By Abraham A. Ungar

This is often the 1st booklet on analytic hyperbolic geometry, absolutely analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics simply as analytic Euclidean geometry regulates classical mechanics. The booklet provides a singular gyrovector house method of analytic hyperbolic geometry, totally analogous to the well known vector area method of Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence periods of directed gyrosegments that upload in keeping with the gyroparallelogram legislation simply as vectors are equivalence periods of directed segments that upload in keeping with the parallelogram legislations. within the ensuing “gyrolanguage” of the booklet one attaches the prefix “gyro” to a classical time period to intend the analogous time period in hyperbolic geometry. The prefix stems from Thomas gyration, that's the mathematical abstraction of the relativistic influence often called Thomas precession. Gyrolanguage seems to be the language one must articulate novel analogies that the classical and the fashionable during this ebook percentage. The scope of analytic hyperbolic geometry that the booklet offers is cross-disciplinary, related to nonassociative algebra, geometry and physics. As such, it really is evidently appropriate with the detailed idea of relativity and, relatively, with the nonassociativity of Einstein pace addition legislation. in addition to analogies with classical effects that the ebook emphasizes, there are extraordinary disanalogies to boot. therefore, for example, not like Euclidean triangles, the edges of a hyperbolic triangle are uniquely made up our minds through its hyperbolic angles. stylish formulation for calculating the hyperbolic side-lengths of a hyperbolic triangle by way of its hyperbolic angles are awarded within the ebook. The publication starts off with the definition of gyrogroups, that's totally analogous to the definition of teams. Gyrogroups, either gyrocommutative and nongyrocommutative, abound in workforce concept. strangely, the possible structureless Einstein speed addition of distinctive relativity seems to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, a few gyrocommutative gyrogroups of gyrovectors develop into gyrovector areas. The latter, in flip, shape the atmosphere for analytic hyperbolic geometry simply as vector areas shape the atmosphere for analytic Euclidean geometry. through hybrid ideas of differential geometry and gyrovector areas, it really is proven that Einstein (Möbius) gyrovector areas shape the surroundings for Beltrami–Klein (Poincaré) ball types of hyperbolic geometry. eventually, novel purposes of Möbius gyrovector areas in quantum computation, and of Einstein gyrovector areas in particular relativity, are awarded.

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**Example text**

19) Let a and b be any two elements of G. 19). 19) share, the gyroautomorphism gyr[a, eb]is called the cogyroautomorphism associated with the gyroautomorphism gyr[a, b]. 21) of the group identity (-a b) (-b + c ) = --a + c. + + Let (G,+) be a gyrogroup. 21) for all a, b, c E G. Proof. 12 (The Gyrotranslation Theorem, I). Let (G, +) be a gyrogroup. 23) for all a , b, c E G. Proof. 21) and a left cancellation. 13 (Gyropolygonal Gyroaddition of Adjacent Sides). Let (G,+) be a gyrogroup, and let ( a , b ) , a , b E G be a pair of two elements of G.

48, p. 280. It presents an extraordinary unexpected hidden beauty that analytic hyperbolic geometry reveals. We thus encounter here one of the remarkable interrelations between truth and beauty, which are abound in the area of analytic hyperbolic geometry. 5 Applications in Quantum and Special Relativity Theory The applicability in physics of the gyrovector space approach to hyperbolic geometry is demonstrated in Chaps. 9 and 10. Chapter 9 demonstrates that Bloch vector of quantum computation theory is, in fact, a gyrovector rather than a vector.

Thus, for instance, gyrolines, gyroangles, and gyrotriangles are associated with their corresponding dual counterparts, cogyrolines, cogyroangles, and cogyrotriangles. Gyrolanguage abuse must be done with care, as the example of the gyrocosine function in Fig. 2 indicates. The definition of the gyrocosine of a gyroangle is presented in Fig. 2. We cannot view it as the “hyperbolic” cosine of a hyperbolic angle since the term “hyperbolic cosine” is already in use in a different sense. Abusing notation, we use the same notation for the trigonometric functions and their gyro-counterparts.