Kristopher Tapp - Symmetry: A Mathematical Exploration

Hexagonal honeycomb structure

The mathematical topics in this book are drawn from diverse fields, including graph theory, abstract algebra, linear algebra, and topology, all of which offer essential tools for undertaking a rigorous study of symmetry. Although some of these topics are advanced, understanding this book requires no background beyond the level of middle-school algebra. The presentation here is intended to be precise and rigorous, yet accessible to a general audience. The only real prerequisite is that you discard any preconceived notions of what math is and isnt, and begin this mathematical journey with an open mind and a willingness to begin actively doing what mathematicians do: discovering patterns, inventing precise language for discussing the mathematical principles underlying those patterns, forming conjectures, and eventually proving beautiful theorems.

This book is primarily intended as a textbook for a math course for math or non-math majors, including humanities majors, with the goal of encouraging effective analytical thinking and exposing students to elegant mathematical ideas. Students find it comforting that there are no prerequisites; previous struggles with math wont come back to haunt them because the content of this book is very different from the math with which they struggled.

The book includes some of the topics that are commonly found in buffet-style sampler textbooks, such as Platonic solids, Eulers formula, irrational numbers, countable sets, permutations, and a proof of the Pythagorean Theorem. But in this book, all of these topics serve a single compelling goal: to understand the mathematical patterns underlying the symmetry that we observe in the physical world. I hope that students from all majors will enjoy the many beautiful mathematical topics herein, and will come to better appreciate the powerful cumulative nature of mathematics as these topics are woven together into a single story about symmetry.