The Principle of Relativity: From Ungar’s Gyrolanguage for Physics to Weaving Computation in Mathematics

Document Type: Special Issue: International Conference on Architecture and Mathematics

Author

Universite Toulouse 1 Capitole,

Abstract

‎This paper extends the scope of algebraic computation based on a non standard $\times$ to ‎the more basic case of a non standard $+$‎, ‎where standard means associative ‎and commutative‎. ‎Two physically meaningful examples of a non standard $+$ are ‎provided by the observation of motion in Special Relativity‎, ‎from either ‎outside (3D) or inside (2D or more)‎, ‎We revisit the ``gyro''-theory of Ungar to present ‎the multifaceted information processing which is created by a metric cloth $W$‎, ‎a relating computational construct framed in a normed vector space $V$‎, ‎and based ‎on a non standard addition denoted $\pluscirc$ whose commutativity and associativity ‎are ruled (woven) by a relator‎, ‎that is a map which assigns to each pair of admissible vectors ‎in $V$ an automorphism in $\Aut W$‎. ‎Special attention is given to the case where the relator is ‎directional‎.

Keywords

Main Subjects


1. M. Allen, The hype over hyperbolic browsers, Online 26 (2002) 20–28.

2. J. P. Auffray, Einstein et Poincaré. Sur les traces de la relativité, 2ème édition, Le Pommier, Paris, 2005.


3. E. Borel, La cinématique dans la théorie de la relativité ,
CR Acad. Sc. Paris, 157 (1913) 703–705.


4. E. Borel,
Introduction géométrique à quelques théories physiques, GauthierVillars, Paris, 1914.


5. C. Calude, F. Chatelin, A dialogue about qualitative computing,
Bull. EATCS 101 (2010) 29–41.


6. F. Chatelin, On relating computation,
Cerfacs Tech. Rep. TR/PA/11/37 (2011) 30 pp.


7. ‎F‎. ‎Chatelin‎, ‎Qualitative Computing‎: ‎a Computational ‎
Journey into Nonlinearity‎, World Scientific‎, ‎Singapore‎, 2012‎.

 

8. F. Chatelin, A computational journey into the mind, Nat. Comput. 11(1) (2012) 67–79.


9. F. Chatelin, About weaving computation in 2 and 3 real dimensions,
Int. J. Unconv. Comp. 9 (2012) 37-59.


10. F. Chatelin, Computation in Mind, in:
Philosophy of Mind, R. J. Jenkins, W. E. Sullivan (Eds.), Nova Science, Hauppage NY, 2012.


11. F. Chatelin, Qualitative Computing, in:
The Human Face of Computing, C. Calude (Ed.), IC Press, London, 2015.


12. F. Chatelin,
Numbers in Mind: The Ways of Multiplication, World Scientific, Singapore, To appear, 2017.


13. T. Damour, What is missing from Minkowski’s “Raum und Zeit” lecture,
Ann. Phys. 17 (2008) 619–630.


14. A. Einstein,
Geometrie und Erfahrung, Erweiterte Fassung des Festvortrages Gehalten an der Preussischen Akademie der Wissenschaften zu Berlin am 27. Januar 1921.


15.
‎D‎. ‎Henderson‎, ‎D‎. ‎Taimina‎, ‎Crocheting the hyperbolic plane‎, Math‎. ‎Intelligencer 23 (2001) 17--28‎.


16. K. Lamotke,
Les Nombres, Vuibert, Paris, 1998.


17. J. Lamping, R. Rao, P. Pirolli, A focus+context technique based on hyperbolic geometry for visualizing large hierarchies,
Proceedings of the ACM SIGCHI Conference on Human Factors in Computing Systems, Denver, May 1995, ACM, 1–8.

18. R. Luneburg, The metric binocular visual space, J. Opt. Soc. Am. 40 (1950) 627–640.


19.
‎J‎. ‎C‎. ‎Maxwell‎, ‎Remarks on the mathematical classification‎‎
of physical quantities‎, ‎ Proc‎. ‎London Math‎. ‎Soc. 3 (1871) 224-233‎.

20. H. Minkowsky, Raum und Zeit, lecture delivered at the 80th Congress of
Naturalists, Köln, 21 September, 1908.


21. M. Paty, Physical geometry and special relativity, Einstein et PoincarÃľ, in:
1830-1930: un siècle géométrie, de C.F Gauss et B. Riemann à H. Poincaré et E. Cartan. Epistémologie, histoire et mathématiques, L. Boi, D. Flament, J. M. Salanski (Eds.), pp. 126-149, Springer Verlag, Berlin, 1992.


22. H. Poincaré,
La science et l’hypothèse, Flammarion, Paris, 1902.


23. H. Poincaré, Sur la dynamique de l’électron,
CR. Acad. Sc. Paris 140 (1905) 1504–1508.


24.
‎L‎. ‎Pyenson‎, ‎Relativity in late Wilhelmian Germany‎:‎
the appeal to a pre-established harmony between mathematics and physics‎, ‎ Arch‎. ‎Hist‎. ‎Exact Sci. 27 (1982) 137-159‎.


25. L. Silberstein,
The Theory of Relativity, MacMillan, London, 1914.


26. L. Thomas, The motion of the spinning electron,
Nature 117 (1926) 514.


27. A. A. Ungar, The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities,
Appl. Math. Lett 1 (1988) 403-405.


28. A. A. Ungar, Thomas precession and its associated group-like structure,
Amer. J. Phys. 59 (1991) 824–834.


29. A. A. Ungar, On the appeal to a pre-established harmony between pure mathematics and relativity physics,
Found. Phys. Lett. 16 (2003) 1–23.


30. A. A. Ungar,
Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity, World Scientific, Singapore, 2008.


31. A. A. Ungar, Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky, in:
Nonlinear Analysis, P. Pardalos, P. Georgiev, H. Srivastava (Eds.), Springer Optimization and Its Applications, Vol 68, Springer, New York, NY, 2012.


32. D. Urribarri, S. Castro, S. Martig, Gyrolayout: a hyperbolic level-of-detail layout,
J. Univ. Comp. Sc. 19 (2013) 132–156.


33. V. Varičak, Anwendung der lobatchefskjschen geometrie in der relativtheorie,
Physikal. Zeitsch. 10 (1910) 826–829.

34. S. Walter, The Non-Euclidean Style of Minkowskian Relativity, in: The Symbolic Universe, J. Gray (Ed.), pp. 91-127, Oxford University Press, 1999.


35.
‎E‎. ‎Wigner‎,
The unreasonable effectivness of mathematics in the natural sciences‎, Comm‎. ‎Pure‎. ‎Appl‎. Math. 13 (1960) 1-14‎.