Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature ()

A. P. Stakhov

International Higher Education Academy of Sciences, Moscow, Russia.

**DOI: **10.4236/jamp.2013.13010
PDF HTML
6,628
Downloads
9,081
Views
Citations

International Higher Education Academy of Sciences, Moscow, Russia.

Recently the new unique
classes of hyperbolic functions-hyperbolic
Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci *l*-functions based on the “metallic
proportions” (*l* is a given natural number), were introduced in mathematics. The principal
distinction of the new classes of hyperbolic functions from the classic hyperbolic
functions consists in the fact that they have recursive properties like the
Fibonacci numbers (or Fibonacci *l*-numbers), which are “discrete” analogs of
these hyperbolic functions. In the classic hyperbolic functions, such relationship with integer
numerical sequences does not exist. This unique property of the new hyperbolic
functions has been confirmed recently by the new geometric theory of
phyllotaxis, created by the Ukrainian researcherOleg
Bodnar(“Bodnar’s hyperbolic geometry). These new hyperbolic functions underlie the original
solution of Hilbert’s Fourth Problem (Alexey Stakhov and Samuil Aranson). These
fundamental scientific results are overturning our views on hyperbolic
geometry, extending fields of its applications (“Bodnar’s hyperbolic geometry”)
and putting forward the challenge for theoretical natural sciences to search
harmonic hyperbolic worlds of Nature. The goal of the present article is to
show the uniqueness of these scientific results and their vital importance for
theoretical natural sciences and extend the circle of readers. Another
objective is to show a deep connection of the new results in hyperbolic
geometry with the “harmonic ideas” of Pythagoras, Plato and Euclid.

Share and Cite:

Stakhov, A. (2013) Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature. *Journal of Applied Mathematics and Physics*, **1**, 60-66. doi: 10.4236/jamp.2013.13010.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | A. P. Stakhov and I. S. Tkachenko, “Fibonacci Hyperbolic Trigonometry (Russian),” Reports of the Academy of Sciences of Ukraine, Vol. 208, No. 7, 1993, pp. 9-14. |

[2] | A. Stakhov and B. Rozin, “On a New Class of Hyperbolic Functions,” Chaos, Solitons & Fractals, Vol. 23, No. 2, 2004, pp. 379-389. http://dx.doi.org/10.1016/j.chaos.2004.04.022 |

[3] | A. Stakhov and B. Rozin, “The Golden Section, Fibonacci series, and New Hyperbolic Models of Nature,” Visual Mathematics, Vol. 8, No. 3, 2006. http://www.mi.sanu.ac.rs/vismath/stakhov/index.html |

[4] | O. Y. Bodnar, “The Golden Section and Non-Euclidean Geometry in Nature and Arts (in Russian),” Svit, Lvov, 1994. |

[5] | O. Y. Bodnar, “Dynamic Symmetry in Nature and Architecture,” Visual Mathematics, Vol. 12, No. 4, 2010. http://www.mi.sanu.ac.rs/vismath/BOD2010/index.html |

[6] | A. P. Stakhov, “Gazale Formulas, a New Class of Hyperbolic Fibonacci and Lucas Functions and the Improved Method of the ‘Golden’ Cryptography,” Academy of Trinitarism, Moscow. http://www.trinitas.ru/rus/doc/0232/004a/023210 63.htm |

[7] | A. P. Stakhov, “On the General Theory of Hyperbolic Functions Based on the Hyperbolic Fibonacci and Lucas Functions and on Hilbert’s Fourth Problem,” Visual Mathematics, Vol. 15, No. 1, 2013. http://www.mi.sanu.ac.rs/vismath/2013stakhov/hyp.pdf |

[8] | A. Stakhov and S. Aranson, “Hyperbolic Fibonacci and Lucas Functions, ‘Golden’ Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem,” Applied Mathematics, Vol. 2, No. 3, 2011, pp. 283-293. |

[9] | A. P. Stakhov, “The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science,” World Scientific, London, 2009. |

[10] | V. G. Shervatov, “Hyperbolic Functions (in Russian),” Fizmatgiz, Moscow, 1958. |

[11] | V. de Spinadel, “The family of Metallic Means,” Visual Mathematics, Vol. 1, No. 3, 1999. http://members.tripod.com/vismath/ |

