Analytic Hyperbolic Geometry in N Dimensions : An Introduction

This book introduces for the first time the hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry. The extension of common Euclidean geometry to N dimensions, with N being any positive integer, results in greater generality and succinctness in related expressions. Using new mathematical tools, the book demonstrates that this is also the case with analytic hyperbolic geometry. For example, the author analytically determines the hyperbolic circumcenter and circumradius of any hyperbolic simplex.

The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry.

Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language that sheds natural light on hyperbolic geometry and special relativity. Several authors have successfully employed the author’s gyroalgebra in their exploration for novel results. Françoise Chatelin noted in her book, and elsewhere, that the computation language of Einstein described in this book plays a universal computational role, which extends far beyond the domain of special relativity.

This book will encourage researchers to use the author’s novel techniques to formulate their own results. The book provides new mathematical tools, such as hyperbolic simplexes, for the study of hyperbolic geometry in n dimensions. It also presents a new look at Einstein’s special relativity theory.