EAN: 9783031012686

SYNTHESIS LECTURES ON MATHEMATICS & STATISTICS ...

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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