Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. This book discusses these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them. It addresses representation theory and reciprocity laws. "To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program."---Lindsay N. Childs, MathSciNet