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From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem
, 1998
"... A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as the "Einstein sum" of the squares of ..."
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A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as the "Einstein sum" of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein.
The Relativistic CompositeVelocity Reciprocity Principle
, 2000
"... Gyrogroup theory [A.A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), pp. 881951] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector add ..."
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Cited by 1 (1 self)
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Gyrogroup theory [A.A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), pp. 881951] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the Relativistic CompositeVelocity Reciprocity Principle. The breakdown of commutativity in the Einstein velocity addition # of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic compositevelocity reciprocity principle, since seemingly (i) on one hand the velocity reciprocal to the composite velocity u#v is (u#v) and (ii) on the other hand it is (v)#(u). But, (iii) (u#v) #= (v)#(u). We remove the confusion in (i), (ii) and (iii) by employing the gyrocommutative gyrogroup structure of Einstein's addition and, subsequ...