By Abraham Albert Ungar
The suggestion of the Euclidean simplex is critical within the research of n-dimensional Euclidean geometry. This ebook introduces for the 1st time the concept that of hyperbolic simplex as an immense idea in n-dimensional hyperbolic geometry.
Following the emergence of his gyroalgebra in 1988, the writer crafted gyrolanguage, the algebraic language that sheds typical gentle on hyperbolic geometry and detailed relativity. numerous authors have effectively hired the author’s gyroalgebra of their exploration for novel effects. Françoise Chatelin famous in her booklet, and somewhere else, that the computation language of Einstein defined during this e-book performs a common computational position, which extends a ways past the area of certain relativity.
This ebook will inspire researchers to take advantage of the author’s novel suggestions to formulate their very own effects. The booklet offers new mathematical tools, such as hyperbolic simplexes, for the examine of hyperbolic geometry in n dimensions. It also presents a brand new examine Einstein’s targeted relativity idea.
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Extra info for Analytic Hyperbolic Geometry in N Dimensions: An Introduction
2), p. 24, and 2. 49), p. 69) for all u, v ∈ Rns where, by definition, 2⊗v = v⊕v. 23), p. 84), p. 194. 10 First Gyrogroup Properties While it is clear how to define a right identity and a right inverse in a gyrogroup, the existence of such elements is not presumed. Indeed, the existence of a unique identity and a unique inverse, both left and right, is a consequence of the gyrogroup axioms, as the following theorem shows, along with other immediate results about gyrogroups. 17 (First Gyrogroup Properties).
59), gyrations keep the norm invariant. z = 0. 53) with w = z, gyr[u, v]z = z. 63) we say that the gyration axis in Rn of the gyration gyr[u, v] : Rn → Rn, generated by u, v ∈ Rns, 38 Analytic Hyperbolic Geometry in N Dimensions is parallel to the vector z. 65) x 0, for any coefficients cu, cv ∈ R, excluding cu = cv = 0. 65). Moreover, we have the following result. 7 (Gyration–Thomas Precession Angle). Let u, v, x ∈ R ns be relativistically admissible velocities such that u −v (so that u⊕v 0).
Specifically, the classical tools are: a) Cartesian coordinates (in Euclidean geometry); b) Barycentric coordinates; c) trigonometry; and d) vector algebra, 16 Analytic Hyperbolic Geometry in N Dimensions and their respective hyperbolic counterparts are: a) Cartesian coordinates (in hyperbolic geometry); b) gyrobarycentric coordinates; c) gyrotrigonometry; and d) gyrovector gyroalgebra. 3. Part III: Hyperbolic Triangles and Circles. Part III of the book, Chapters 8–9, employs the tools developed in Part II for the discovery of properties of hyperbolic triangles (gyrotriangles) and hyperbolic circles (gyrocircles).