A Gyrovector Space Approach to Hyperbolic Geometry | Date: 04 May 2011, 07:11
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A Gyrovector Space Approach to Hyperbolic Geometry (Synthesis Lectures on Mathematics and Statistics)
By Abraham Ungar
ABSTRACT
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 fromThomas 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.
KEYWORDS
hyperbolic geometry, Analogies between Euclidean and hyperbolic geometry, Poincare ball
model of hyperbolic geometry, Beltrami-Klein ball model of hyperbolic geometry, Mobius
transformations of the complex open unit disc, Mobius addition, from Mobius to gyrogroups,
gyrogroups, gyrocommutative gyrogroups, gyrovectors, gyrovector spaces, gyrotrigonometry,
Einstein relativistic velocity addition, relativistic stellar aberration, dark
matter
* Publisher: Morgan&Claypool Publishers
* Number Of Pages: 194
* Publication Date: 2009-03-15
* ISBN-10 / ASIN: 1598298224
* ISBN-13 / EAN: 9781598298222
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