Abstract
The present paper proposes a generation mechanism of a fractal pattern related to a coin system. The problem is formulated in terms of a situation of dividing coins among people. Remarkably, a fractal pattern like the Sierpinski gasket is obtained, by marking all the possible division of coins as a point set. The mechanism for this fractal structure is reduced to nested relations, owing to a hierarchical property of coin denominations. Relevance to dynamical systems is also discussed.
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References
Mandelbrot, B.: The Fractal Geometry of Nature. WH Freeman, San Francisco (1982)
Richardella, A., Roushan, P., Mack, S., Zhou, B., Huse, D.A., Awschalom, D.D., Yazdani, A.: Science 327, 665 (2010)
Matsushita, M., Fujikawa, H.: Physica A 168, 498 (1990)
Martinez, V.J., Saar, E.: Statistics of the Galaxy Distribution. CRC, Boca Raton (2001)
Kobayashi, N., Yamazaki, Y., Kuninaka, H., Katori, M., Matsushita, M., Matsushita, S., Chiang, L.-Y.: J. Phys. Soc. Jpn. 80, 074003 (2011)
Feder, J.: Fractals. Plenum, New York (1988)
Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer, New York (1996)
Sornette, D.: Critical Phenomena in Natural Science. Springer, Berlin (2004)
Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)
Yamamoto, K., Yamazaki, Y.: Chaos Solitons Fractals 45, 1058 (2012)
Matsushita, M.: J. Phys. Soc. Jpn. 54, 857 (1985)
Peitgen, H.-O., Jürgens, H., Saupe, D.: Chaos and Fractals: New Frontiers of Science. Springer, New York (1992)
Jeffrey, H.J.: Nucleic Acids Res. 18, 2163 (1990)
Yu, Z.-G., Anh, V., Lau, K.-S.: J. Theor. Biol. 226, 341 (2004)
Matsushita, R., Gleria, I., Figueiredo, A., Da Silva, S.: Physica A 378, 427 (2007)
Barnsley, M.F., Hurd, L.P.: Fractal Image Compression. A K Peters, Boston (1993)
Maimon, O., Rokach, L.: Data Mining and Knowledge Discovery Handbook, 2nd edn. Springer, New York (2010)
Barnsley, M.F.: Fractals Everywhere. Academic Press, New York (1993)
Ball, P.: Branches: Nature’s Patterns: A Tapestry in Three Parts. Oxford University Press, Oxford (2009)
Acknowledgements
The author is grateful to Dr. Makoto Katori, Dr. Yoshihiro Yamazaki, and Dr. Jun-ichi Wakita for their fruitful and stimulating discussions and comments.
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Yamamoto, K. Emergence of the Sierpinski Gasket in Coin-Dividing Problems. J Stat Phys 152, 534–540 (2013). https://doi.org/10.1007/s10955-013-0776-y
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DOI: https://doi.org/10.1007/s10955-013-0776-y