By Abraham Albert Ungar
This ebook introduces for the 1st time the hyperbolic simplex as a major notion in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any optimistic integer, leads to better generality and succinctness in similar expressions. utilizing new mathematical instruments, the ebook demonstrates that this can be additionally the case with analytic hyperbolic geometry. for instance, the writer analytically determines the hyperbolic circumcenter and circumradius of any hyperbolic simplex.
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Extra info for Analytic Hyperbolic Geometry in N Dimensions : An Introduction
50) for all u, v, w ∈ Rns. 50) that the departure of Einstein addition, ⊕, from commutativity and associativity is strictly controlled by gyrations. 50) present important gyration identities. 107), p. 47. Einstein addition plays in hyperbolic geometry the role that vector addition plays in the vector space approach to Euclidean geometry. Einstein addition is more complex than vector addition, but richer in structure. Hence, a computer algebra system, like Mathematica or Maple, is an indispensable tool.
The relativistic, velocity dependent mass plays an important role since it enables the adaptation of barycentric coordinates, commonly used in Euclidean geometry, for use in hyperbolic geometry. The study of analytic hyperbolic geometry in n dimensions, guided by analogies with classical results, thus begins with the study of Einstein gyrogroups and gyrovector spaces in Part I. 6 Einstein Addition vs. 1 Introduction Einstein’s addition law of three-dimensional relativistically admissible velocities is the corner stone  of Einstein’s three-vector formalism of the special theory of relativity that he founded in 1905 [29, 71].
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.
Analytic Hyperbolic Geometry in N Dimensions : An Introduction by Abraham Albert Ungar