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 element of human fulfillment is a closed door to them. The project of this publication is to open that door via making the hyperbolic geometry of Bolyai and Lobachevsky, in addition to the specified relativity concept of Einstein that it regulates, available to a much broader viewers when it comes to novel analogies that the fashionable and unknown proportion with the classical and generic. those novel analogies that this ebook captures stem from Thomas gyration, that is the mathematical abstraction of the relativistic influence referred to as Thomas precession. Remarkably, the mere creation of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and divulges mystique analogies that the 2 geometries proportion. for that reason, Thomas gyration provides upward thrust to the prefix "gyro" that's broadly utilized in the gyrolanguage of this ebook, 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 sessions that upload based on the gyroparallelogram legislations 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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2, p. 35. Hence, it is valid in gyrocommutative gyrogroups. 30) are independent of c. 3, p. 112. 30) in commutative groups are immediate. However, they play important role in the translation of vectors in Euclidean geometry. In full analogy, we will ﬁnd in Chaps. 11 in gyrocommutative gyrogroups play important role in the hyperbolic translation of hyperbolic vectors in hyperbolic geometry, where they are called gyrovectors. Indeed, classically, vectors are invariant under Euclidean translations.

Follows from (10) by Def. 9, p. 7, of the gyrogroup cooperation . 41, p. 93. 2. 2 43 MÖBIUS GYROGROUPS As suggested in Sec. 3, Möbius addition ⊕M in the ball Vs of any real inner product space gives rise to a gyrocommutative gyrogroup (Vs , ⊕M ) called a Möbius gyrogroup. It is assumed that the reader is familiar with real inner product spaces, the formal deﬁnition of which will be presented in Def. 1 in Chap. 3, p. 55. The deﬁnition of Möbius addition, ⊕M , in the ball follows. (Möbius Addition in the Ball).

GYROCOMMUTATIVE GYROGROUPS Proof. 19, p. 16, with a replaced by −a. Hence, it is valid in nongyrocommutative gyrogroups as well. 2, p. 35. Hence, it is valid in gyrocommutative gyrogroups. 30) are independent of c. 3, p. 112. 30) in commutative groups are immediate. However, they play important role in the translation of vectors in Euclidean geometry. In full analogy, we will ﬁnd in Chaps. 11 in gyrocommutative gyrogroups play important role in the hyperbolic translation of hyperbolic vectors in hyperbolic geometry, where they are called gyrovectors.