Analytic Hyperbolic Geometry in N Dimensions An Introduction

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.

List of Figures. Preface. Author’s Biography. Introduction. Einstein Gyrogroups and Gyrovector Spaces. Einstein Gyrogroups. Problems. Einstein Gyrovector Spaces. Problems. Relativistic Mass Meets Hyperbolic Geometry. Problems. Mathematical Tools for Hyperbolic Geometry. Barycentric and Gyrobarycentric Coordinates. Problems. Gyroparallelograms and Gyroparallelotopes. Problems. Gyrotrigonometry. Problems. Hyperbolic Triangles and Circles. Gyrotriangles and Gyrocircles. Problems. Gyrocircle Theorems. Problems. Hyperbolic Simplices, Hyperplanes and Hyperspheres in N Dimensions. Gyrosimplices. Problems. Gyrosimplex Gyrovolume. Problems. Hyperbolic Ellipses and Hyperbolas. Gyroellipses and Gyrohyperbolas. Problems. VI Thomas Precession. Thomas Precession. Problems. Bibliography. Index.