and explores their properties, did not emerge from the study of symmetries at all. Rather, this astonishingly unifying idea of modern thought emanated from a most unlikely source -- an equation that couldn't be solved. The dramatic and tortuous history of this equation is an essential part of this intellectual saga. At the same time, this tale will shed light on the loneliness of genius and on the tenacity of the human intellect in the face of seemingly insurmountable challenges. I have put a tremendous effort into trying to solve the two-centuries-old mystery of the death of the protagonist of this story -- the brilliant mathematician variste Galois....