Share This Article:

The Mathematics of Harmony, Hilbert’s Fourth Problem and Lobachevski’s New Geometries for Physical World

Abstract Full-Text HTML Download Download as PDF (Size:1330KB) PP. 457-494
DOI: 10.4236/jamp.2014.27056    3,570 Downloads   4,494 Views   Citations

ABSTRACT

We suggest an original approach to Lobachevski’s geometry and Hilbert’s Fourth Problem, based on the use of the “mathematics of harmony” and special class of hyperbolic functions, the so-called hyperbolic Fibonacci l-functions, which are based on the ancient “golden proportion” and its generalization, Spinadel’s “metallic proportions.” The uniqueness of these functions consists in the fact that they are inseparably connected with the Fibonacci numbers and their generalization Fibonacci l-numbers (l > 0 is a given real number) and have recursive properties. Each of these new classes of hyperbolic functions, the number of which is theoretically infinite, generates Lobachevski’s new geometries, which are close to Lobachevski’s classical geometry and have new geometric and recursive properties. The “golden” hyperbolic geometry with the base

(“Bodnar’s geometry) underlies the botanic phenomenon of phyllotaxis. The “silver” hyperbolic geometry with the base  has the least distance to Lobachevski’s classical geometry. Lobachevski’s new geometries, which are an original solution of Hilbert’s Fourth Problem, are new hyperbolic geometries for physical world.


Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Stakhov, A. and Aranson, S. (2014) The Mathematics of Harmony, Hilbert’s Fourth Problem and Lobachevski’s New Geometries for Physical World. Journal of Applied Mathematics and Physics, 2, 457-494. doi: 10.4236/jamp.2014.27056.

