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. Francoise 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.

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. Francoise 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 Gyrovector Spaces in the Service of Abalytic Hyperbolic Geometry When Two Counterintuitive Theories Meet The Fascinating Rich Mathematical Life of Einstein's Velocity Addition Law Parts of the Book Einstein Gyrogroups and Gyrovector Spaces Einstein Gyrogroups Introduction Einstein Velocity Addition Einstein Addition for Computer Algebra Thomas Precession Angle Einstein Addition With Respect to Cartesian Coordinates Einstein Addition Vs. Vector Addition Gyrations Gyration Angles From Einstein Velocity Addition to Gyrogroups Gyrogroup Cooperation (Coaddition) First Gyrogroup Properties Elements of Gyrogroup Theory The Two Basic Gyrogroup Equations The Basic Gyrogroup Cancellation Laws Automorphisms and Gyroautomorphisms Gyrosemidirect Product Basic Gyration Properties An Advanced Gyrogroup Equation Gyrocommutative Gyrogroups Problems Einstein Gyrovector Spaces 65 The Abstract Gyrovector Space Einstein Gyrovector Spaces Einstein Addition and Differential Geometry Euclidean Lines Gyrolines - The Hyperbolic Lines Gyroangles - The Hyperbolic Angles Euclidean Isometries The Group of Euclidean Motions Gyroisometries - The Hyperbolic Isometries Gyromotions - The Motions of Hyperbolic Geometry Problems Relativistic Mass Meets Hyperbolic Geometry Lorentz Transformation and Einstein Addition Mass of Particle Systems Resultant Relativistically Invariant Mass Problems Mathematical Tools for Hyperbolic Geometry Barycentric and Gyrobarycentric Coordinates Barycentric Coordinates Segments Gyrobarycentric Coordinates Uniqueness of Gyrobarycentric Representations Gyrovector Gyroconvex Span Gyrosegments Triangle Centroid Gyromidpoint Gyroline Boundary points Gyrotriangle Gyrocentroid Gyrodistance in Gyrobarycentric Coordinates Gyrolines in Gyrobarycentric Coordinates Problems Gyroparallelograms and Gyroparallelotopes The Parallelogram Law Einstein Gyroparallelograms The Gyroparallelogram Law The Higher-Dimensional Gyroparallelotope Law Gyroparallelotopes Gyroparallelotope Gyrocentroid Gyroparallelotope Formal Definition and Theorem Low Dimensional Gyroparallelotopes Hyperbolic Plane Separation GPSA for the Einstein Gyroplane Problems Gyrotrigonometry Gyroangles Gyroangle - Angle Relationship The Law of Gyrocosines The SSS to AAA Conversion Law Inequalities for Gyrotriangles The AAA to SSS Conversion Law The Law of Sines/Gyrosines The Law of Gyrosines The ASA to SAS Conversion Law Gyrotriangle Defect Right Gyrotriangles Gyrotrigonometry Gyroangle of Parallelism Useful Gyrotriangle Gyrotrigonometric Identities A Determinantal Pattern Problems Hyperbolic Triangles and Circles Gyrotriangles and Gyrocircles Gyrocircles Gyrotriangle Circumgyrocenter Triangle Circumcenter Gyrotriangle Circumgyroradius Triangle Circumradius The Gyrocircle Through Three Points The Inscribed Gyroangle Theorem I The Inscribed Gyroangle Theorem II Gyrocircle Gyrotangent Gyrolines Semi-Gyrocircle Gyrotriangles Problems Gyrocircle Theorems The Gyrotangent-Gyrosecant Theorem The Intersecting Gyrosecants Theorem Gyrocircle Gyrobarycentric Representation Gyrocircle Interior and Exterior Points Circle Barycentric Representation Gyrocircle Gyroline Intersection Gyrocircle-Gyroline Tangency Points Gyrocircle Gyrotangent Gyrolength Circle-Line Tangency Points Circumgyrocevians Gyrodistances Related to the Gyrocevian A Gyrodistance Related to the Circumgyrocevian Circumgyrocevian Gyrolength The Intersecting Gyrochords Theorem Problems Hyperbolic Simplices, Hyperplanes and Hyperspheres in N Dimensions Gyrosimplices Gyrotetrahedron Circumgyrocenter Gyrotetrahedron Circumgyroradius Gyrosimplex Gyrocentroid Gamma Matrices Gyrosimplex Gyroaltitudes Gyrosimplex Circumhypergyrosphere The Gyrosimplex Constant Point to Gyrosimplex Gyrodistance Cramer's Rule Point to Gyrosimplex Perpendicular Projection Gyrosimplex In-Exgyrocenters and In-Exgyroradii Gyrotriangle In-Exgyrocenters Gyrosimplex Gyrosymmedian Problems Gyrosimplex Gyrovolume Gyrovolume Problems Hyperbolic Ellipses and Hyperbolas Gyroellipses and Gyrohyperbolas Gyroellipses - A Gyrobarycentric Representation Gyroellipses - Gyrotrigonometric Gyrobarycentric Representation Gyroellipse Major Vertices Gyroellipse Minor Vertices Canonical Gyroellipses Gyrobarycentric Representation of Canonical Gyroellipses Barycentric Representation of Canonical Ellipses Some Properties of Canonical Gyroellipses Canonical Gyroellipses and Ellipses Canonical Gyroellipse Equation A Gyrotrigonometric Constant of the Gyroellipse Ellipse Eccentricity Gyroellipse Gyroeccentricity Gyrohyperbolas - A Gyrobarycentric Representation Problems Thomas Precession Thomas Precession Introduction The Gyrotriangle Defect and Thomas Precession Thomas Precession Thomas Precession Matrix Thomas Precession Graphical Presentation Thomas Precession Angle Thomas Precession Frequency Thomas Precession and Boost Composition Thomas Precession Angle and Generating Angle have Opposite Signs Problems Bibliography Index

Anyone who is concerned with hyperbolic geometry should use this wonderful and comprehensive book as a helpful compendium. -Zentralblatt MATH 1312