Contents Preface 1 Gyrogroups 1.1 From M6bius to Gyrogroups 1.2 Groupoids, Loops, Groups, and Gyrogroups 1.3 M6bius Gyrogroups: From the Disc To The Ball 1.4 First Gyrogroup Theorems 1.5 The Two Basic Equations of Gyrogroups 1.6 The Basic Cancellation Laws of Gyrogroups 1.7 Commuting Automorphisms with Gyroautomorphisms 1.8 The Gyrosemidirect Product 1.9 Basic Gyration Properties 1.10 An Advanced Gyrogroup Equation 1.11 Exercises 2 Gyrocommutative Gyrogroups 2.1 Gyrocommutative Gyrogroups 2.2 Mobius Gyrogroups 2.3 Einstein Gyrogroups 2.4 Gyrogroup Isomorphism 2.5 Exercises 3 Gyrovector Spaces 3.1 Definition and First Gyrovector Space Theorems 3.2 Gyrolines 3.3 Gyromidpoints 3.4 Analogies Between Gyromidpoints and Midpoints 3.5 Gyrogeodesics 3.6 Mobius Gyrovector Spaces 3.7 Mobius Gyrolines 3.8 Einstein Gyrovector Spaces 3.9 Einstein Gyrolines 3.10 Einstein Gyromidpoints and Gyrotriangle Gyrocentroids 3.11 Mobius Gyrotriangle Gyromedians and Gyrocentroids 3.12 The Gyroparallelogram 3.13 Points, Vectors, and Gyrovectors 3.14 The Gyroparallelogram Addition Law of Gyrovectors 3.15 Gyrovector Gyrotranslation 3.16 Gyrovector Gyrotranslation Composition 3.17 Gyrovector Gyrotranslation and the GyroparaUelogram Law 3.18 The Mobius Gyrotriangle Gyroangles 3.19 Exercises 4 Gyrotrigonometry 4.1 The Gyroangle 4.2 The Gyrotriangle 4.3 The Gyrotriangle Addition Law 4.4 Cogyrolines, Cogyrotriangles, and Cogyroangles 4.5 The Law of Gyrocosines 4.6 The SSS to AAA Conversion Law 4.7 Inequalities for Gyrotriangles 4.8 The AAA to S8S Conversion Law 4.9 The Law of Gyrosines 4.10 TheASA to 8AS Conversion Law 4.11 The Gyrotriangle Defect 4.12 The Right Gyrotriangle 4.13 Gyrotrigonometry 4.14 Gyrodistance Between a Point and a Gyroline 4.15 The Gyrotriangle Gyroaltitude 4.16 The Gyrotriangle Gyroarea 4.17 Gyrotriangle Similarity 4.18 The Gyroangle Bisector Theorem 4.19 The Hyperbolic Steiner-Lehmus Theorem 4.20 The Urquhart theorem 4.21 The Hyperbolic Urquhart theorem 4.22 The Gyroparallelogram Gyroangles 4.23 Relativistic Mechanical Interpretation 4.24 Gyro-Analogies that May Reveal the Origin of Dark Matter 4.24.1 Newtonian Systems of Particles 4.24.2 Einsteinian Systems of Particles 4.24.3 The Relativistic Invariant Mass Paradox 4.25 Exercises Bibliography Index ��݋��ӛ