●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
編輯手記

本書共分為四部分，第一部分介紹了陀螺組；第二部分介紹了陀螺交換陀螺組；第三部分介紹了陀螺矢量空間，陀螺儀矢量是根據陀螺儀平行四邊形法則添加的定向陀螺儀段的等價類，正如矢量是根據平行四邊形法則添加的定向段的等價類；第四部分介紹了回轉三角法。本書捕捉到的這些新奇的類比源於托馬斯旋轉，這是相對論效應的數學抽像，稱為托馬斯旋進。值得注意的是，本書僅僅引入托馬斯？回轉就把歐幾裡得幾何變成了雙曲幾何，並揭示了這兩種幾何所共有的神秘類比。