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- Abraham Albert Ungar
- A Gyrovector Space Approach to Hyperbolic…
- 2009

Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincaré ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincaré ball model of 3-dimensional hyperbolic geometry is becoming… (More)

We show that the algebra of the group SL(2; C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. The superiority of the use of the gyrogroup formalism over the use of theSL(2; C) formalism for dealing with the Lorentz group in some cases is indicated by (i) the… (More)

- Tuval Foguel, Abraham A. Ungar
- 2000

Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity… (More)

- Tuval Foguel, Abraham A. Ungar
- 2007

An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are grouplike structures which rst arose in the study of Einstein's… (More)

- Abraham A. Ungar
- 1998

new form of the Hyperbolic Pythagorean Theorem, which has a striking t e intuitive appeal and offers a strong contrast to its standard form, is presented. I xpresses the square of the hyperbolic length of the hypotenuse of a hyperbolic s o right angled triangle as the "Einstein sum" of the squares of the hyperbolic length f the other two sides, Fig. 1, thus… (More)

- Abraham Albert Ungar
- The American Mathematical Monthly
- 2008

- Abraham Albert Ungar
- Quantum Information & Computation
- 2002

- Abraham Albert Ungar
- Computers & Mathematics with Applications
- 2007

Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean… (More)

- Abraham A. Ungar
- 2010

Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the… (More)