[12] | Pell Numbers. http://en.wikipedia.org/wiki/Pell_number |

[13] | M. J. Gazale, “Gnomon. From Pharaohs to Fractals,” Princeton University Press, Princeton, 1999. |

[14] | Hilbert’s Problems. http://en.wikipedia.org/wiki/Hilbert’s_problems |

[15] | Hilbert’s Fourth Problem. http://en.wikipedia.org/wiki/Hilbert’s_fourth_problem |

[16] | D. Hilbert, “Mathematical Problems”. http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob4 |

[17] | A. V. Pogorelov, “Hilbert’s Fourth Problem (in Russian),” Nauka, Moscow, 1974. |

[18] | B. H. Yandell, “The Honours Class—Hilbert’s Problems and Their Solvers,” A K Peters/CRC Press, Natick, 2003. |

[19] | A. P. Stakhov, “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 |

[20] | Fullerene. http://en.wikipedia.org/wiki/Fullerene |

[21] | Quasicrystal. http://en.wikipedia.org/wiki/Quasicrystal |

[22] | S. Petoukhov, “Matrix Genetics, Algebra of the Genetic Code, Noise Immunity (in Russian),” Regular and Chaotic Dynamics, Moscow, 2008. |

[23] | A. P. Stakhov, “Introduction into Algorithmic Measurement Theory (in Russian),” Soviet Radio, Moscow, 1977. |

[24] | A. P. Stakhov, “Algorithmic Measurement Theory (News in Life, Science and Technology, Series “Mathematics and Cybernetics) (in Russian),” Knowledge, Moscow, 1979. |

[25] | A. P. Stakhov, “The Golden Section in the Measurement Theory,” Computers & Mathematics with Applications, 1989, Vol. 17, No. 4-6, pp. 613-638. http://dx.doi.org/10.1016/0898-1221(89)90252-6 |

[26] | G. Bergman, “A Number System with an Irrational Base,” Mathematics Magazine, Vol. 31, No. 2, 1957, pp. 98-119. http://dx.doi.org/10.2307/3029218 |

[27] | A. P. Stakhov, “Codes of Golden Proportion (in Russian),” Radio and Communications, Moscow, 1984. |

[28] | A. P. Stakhov, “Generalized Golden Sections and a New Approach to Geometric Definition of a Number,” Ukrainian Mathematical Journal, Vol. 56, No. 8, 2004, pp. 1143-1150. http://dx.doi.org/10.1007/s11253-005-0064-3 |

[29] | A. P. Stakhov, “Brousentsov’s Ternary Principle, Bergman’s Number System and Ternary Mirror-Symmetrical Arithmetic,” The Computer Journal, Vol. 45, No. 2, 2002, pp. 221-236. http://dx.doi.org/10.1093/comjnl/45.2.221 |

[30] | A. P. Stakhov, “A Generalization of the Fibonacci Q-matrix,” Reports of the National Academy of Sciences of Ukraine, Vol. 9, 1999, pp. 46-49. |

[31] | A. Stakhov “Fibonacci Matrices, a Generalization of the ‘Cassini formula,’ and a New Coding Theory,” Chaos, Solitons & Fractals, Vol. 30, No. 1, 2006, pp. 56-66. http://dx.doi.org/10.1016/j.chaos.2005.12.054 |

[32] | A. P. Stakhov, “The Mathematics of Harmony: Clarifying the Origins and Development of Mathematics,” Congressus Numerantium, Vol. 193, 2008, pp. 5-48. |

[33] | A. P. Stakhov, “The ‘Golden’ Matrices and a New Kind of Cryptography,” Chaos, Solitons & Fractals, Vol. 32, No. 3, 2007, pp. 1138-1146. http://dx.doi.org/10.1016/j.chaos.2006.03.069 |

[34] | A. Stakhov and S. Aranson, “‘Golden’ Fibonacci Goniometry. Fibonacci-Lorentz Transformations, and Hilbert’s Fourth Problem,” Congressus Numerantium, Vol. 193, 2008, pp. 119-156. |

[35] | A. P. Stakhov, “The Golden Section and Modern Harmony Mathematics,” Applications of Fibonacci Numbers, Vol. 7, 1998, pp. 393-399.. |

Journals Menu

Contact us

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2022 by authors and Scientific Research Publishing Inc.

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