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 component to human fulfillment is a closed door to them. The challenge of this e-book is to open that door via making the hyperbolic geometry of Bolyai and Lobachevsky, in addition to the precise relativity conception of Einstein that it regulates, available to a much wider viewers by way of novel analogies that the trendy and unknown proportion with the classical and conventional. those novel analogies that this e-book captures stem from Thomas gyration, that is the mathematical abstraction of the relativistic impact often called Thomas precession. Remarkably, the mere advent 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 thrust to the prefix "gyro" that's greatly utilized in the gyrolanguage of this ebook, giving upward thrust 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 in line with the gyroparallelogram legislations in complete analogy with vectors, that are equivalence sessions that upload in response to 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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We will find in this section that Einstein and Möbius gyrogroups are isomorphic. 91) so that if v is an element of an Einstein gyrogroup (Vs , ⊕E ) we have 2⊗v = v⊕E v, and if v is an element of a Möbius gyrogroup (Vs , ⊕M ) we have 2⊗v = v⊕M v. The unique solution to the equation 2⊗v = u for the unknown v in an Einstein gyrogroup (Vs , ⊕E ) is the Einstein half, u = (1/2)⊗v, given by the equation 1 2 ⊗v = γv v. 93) =v. 94) for all ue , ve ∈ (Vs , ⊕E ), and all um , vm ∈ (Vs , ⊕M ). 97) 52 CHAPTER 2.

A + b) = −a − b . 1) (The Gyroautomorphic Inverse Property). A gyrogroup is gyrocommutative if and only if it possesses the gyroautomorphic inverse property. 2. Proof. Let (G, +) be a gyrogroup possessing the gyroautomorphic inverse property. 34), p. 13(12), p. 11, to the gyrocommutative law (G6) in Def. 8, p. 2) = gyr[a, b](b + a) for all a, b ∈ G. 34), p. 3) and inverting the gyro-sign we recover the gyroautomorphic inverse property, −(b + a) = −b − a for all a, b ∈ G. 4) ✷ 36 CHAPTER 2. 3.

Proof. 32, p. 30, a b = −(−b − gyr[b, −a]a) . 5) But by definition, b a = b + gyr[b, −a]a . 6). 35, p. 33. 8) holds for all b, c∈G. 2, is equivalent to the gyrocommutativity of the gyrogroup (G, +). 7) holds for all a, b∈G if and only if the gyrogroup (G, +) is ✷ gyrocommutative. 4. Let (G, +) be a gyrocommutative gyrogroup. 11) for all a, b, c ∈ G. Proof. 21, p. 38), p. 12) = gyr[a, b + c]gyr[b, c] . 1. 5. d = (b c) − a . 14) for all a, b, c ∈ G. Proof. 11), along with an application of the right and the left loop property, we have gyr[a , b + a ]gyr[b + a , c ] = gyr[a , b + c ]gyr[b + c , c ] .

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