Golden Ratio‎: ‎The Mathematics of Beauty

Document Type : Technical Paper

Author

Faculty of Mathematical Sciences, Department of Statistics, University of Kashan, Kashan, Iran

Abstract

Historically, mathematics and architecture have been associated with one another. Ratios are good example of this interconnection. The origin of ratios can be found in nature, which makes the nature so attractive. As an example, consider the architecture inspired by flowers which seems so harmonic to us. In the same way, the architectural plan of many well-known historical buildings such as mosques and bridges shows a rhythmic balance which according to most experts the reason lies in using the ratios. The golden ratio has been used to analyze the proportions of natural objects as well as building’s harmony. In this paper, after recalling the (mathematical) definition of the golden ratio, its ability to describe the harmony in the nature is discussed. When teaching mathematics in the schools, one may refer to this interconnection to encourage students to feel better with mathematics and deepen their understanding of proportion. At the end, the golden ratio decimals as well as its binary digits has been statistically examined to confirm their behavior as a random number generator.

Keywords

Main Subjects


[1] Md. Akhtaruzzaman and Amir A. Shafie, Geometrical substantiation of Phi, the golden ratio and the baroque of nature, architecture, design and engineering, Int. J. Arts 1 (1) (2011) 1 − 22.
[2] D. Biebighauser, Testing random number generators, unpublished research paper, University of Minnesota, (2000). www- users.math.umn.edu/garrett/students/reu/pRNGs.pdf.
[3] L. Corbusier, Modulor I and II, Translated by P. de Francia and R. A. Bostock, Harvard University Press, Cambridge, 1980.
[4] L. M. Dabbour, Geometric proportions, The underlying structure of design process for Islamic geometric patterns, Front. Archit. Res. 1 (4) (2012) 380 − 391.
[5] G. Dóczi, The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture, Shambhala publications, Buenos Aires, 1996
[6] R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing Co., New Jersey, 1997.
[7] U. Eco, Art and Beauty in the Middle Ages, Translated by H. Bredin, Yale University Press, New Haven, 2002.
[8] K. Fink, W. W. Beman and D. E. Smith, A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink’s Geschichte der Elementar-Mathematik, The Open Court Pub. Co., Chicago, 1900.
[9] R. H. Fischler, The Shape of the Great Pyramid, Wilfrid Laurier University Press, Waterloo, 2000.
[10] R. Fletcher, Golden proportions in a great house: Palladios Villa Emo. In M. J. Ostwald (eds.) Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future, chapter 55, 131 − 138, Birkhäuser Basel, New York, 2015.
[11] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979.
[12] M. Hejazi, Geometry in nature and Persian architecture, Build. Environ. 40 (2005) 1413 − 1427.
[13] M. Hejazi and F. Mehdizadeh Saradj, Persian Architectural Heritage: Architecture, Structure and Conservation, WIT Press, Southampton, 2014.
[14] G. L. Hersey, Architecture and Geometry in the Age of the Baroque, The University of Chicago Press, Chicago, 2002.
[15] R. Hillenbrand, Islamic Art and Architecture, Thames and Hudson, Ltd., New York, 1999.
[16] S. Iwamoto and K. Akifumi, On golden inequalities, 1504 (2006) 168−176.
[17] R. V. Jean, Phyllotaxis: A Systemic Study in Plant Morphogenesis, Cambridge University Press, Cambridge, 1994.
[18] J. Kappraff, A course in the mathematics of design, Comput. Math. Appl. 12 (3-4) (Part 2) (1986) 913 − 948.
[19] E. Kilic, The Binet formula, sums and representations of generalized Fibonacci p-numbers, Eur. J. Comb. 29 (3) (2008) 701 − 711.
[20] D. E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley Professional Press, Massachusetts, 1997.
[21] M. Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, Broadway Press, New York, 2008.
[22] N. A. Megahed, Towards math-based architectural education in Egyptian engineering faculties, Nexus Netw. J. 15 (3) (2013) 565 − 581.
[23] F. Nabavi and Y. Ahmad, Is there any geometrical golden ratio in traditional Iranian courtyard houses, Int. J. Architectural Res. 10 (1) (2016) 143 − 154.
[24] D. Persaud-Sharma and J. P ÓLeary, Fibonacci series, golden proportions, and the human biology , Austin J. Surgery 2 (5) (2015) 1066 − 1073.
[25] A. S. Posamentier and I. Lehmann, The Fabulous Fibonacci Numbers, Prometheus Books publisher, New York, 2007.
[26] P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, New York, 2004.
[27] A. Samalavičius, Ideas and Structures: Essays in Architectural History, Eugene, Oregon: Resource Publications/An Imprint of Wipf and Stock Publishers, Portland, 2011.
[28] S. K. Sen and R. P. Agarwal, Golden ratio in science, as random sequence source, its computation and beyond, Comput. Math. Appl. 56 (2) (2008) 469 − 498.
[29] I. Stewart, Nature’s Numbers: The Unreal Reality of Mathematics, Basic Books Press, New York, 2008.
[30] A. F. Vico-Prieto, A. Cagigas, J. M. Rosas and J. E. Callejas-Aguilera, Ex perimental approach to the study of beauty: The role of golden proportion, Psicológica 37 (2) (2016) 187207.
[31] Wikipedia contributors, Euclid-Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/wiki/Euclid.
[32] K. Williams (eds.), Persian Architecture and Mathematics, Birkhäuser Basel, Turin, 2012.