By Abraham Albert Ungar
The inspiration of the Euclidean simplex is necessary within the research of n-dimensional Euclidean geometry. This e-book introduces for the 1st time the concept that of hyperbolic simplex as an incredible idea in n-dimensional hyperbolic geometry.
Following the emergence of his gyroalgebra in 1988, the writer crafted gyrolanguage, the algebraic language that sheds average gentle on hyperbolic geometry and targeted relativity. numerous authors have effectively hired the author’s gyroalgebra of their exploration for novel effects. Françoise Chatelin famous in her ebook, and in different places, that the computation language of Einstein defined during this publication performs a common computational position, which extends a ways past the area of specified relativity.
This booklet will motivate researchers to exploit the author’s novel ideas to formulate their very own effects. The booklet presents new mathematical tools, such as hyperbolic simplexes, for the research of hyperbolic geometry in n dimensions. It also presents a brand new examine Einstein’s distinctive relativity concept.
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Extra info for Analytic Hyperbolic Geometry in N Dimensions: An Introduction
The resulting gyroalgebra stems from the notions of a) the gyrogroup, which is a natural generalization of the group concept in algebra; and b) the gyrovector space, which is a natural generalization of the vector space concept in algebra. It is demonstrated that gyroalgebra regulates Einstein addition and, hence, sheds a natural light on the special relativity theory of Einstein and on its underlying hyperbolic geometry of Lobachevsky and Bolyai. As such, gyroalgebra is used extensively in the book in the study of analytic hyperbolic geometry in n dimensions.
There is an element 0 ∈ G satisfying axiom (G1) such that for each a ∈ G there is an element −a ∈ G, called a left inverse of a, satisfying (G2) −a + a = 0. Moreover, the binary operation obeys the associative law (G3) (a + b) + c = a + (b + c) for all a, b, c ∈ G. Groups are classified into commutative and noncommutative groups. 11 (Commutative Groups). A group (G, +) is commutative if its binary operation obeys the commutative law (G6) a+b=b+a for all a, b ∈ G. 12 (Subgroups). A subset H of a group (G, +) is a subgroup of G if it is nonempty, and H is closed under group compositions and inverses in G, that is, x, y ∈ H implies x + y ∈ H and −x ∈ H.
We naturally use the abbreviation u v = u⊕(−v) for Einstein subtraction, so that, for instance, v v = 0 and v = 0 v = −v. 8) and for all u, v in the ball Rns, in full analogy with vector addition and subtraction in Rn. 8) is called the left cancellation law of Einstein addition. 8) since, in general, (u⊕v) v u. 112), p. 49. 10). Rns. 11) are γ γ u·v = −1 + u⊕v = 1 − u⊕v . 10) that signaled the emergence of the link between hyperbolic geometry and special relativity. It was first studied by Sommerfeld  and Varičak [139, 140] in terms of rapidities, a term coined by Robb .
Analytic Hyperbolic Geometry in N Dimensions: An Introduction by Abraham Albert Ungar