From Pythagoras To Einstein: m The Hyperbolic Pythagorean Theore
نویسنده
چکیده
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 completing the long path from Pythagoras to Einstein. Following the pioneering work of Varicak it is well known that relativistic v
منابع مشابه
Sums of Squares in Function Fields of Quadrics and Conics
For a quadric Q over a real field k, we investigate whether finiteness of the Pythagoras number of the function field k(Q) implies the existence of a uniform bound on the Pythagoras numbers of all finite extensions of k. We give a positive answer if the quadratic form that defines Q is weakly isotropic. In the case where Q is a conic, we show that the Pythagoras number of k(Q) is 2 only if k is...
متن کاملThe Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces
The only justification for the Einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in Einstein addition remained for a long time a mystery to be conquered. Accordingly, the aim of this expository article is to present (i) the Einstein relativistic vector addition, (ii) the resulting Einstein scalar multiplication, (iii) th...
متن کاملConsistency and Pythagoras
Pythagorean win share has been one of the fundamental contributions to Sabermetrics. Several hundred articles, both academic and non-academic, have explored variations on Bill James’ original formula and its fit to empirical data. This paper considers a variation that is previously unexplored on any systematic level, consistency. After discussing several important contributions to the line of l...
متن کاملOn the Pythagoras numbers of real analytic set germs
We show that: (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive ...
متن کاملNon-euclidean Pythagorean Triples, a Problem of Euler, and Rational Points on K3 Surfaces
We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler’s work on the second problem with a mo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998