By Abraham Albert Ungar

This e-book introduces for the 1st time the hyperbolic simplex as an enormous suggestion in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any optimistic integer, leads to larger generality and succinctness in comparable expressions. utilizing new mathematical instruments, the booklet 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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**Sample text**

Gamma factors γij of gyrosides play an important role. 3. between the parallelogram law of vector addition and the gyroparallelogram law of gyrovector addition, illustrated in Figs. 6; and 4. between barycentric coordinates and gyrobarycentric coordinates, studied in Chapter 5, and employed in Chapters 5–12. The formal link between Einstein addition and the differential geometry that underlies the Beltrami-Klein model of the hyperbolic geometry of Lobachevsky and Bolyai is presented in Sect. 4.

463, lim s2(N −1) Det ΓN = − s→∞ 1 Det MN. 25) Accordingly, the gamma determinant, Det ΓN, that we use in the study of higher dimensional hyperbolic geometry is the hyperbolic counterpart of the well-known Cayley–Menger determinant, Det MN. Yet, undoubtedly, our gamma matrix ΓN appears to be more elegant than its Euclidean counterpart, the Cayley– Menger matrix MN. By discovering the hyperbolic counterpart of Cayley–Menger determinant, we pave the road to the study of analytic hyperbolic geometry in n dimensions, guided by analogies with the common study of analytic Euclidean geometry in n dimensions.

Just after Varičak’s first exposé of the non-Euclidean style ([140], 1910), Sommerfeld completed his signal work on the four-dimensional vector calculus for the Annalen der Physik. In a footnote to his work, Sommerfeld remarked that the geometrical relations he presented in terms of three real and one imaginary coordinate could be reinterpreted in terms of non-Euclidean geometry. The latter approach, Sommerfeld cautioned in [102, p. 752], could “hardly be recommended”. Furthermore, Walter notes in [143] that following the competition between the two geometrical approaches to relativity physics: Minkowski neither mentioned the [Einstein] law of velocity addition, nor expressed it in formal terms.