A Sangaku Revived

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Zsófia Ruttkay: A Sangaku Revived. In: Bridges 2008. Pages 155–162



In this paper we give an account on our mathematical and visual explorations inspired by a sangaku. First we introduce sangakus – traditional Japanese mathematical tablets. Then we give four examples of our static contemporary variants. Finally, we discuss in detail how a fifth sangaku led us to simulate the growth of water lilies, as a means of visualizing the problem. This approach lead to the mathematical field of circle packing, and made it possible to experience the visually intriguing process with different settings of the algorithm.

Extended Abstract


Used References

Sangaku from 1873, solved by 11 year old boy http://www.kurims.kyoto-u.ac.jp/~okamoto/sangaku/sangaku.html

H. Fukagawa and D. Pedoe, Japanese temple geometry problems = Sangaku, Charles Babbage Research Centre, Winnipeg, Canada, 1989.

T. Rothman, Japanese temple geometry, Scientific American, 5, 1998.

H. Fukagawa and T. Rothman, Sacred Mathematics, to appear http://www.princeton.edu/main/news/archive/S15/04/04O77/index.xml

Sangaku Diagrams and Kinetigrams: http://interactive-mathvision.com/PaisPortfolio/Sangaku/SangakuFrames.html

Japanese mathematics: http://www.ballstructure.com/Japanese_Math/

Map of Sangaku locations: http://www.wasan.jp/english/

Photos of some tablets: http://www.sangaku.info/

I. Peterson, Temple Circles, http://www.maa.org/mathland/mathtrek_4_23_01.html

G. Hart, A Modern Day Sangaku http://www.georgehart.com/sangaku/

Solution of the 4 sangakus (in Dutch) http://www.arsetmathesis.nl/arthesis/sangopl.pdf

Monet: Les Nympéas, Paris, L’ensemble de l’Orangerie, http://www.musee-orangerie.fr/homes/home_id24799_u1l2.htm

Kepler’s sphere packing problem, http://mathworld.wolfram.com/SpherePacking.html

Kepler’s Conjecture, http://mathworld.wolfram.com/KeplerConjecture.html

T. Tarnai, Packing of equal circles in a circle in: Structural Morphology: Toward the New Millenium, The University of Nottingham, Nottingham, UK, 217-224, 1997.

Best known circle packing up to 2005: http://www.packomania.com/

K. Stephenson, Circle Packing: A Mathematical Tale , Notices of the American Mathematical Society, 2003. http://www.ams.org/notices/200311/fea-stephenson.pdf

P. G. Szabó, M. Cs. Markót, T. Csendes, E. Specht, L. G. Casado, I. García, New Approaches to Circle Packing in a Square. With Program Codes (Springer Optimization and Its Applications, Vol. 6). New York. 2007. Springer.

Zs Ruttkay, The simulation program as Java applet: http://wwwhome.cs.utwente.nl/~zsofi/sangaku/SanSim.html


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