By Abraham Albert Ungar

This e-book introduces for the 1st time the hyperbolic simplex as a massive notion in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any confident integer, leads to better generality and succinctness in comparable expressions. utilizing new mathematical instruments, the booklet demonstrates that this is often 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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**Sample text**

According to Kiss [62], Lajos Dávid drew attention in a 1924 series of articles in Italian journals to the precursory role which János Bolyai played in the constructions of Einstein’s relativity theory. I. Miller [79, p. 266], one of the first demonstrations that non-Euclidean geometry could be used to present concisely results of relativity theory was obtained by Sommerfeld in 1909 [101] when he was led to the result that relativistically admissible velocities add according to a spherical geometry.

Soon later, the link between Einstein’s special theory of relativity and hyperbolic geometry was discovered and developed during the period 1908–1912 by Varičak, Robb, Wilson and Lewis, and Borel [143]. The subsequent major development that followed 1912 appeared about 80 years later, in 2001 [119]. Following the emergence of gyroalgebra since 1988 [111, 112, 113], the author has crafted gyrolanguage, the algebraic language that sheds natural light on hyperbolic geometry and special relativity, in several books [119, 122, 129, 131, 133, 134], [144, 89].

Among outstanding exceptions we note the relativity physics books by Fock [39] and by Sexl and Urbantke [96]. ) be the Euclidean n-space, n = 1, 2, 3, . , equipped with the common vector addition, +, and inner product, .. The home of all n-dimensional Einsteinian velocities is the s-ball Rns = {v ∈ Rn : ||v|| < s}. 1) 24 Analytic Hyperbolic Geometry in N Dimensions The s-ball Rns is the open ball of radius s, centered at the origin of Rn, consisting of all vectors v in Rn with magnitude ||v|| smaller than s.