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

A. P. Stakhov^{*}

Show more

References

[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..