Analytic Hyperbolic Geometry in N Dimensions: An Introduction 1st Edition by Abraham Albert Ungar (PDF)

Description

The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry. Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language that sheds natural light on hyperbolic geometry and special relativity. Several authors have successfully employed the author’s gyroalgebra in their exploration for novel results. Françoise Chatelin noted in her book, and elsewhere, that the computation language of Einstein described in this book plays a universal computational role, which extends far beyond the domain of special relativity. This book will encourage researchers to use the author’s novel techniques to formulate their own results. The book provides new mathematical tools, such as hyperbolic simplexes, for the study of hyperbolic geometry in n dimensions. It also presents a new look at Einstein’s special relativity theory.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book has several forerunners in which the author, A.A. Ungar, developsnovel tools and techniques to study Analytic Hyperbolic Geometry in a way guidedby analogies with tools and techniques to study Analytic Euclidean Geometry. Infact, Ungar’s novel tools and techniques result from the adaptation of well-knowntools and techniques in Euclidean geometry for use in hyperbolic geometry. Specifically,(1) Cartesian coordinates, (2) barycentric coordinates, (3) trigonometry, and(4) vector algebra, commonly used in the study of Euclidean geometry, are adaptedfor use in hyperbolic geometry as well. The use of these tools and techniques enablesfor the first time several important theorems in Euclidean geometry to be translatedinto their hyperbolic counterparts. Specifically, the book presents the translationinto hyperbolic geometry of the following well-known theorems in Euclidean geometry:(1) the Inscribed Angle Theorem, (2) the Tangent-Secant Theorem, (3) theIntersecting Secants Theorem, and (4) the Intersecting Chords Theorem. Moreover,in the study of Euclidean geometry in higher dimensions one commonly assigns aso called Cayley-Menger matrix to each simplex. In full analogy, the author assignsin the book a gamma matrix to each hyperbolic simplex, enabling novel results inhigher-dimensional hyperbolic geometry to be discovered. I strongly recommendthe book for all students and researchers who are interested in the study of hyperbolicgeometry by means of novel tools and techniques. Moreover, I stronglyrecommend the book for everyone who loves elegant algebra and who is familiarwith the basic elements of vector space approach to Euclidean geometry.

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