References

[1] Wikipedia, the Free Encyclopaedia (2014) Hyperbolic Geometry.
http://en.wikipedia.org/wiki/Hyperbolic_geometry
[2] Kolmogorov, A.N. (1991) Mathematics in Its Historical Development (in Russian). Nauka, Moscow.
[3] Stakhov, A.P. (2009) The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science. International Publisher “World Scientific”, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai.
[4] Stakhov, A.P. (2008) The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics. Congressus Numerantium, CXCIII, 5-48.
[5] Wikipedia, the Free Encyclopaedia (2014) Fullerene. http://en.wikipedia.org/wiki/Fullerene
[6] Wikipedia, the Free Encyclopaedia (2014) Quasi-Crystal. http://en.wikipedia.org/wiki/Quasicrystal
[7] Soroko, E.M. (1984) Structural Harmony of Systems (in Russian). Publishing House “Nauka i Tekhnika,” Minsk.
[8] Bodnar, O.Y. (1994) The Golden Section and Non-Euclidean Geometry in Nature and Art (in Russian). Publishing House “Svit”, Lvov.
[9] Bodnar, O. (2010) Dynamic Symmetry in Nature and Architecture. Visual Mathematics, 12, 4.
http://www.mi.sanu.ac.rs/vismath/BOD2010/index.html
[10] Bodnar, O. (2011) Geometric Interpretation and Generalization of the Non-Classical Hyperbolic Functions. Visual Mathematics, 13, 2. http://www.mi.sanu.ac.rs/vismath/bodnarsept2011/SilverF.pdf
[11] Bodnar, O. (2012) Minkovski’s Geometry in the Mathematical Modeling of Natural Phenomena. Visual Mathematics, 14, 1. http://www.mi.sanu.ac.rs/vismath/bodnardecembar2011/mink.pdf
[12] Wikipedia, the Free Encyclopedia (2014) Hilbert’s Tenth Problem.
http://en.wikipedia.org/wiki/Hilbert’s_tenth_problem
[13] Petoukhov, S.V. (2006) Metaphysical Aspects of the Matrix Analysis of Genetic Code and the Golden Section. Metaphysics: Century XXI (in Russian). Publishing House “BINOM”, Moscow, 216-250.
[14] Stakhov, A.P. and Tkachenko, I.S. (1993) Hyperbolic Fibonacci Trigonometry (in Russian). Reports of the Ukrainian Academy of Sciences, 208, 9-14.
[15] Stakhov, A. and Rozin, B. (2004) On a New Class of Hyperbolic Function. Chaos, Solitons & Fractals, 23, 379-389.
[16] Stakhov, A. and Rozin, B. (2007) The “Golden” Hyperbolic Models of Universe. Chaos, Solitons & Fractals, 34, 159-171.
[17] Stakhov, A.P. and Rozin, B.N. (2006) The Golden Section, Fibonacci Series and New Hyperbolic Models of Nature. Visual Mathematics, 8, 3. http://www.mi.sanu.ac.rs/vismath/stakhov/index.html
[18] Stakhov, A.P. (2006) Gazale formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the “Golden” Cryptography. Academy of Trinitarizam, Moscow.
http://www.trinitas.ru/rus/doc/0232/004a/02321063.htm
[19] Stakhov, A. (2013) On the General Theory of Hyperbolic Functions Based on the Hyperbolic Fibonacci and Lucas Functions and on Hilbert’s Fourth Problem. Visual Mathematics, 15, 1.
http://www.mi.sanu.ac.rs/vismath/pap.htm)
[20] Stakhov, A. and Aranson, S. (2011) Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem. Part I. Applied Mathematics, 2, 74-84. http://www.scirp.org/journal/am/
http://dx.doi.org/10.4236/am.2011.21009
[21] Stakhov, A. and Aranson, S. (2011) Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem. Part II. Applied Mathematics, 2, 181-188.
http://www.scirp.org/journal/am/ http://dx.doi.org/10.4236/am.2011.22020
[22] Stakhov, A. and Aranson, S. (2011) Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem. Part III. Applied Mathematics, 2, 283-293.
http://www.scirp.org/journal/am/ http://dx.doi.org/10.4236/am.2011.23033
[23] Stakhov, A.P. (2013) Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature. Applied Mathematics and Physics, 1, 60-66. http://www.scirp.org/journal/jamp/.
[24] Stakhov, A.P. (1984) Codes of the Golden Proportion (in Russian). Publishing House “Radio and Communications”, Moscow, 1984.
[25] Shervatov, V.G. (1958) Hyperbolic Functions (in Russian). Fizmatgiz, Moscow.
[26] de Spinadel, V.W. (2004) From the Golden Mean to Chaos. 2nd Edition, Nueva Libreria, Nobuko.
[27] Gazale, M.J. (1999) Gnomon. From Pharaohs to Fractals. Princeton University Press, Princeton.
