# A gyrovector space approach to hyperbolic geometry by Abraham Ungar

By Abraham Ungar

The mere point out of hyperbolic geometry is sufficient to strike worry within the center of the undergraduate arithmetic and physics scholar. a few regard themselves as excluded from the profound insights of hyperbolic geometry in order that this huge, immense section of human success is a closed door to them. The project of this e-book is to open that door through making the hyperbolic geometry of Bolyai and Lobachevsky, in addition to the targeted relativity idea of Einstein that it regulates, obtainable to a much broader viewers by way of novel analogies that the trendy and unknown proportion with the classical and ordinary. those novel analogies that this ebook captures stem from Thomas gyration, that is the mathematical abstraction of the relativistic impression often called Thomas precession. Remarkably, the mere creation of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and divulges mystique analogies that the 2 geometries proportion. as a result, Thomas gyration offers upward push to the prefix "gyro" that's largely utilized in the gyrolanguage of this booklet, giving upward push to phrases like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector areas. Of specific significance is the advent of gyrovectors into hyperbolic geometry, the place they're equivalence periods that upload based on the gyroparallelogram legislation in complete analogy with vectors, that are equivalence sessions that upload in accordance with the parallelogram legislation. A gyroparallelogram, in flip, is a gyroquadrilateral the 2 gyrodiagonals of which intersect at their gyromidpoints in complete analogy with a parallelogram, that's a quadrilateral the 2 diagonals of which intersect at their midpoints. desk of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector areas / Gyrotrigonometry

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21, p. 20, to u v in Def. 9, p. 7. 146) 35 CHAPTER 2 Gyrocommutative Gyrogroups Some gyrocommutative gyrogroups give rise to gyrovector spaces, which are the framework for analytic hyperbolic geometry just as some commutative groups give rise to vector spaces, which are the framework for analytic Euclidean geometry. To pave the way for gyrovector spaces we, therefore, study gyrocommutative gyrogroups in this chapter. In gyrocommutative gyrogroups the gyrogroup operation is gyrocommutative, by Def.

4) Follows from (3) by the left gyroassociative law. Indeed, an application of the left gyroassociative law to (4) results in (3). (5) Follows from (4) since gyr[b, a]a is the unique inverse of gyr[b, a]a. (6) Follows from (5) since 0 is the unique identity element of the gyrogroup (G, ⊕). 6. THE BASIC CANCELLATION LAWS OF GYROGROUPS 19 Formalizing the results of this section, we have the following theorem. (The Two Basic Equations Theorem). Let (G, ⊕) be a gyrogroup, and let a, b ∈ G. 20. 67) in G for the unknown x is x=b a.

54) which is both commutative and associative. 54) is a binary operation between real numbers. 3. EINSTEIN GYROGROUPS 45 for all u, v ∈Vs . 55) gives rise to the Möbius gyrotriangle inequality in the following theorem. 17. (The Möbius Gyrotriangle Inequality). 57) for all u, v in a Möbius gyrogroup (Vs , ⊕). Proof. 58) 2 = γu⊕v =γ u⊕v for all u, v in the Möbius gyrogroup (Vs , ⊕). But γx = γ function of x , 0 ≤ x < s. 58) implies x , x ∈ Vs , is a monotonically increasing u⊕v ≤ u ⊕ v for all u, v in any Möbius gyrogroup (Vs , ⊕).