By Abraham Albert Ungar

This e-book introduces for the 1st time the hyperbolic simplex as an incredible thought 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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**Example text**

9. If a⊕b = a⊕c, then b = c (general left cancellation law; see Item (9) below). gyr[0, a] = I for any left identity 0 in G. gyr[x, a] = I for any left inverse x of a in G. gyr[a, a] = I There is a left identity which is a right identity. There is only one left identity. Every left inverse is a right inverse. There is only one left inverse, a, of a, and ( a) = a. The Left Cancellation Law: a⊕(a⊕b) = b. 70) 10. 71) 11. gyr[a, b]0 = 0. 12. gyr[a, b]( x) = gyr[a, b]x. 13. gyr[a, 0] = I. Proof. 1. Let x be a left inverse of a corresponding to a left identity, 0, in G.

21],[67]. Einstein addition captures the notion of the Einsteinian velocity vector, u = A⊕B, called a velocity gyrovector. Interestingly, it is Einstein coaddition that captures the hyperbolic parallelogram (gyroparallelogram) law, u ⊞ v, for the composition of Einsteinian velocity gyrovectors. Experimental evidence that supports the physical significance of Einstein gyroparallelogram law of velocity addition is provided by the relativistic interpretation of the cosmological stellar aberration phenomenon, as explained in Sect.

53) the domain of w from Rns to Rn, gyrations gyr[u, v] are expanded from maps of Rns to linear maps of Rn for any u, v ∈ Rns, gyr[u, v] : Rn → Rn. In each of the three special cases when (i) u = 0, or (ii) v = 0, or (iii) u and v are parallel in Rn, u||v, we have Au + Bv = 0 so that gyr[u, v] is trivial. 54) u||v, Rns, for all u, v ∈ u||v in the third equation, and all w ∈ Rn. 56) Rns, for all u, v ∈ where I denotes the trivial map, also called the identity map. Hence, gyrations are invertible linear maps of Rn, the inverse, gyr−1[u, v], of gyr[u, v] being gyr[v, u].