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The rules of hyperbolic geometry seem to contradict and fly in the face of immutable truths, and yet its principles help us understand the world and can be physically created in front of our very eyes. This textbook aims to explain the principles, propositions, and proofs necessary for a foundation and understanding in the growing field of geometry. First, Rothe introduces the groundwork laid by ideas from mathematicians such as Henri Poincaré, Nikolai Lobachevsky, and Carl Friedrich Gauss. He also provides a significant explanation of the axioms and theorems of David Hilbert. From here, the divergence between Euclidean geometry and hyperbolic geometry is explained by proving the key difference that the former requires but not the latter: the parallel postulate. Each chapter introduces more concepts and formulas that will make the reader fluent in this specific area of mathematics.
After the history and background have been presented, the teaching advances topic by topic to illustrate how hyperbolic geometry affects circles, triangles, and unique figures, such as the pseudo-sphere, half-plane, and the Poincaré disk. Lessons feature numerous proofs and formulas, as well as interactive exercises to engage students in the learning process and help them retain each concept. This book is written with the appropriate level of student in mind, and readers should expect to be well-versed in advanced geometric and mathematical concepts. As a result, the text skips the filler and gets straight to the relevant information, making it easier to understand and study. Those studying advanced mathematics or who have a fascination for the higher levels of thought that go beyond basic education will find this textbook rich with information and presented in a straightforward, direct manner.
The succinct, well-organized information and ideas in this book make it a very useful study guide for those who are just picking this book up for the first time, as well as those who have been studying the previous books in this series. Each page is packed with proofs, graphs, and formulas that compound and expand on previous ones, focusing more time and energy on the meat of the subject rather than lengthy, surface-level explanations of the ideas for the casual reader. A searchable index included at the end of the book can help those who need help with a single specific aspect, though the book itself is organized in such a way that those who are looking at the subject of hyperbolic geometry as a whole can see it unfold from cover to cover.
The inclusion of some of the history of hyperbolic geometry's development and the thinkers behind it provides some interesting context, and those familiar with the significant mathematicians involved will surely see how these ideas took shape and rose to acceptance. At its core, however, this is a mathematics textbook, and so one should expect the majority of the content here to be focused on study and comprehension more than any biographical details or historical retelling. Although the concepts contained within can appear daunting and complicated, the author adheres to the structure of math education, and through this book, hyperbolic geometry is explained thoroughly, provided the audience possesses the necessary subject literacy to follow the proofs and formulas. Steady but not overwrought, comprehensive but not dense, this book will help those interested in the subject to tackle yet another facet of geometric study.