[28] Kappraff, J. (2001) Connections. The Geometric Bridge between Art and Science. 2nd Edition, World Scientific, Singapore, New Jersey, London, Hong Kong.
[29] Tatarenko, A. (2005) The Golden -Harmonies’ and -fractals (in Russian). Academy of Trinitarism, Moscow.
http://www.trinitas.ru/rus/doc/0232/009a/02320010.htm
[30] Arakelyan, H. (1989) The Numbers and Magnitudes in Modern Physics (in Russian). Publishing House “Armenian Academy of Sciences”, Yerevan.
[31] Shenyagin, V.P. (2011) Pythagoras, or How Everyone Creates His Own Myth. The Fourteen Years after the First Publication of the Quadratic Mantissa’s Proportions (in Russian). Academy of Trinitarism, Moscow.
http://www.trinitas.ru/rus/doc/0232/013a/02322050.htm
[32] Kosinov, N.V. (2007) The Golden Ratio, Golden Constants, and Golden Theorems (in Russian). Academy of Trinitarism, Moscow. http://www.trinitas.ru/rus/doc/0232/009a/02321049.htm
[33] Falcon, S. and Plaza, A. (2007) On the Fibonacci k-numbers. Chaos, Solitons & Fractals, 32, 1615-1624.
http://dx.doi.org/10.1016/j.chaos.2006.09.022
[34] Wikipedia, the Free Encyclopedia (2014) Pell Number. http://en.wikipedia.org/wiki/Pell_number
[35] Stakhov, A.P. (2012) A Generalization of the Cassini Formula. Visual Mathematics, 14, 2.
http://www.mi.sanu.ac.rs/vismath/stakhovsept2012/cassini.pdf
[36] Aranson, S.Kh. (2000) Qualitative Properties of Foliations on Closed Surfaces. Journal of Dynamical and Control Systems, 6, 127-157. http://dx.doi.org/10.1023/A:1009525823422
[37] Aranson, S.Kh. and Zhuzoma, E.V. (2004) Nonlocal Properties of Analytic Flows on Closed Orientable Surfaces. Proceedings of the Steklov Institute of Mathematics, 244, 2-17.
[38] Aranson, S., Medvedev, V. and Zhuzhoma, E. (2000) Collapse and Continuity of Geodesic Frameworks of Surface Foliations. In: Andronov-Leontovich, E.A., Ed., Methods of Qualitative Theory of Differential Equations and Related Topics, American Mathematical Society, 200, 35-49.
[39] Aranson, S.Kh., Belitsky, E.V. and Zhuzhoma (1996) Introduction to the Qualitative Theory of Dynamical Systems on surfaces. American Mathematical Society.
[40] Anosov, D.V., Aranson, S.Kh., Arnold, V.I., Bronshtein, I.U., Grines, V.Z. and Il’yashenko, Yu.S. (1997) Ordinary Differential Equations and Smooth Dynamical Systems. Springer, Berlin.
[41] Anosov, D.V., Aranson, S.Kh., Grines, V.Z., Plykin, R.V., Safonov, A.V., Sataev, E.A., Shlyachkov, S.V., Solodov, V.V., Starkov, A.N. and Stepin, A.M. (1995) Dynamical Systems with Hyperbolic Behaviour. Encyclopaedia of Mathematical Sciences. Dynamical Systems IX, 66, Springer, Berlin, 1-235. http://dx.doi.org/10.1007/978-3-662-03172-8
[42] Hilbert, D. (1976) Mathematical Developments Arising from Hilbert’s Problems, American Mathematical Society.
http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob4
[43] Alexandrov’s, P.S. (1969) Hilbert’s Problems. Nauka, Moscow.
[44] Wikipedia, the Free Encyclopaedia (2003) Hilbert’s Fourth Problem.
[45] Busemann, H. (1966) On Hilbert’s Fourth Problem. Russian Mathematical Surveys, 21.
[46] Busemann, H. (1966) On Hilbert’s Fourth Problem (in Russian). Uspechi mathematicheskich nauk, 21, 155-164.
[47] Pogorelov, A.V. (1974) Hilbert’s Fourth Problem (in Russian). Nauka, Moscow.
[48] Aranson, S.Kh. (2009) Once again on Hilbert’s Fourth Problem (in Russian). Academy of Trinitarizm, Мoscow.
http://www.trinitas.ru/rus/doc/0232/009a/02321180.htm
[49] Yandell, B.H. (2003) The Honors Class-Hilbert’s Problems and Their Solvers.
[50] Stakhov, A.P. (2013) Non-Euclidean Geometries. From the “Game of Postulates” to the “Game of Function (in Russian).” Academy of Trinitarizm, Мoscow.
http://www.trinitas.ru/rus/doc/0016/001d/00162125.htm
[51] Dubrovin, B.A., Novikov, S.P. and Fomenko, A.T. (1979) Modern Geometry. Methods and Applications (in Russian). Nauka, Moscow.
[52] Arnold, V.I., II’yashenko, Yu.S., Anosov, D.V., Bronshtein, I.U., Aranson, S.Kh. and Grines, V.Z. (1998) Dynamical Systems I, Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin.
[53] Kovantsov, N.I., et al. (1982) Differential Geometry, Topology, Tensor Analysis (in Russian). Higher School, Kiev.

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.