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Analytic Hyperbolic Geometry in N Dimensions: An Introduction, by Abraham Albert Ungar
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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. Fran�oise 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.
- Sales Rank: #6407741 in Books
- Published on: 2014-12-17
- Original language: English
- Number of items: 1
- Dimensions: 9.10" h x 1.40" w x 6.10" l, .0 pounds
- Binding: Hardcover
- 622 pages
Review
"Anyone who is concerned with hyperbolic geometry should use this wonderful and comprehensive book as a helpful compendium."
―Zentralblatt MATH 1312
Most helpful customer reviews
3 of 3 people found the following review helpful.
Not Just for the Geometric Community
By MathReviewer
This book has several forerunners in which the author, A.A. Ungar, develops
novel tools and techniques to study Analytic Hyperbolic Geometry in a way guided
by analogies with tools and techniques to study Analytic Euclidean Geometry. In
fact, Ungar’s novel tools and techniques result from the adaptation of well-known
tools 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 adapted
for use in hyperbolic geometry as well. The use of these tools and techniques enables
for the first time several important theorems in Euclidean geometry to be translated
into their hyperbolic counterparts. Specifically, the book presents the translation
into hyperbolic geometry of the following well-known theorems in Euclidean geometry:
(1) the Inscribed Angle Theorem, (2) the Tangent-Secant Theorem, (3) the
Intersecting Secants Theorem, and (4) the Intersecting Chords Theorem. Moreover,
in the study of Euclidean geometry in higher dimensions one commonly assigns a
so called Cayley-Menger matrix to each simplex. In full analogy, the author assigns
in the book a gamma matrix to each hyperbolic simplex, enabling novel results in
higher-dimensional hyperbolic geometry to be discovered. I strongly recommend
the book for all students and researchers who are interested in the study of hyperbolic
geometry by means of novel tools and techniques. Moreover, I strongly
recommend the book for everyone who loves elegant algebra and who is familiar
with the basic elements of vector space approach to Euclidean geometry.
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