A gyrovector space approach to hyperbolic geometry

Abraham Ungar9781598298222, 1598298224

The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix “gyro” that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry

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Table of contents :
Synthesis Lectures on Mathematics and Statisticsheight 0in depth .3in width 0pt……Page 3
Contents……Page 7
Preface……Page 11
From Möbius to Gyrogroups……Page 13
Groupoids, Loops, Groups, and Gyrogroups……Page 16
Möbius Gyrogroups: From the Disc To The Ball……Page 19
First Gyrogroup Theorems……Page 23
The Two Basic Equations of Gyrogroups……Page 29
The Basic Cancellation Laws of Gyrogroups……Page 31
Commuting Automorphisms with Gyroautomorphisms……Page 32
The Gyrosemidirect Product……Page 33
Basic Gyration Properties……Page 37
An Advanced Gyrogroup Equation……Page 43
Exercises……Page 45
Gyrocommutative Gyrogroups……Page 47
Möbius Gyrogroups……Page 55
Einstein Gyrogroups……Page 57
Gyrogroup Isomorphism……Page 63
Exercises……Page 65
Definition and First Gyrovector Space Theorems……Page 67
Gyrolines……Page 74
Gyromidpoints……Page 81
Analogies Between Gyromidpoints and Midpoints……Page 83
Gyrogeodesics……Page 86
Möbius Gyrovector Spaces……Page 87
Möbius Gyrolines……Page 91
Einstein Gyrovector Spaces……Page 93
Einstein Gyrolines……Page 94
Einstein Gyromidpoints and Gyrotriangle Gyrocentroids……Page 96
Möbius Gyrotriangle Gyromedians and Gyrocentroids……Page 100
The Gyroparallelogram……Page 103
Points, Vectors, and Gyrovectors……Page 107
The Gyroparallelogram Addition Law of Gyrovectors……Page 109
Gyrovector Gyrotranslation……Page 112
Gyrovector Gyrotranslation Composition……Page 118
Gyrovector Gyrotranslation and the Gyroparallelogram Law……Page 120
The Möbius Gyrotriangle Gyroangles……Page 121
Exercises……Page 122
The Gyroangle……Page 123
The Gyrotriangle……Page 128
The Gyrotriangle Addition Law……Page 131
Cogyrolines, Cogyrotriangles, and Cogyroangles……Page 134
The Law of Gyrocosines……Page 136
The SSS to AAA Conversion Law……Page 137
Inequalities for Gyrotriangles……Page 139
The AAA to SSS Conversion Law……Page 141
The ASA to SAS Conversion Law……Page 145
The Gyrotriangle Defect……Page 146
The Right Gyrotriangle……Page 148
Gyrotrigonometry……Page 149
Gyrodistance Between a Point and a Gyroline……Page 153
The Gyrotriangle Gyroaltitude……Page 158
The Gyrotriangle Gyroarea……Page 160
Gyrotriangle Similarity……Page 161
The Gyroangle Bisector Theorem……Page 162
The Hyperbolic Steiner–Lehmus Theorem……Page 163
The Urquhart theorem……Page 165
The Hyperbolic Urquhart theorem……Page 168
The Gyroparallelogram Gyroangles……Page 170
Relativistic Mechanical Interpretation……Page 175
Newtonian Systems of Particles……Page 179
Einsteinian Systems of Particles……Page 180
The Relativistic Invariant Mass Paradox……Page 181
Exercises……Page 183
Bibliography……Page 185
Index……Page 191