By Abraham Albert Ungar

This publication introduces for the 1st time the hyperbolic simplex as an incredible inspiration in n-dimensional hyperbolic geometry. The extension of universal Euclidean geometry to N dimensions, with N being any confident integer, leads to larger generality and succinctness in comparable expressions. utilizing new mathematical instruments, the publication 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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Accordingly, this part of the book illustrates the physical background of gyrations in hyperbolic geometry, and the usefulness of the study of special relativity theory and hyperbolic geometry under the same umbrella. Introduction 17 The study of special relativity theory and hyperbolic geometry under the same umbrella is rewarding. It reveals, for instance, that the Einstein relativistic, velocity dependent mass conforms with the Minkowskian formalism of special relativity theory, as explained in Chapter 4.

319; c) the Intersecting Secants Theorem, p. 320; and d) the Intersecting Chords Theorem, p. 359, are translated into their counterparts in hyperbolic geometry. The resulting counter-part theorems in hyperbolic geometry, respectively, are: a) the Inscribed Gyroangle Theorem, p. 304, 305; b) the Gyrotangent–Gyrosecant Theorem, p. 313, 318; c) the Intersecting Gyrosecants Theorem, p. 319; and d) the Intersecting Gyrochords Theorem, p. 358. 4. Part IV: Hyperbolic Simplices, Hyperplanes and Hyperspheres in n Dimensions.

Like vectors, a gyrovector A⊕B in an Einstein gyrovector space (Rns, ⊕, ⊗), n = 2, 3, is described graphically as a straight arrow from the tail A to the head B with gyrolength || A ⊕ B||. For the sake of comparison with its hyperbolic counterpart in Fig. 6, Fig. 5 depicts the well-known parallelogram law of vector addition, (−A + B) + (−A + C) = (−A + D). 21) In Fig. 5 we see arbitrarily selected three noncollinear points A, B, C ∈ R2, together with a fourth point D ∈ R2, which satisfies the parallelogram condition, D = B + C − A.