| Description |
1 online resource (xii, 182 pages) : illustrations |
| Series |
Synthesis lectures on mathematics and statistics, 1930-1751 ; #4 |
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Synthesis lectures in mathematics and statistics (Online) ; #4.
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| Contents |
Gyrogroups -- From möbius to gyrogroups -- Groupoids, loops, groups, and gyrogroups -- Möbius gyrogroups: from the disc to the ball -- First gyrogroup theorems -- The two basic equations of gyrogroups -- The basic cancellation laws of gyrogroups -- Commuting automorphisms with gyroautomorphisms -- The gyrosemidirect product -- Basic gyration properties -- An advanced gyrogroup equation -- Exercises -- Gyrocommutative gyrogroups -- Gyrocommutative gyrogroups -- Möbius gyrogroups -- Einstein gyrogroups -- Gyrogroup isomorphism -- Exercises -- Gyrovector spaces -- Definition and first gyrovector space theorems -- Gyrolines -- Gyromidpoints -- Analogies between gyromidpoints and midpoints -- Gyrogeodesics -- Möbius gyrovector spaces -- Möbius gyrolines -- Einstein gyrovector spaces -- Einstein gyrolines -- Einstein gyromidpoints and gyrotriangle gyrocentroids -- Möbius gyrotriangle gyromedians and gyrocentroids -- The gyroparallelogram -- Points, vectors, and gyrovectors -- The gyroparallelogram addition law of gyrovectors -- Gyrovector gyrotranslation -- Gyrovector gyrotranslation composition -- Gyrovector gyrotranslation and the gyroparallelogram law -- The möbius gyrotriangle gyroangles -- Exercises -- Gyrotrigonometry -- The gyroangle -- The gyrotriangle -- The gyrotriangle addition law -- Cogyrolines, cogyrotriangles, and cogyroangles -- The law of gyrocosines -- The SSS to AAA conversion law -- Inequalities for gyrotriangles -- The AAA to SSS conversion law -- The law of gyrosines -- The ASA to SAS conversion law -- The gyrotriangle defect -- The right gyrotriangle -- Gyrotrigonometry -- Gyrodistance between a point and a gyroline -- The gyrotriangle gyroaltitude -- The gyrotriangle gyroarea -- Gyrotriangle similarity -- The gyroangle bisector theorem -- The hyperbolic Steiner-Lehmus theorem -- The Urquhart theorem -- The hyperbolic Urquhart theorem -- The gyroparallelogram gyroangles -- Relativistic mechanical interpretation -- Gyro-analogies that may reveal the origin of dark matter -- Newtonian systems of particles -- Einsteinian systems of particles -- The relativistic invariant mass paradox -- Exercises |
| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 173-177) and index |
| Subject |
Geometry, Hyperbolic.
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MATHEMATICS -- Geometry -- Non-Euclidean.
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Geometry, Hyperbolic
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| Form |
Electronic book
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| ISBN |
9781598298239 |
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1598298232 |
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9783031023965 |
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303102396X |
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