Abstract
After giving a simple introduction of Japanese traditional mathematics called Wasan and Wasan geometry, we consider problems in Wasan geometry in details in three sections and show that the problems are rich source for mathematical study today, although many people consider Wasan to be a historical mathematics. In the first section, we consider problems involving several congruent circles. Those figures have not been considered elsewhere, though they have interesting properties, and there are few expository writings dealing with those problems today. In the second section, we consider problems involving an arbelos formed by mutually touching three circles with collinear centers. Since it is one of the most well-known plane figures and has been studied by many mathematicians, it is a very good example to see the different approaches to studying the same figure between the East and the West. In the third section, we consider simple application of recently made definition of division by zero to Wasan geometry. At the end of this chapter, we see the practical side of Wasan and Wasan geometry briefly and give a simple history of the study of Wasan geometry.
References
Aida () (ed) (no date) Sampō Chikusaku Jutsu (), vol 2. Tohoku University Digital Collection
Aida () (ed) (1788) Sampō Tenshōhō (). Tohoku University Digital Collection
Akabane () et al (ed) (1998) The sangaku in Nagano prefecture (enlarged and revised edition) (). Kyōiku Shokan ()
Akita () (ed) (1833) Sampō Tenzan Tebikigusa Furoku (). Tohoku University Digital Collection
Bankoff L (1974) Are the twin circles of Archimedes really twins?. Math Mag 72: 214–218
Dodge C W, Schoch T, Woo P Y, Yiu P (1999) Those ubiquitous Archimedean circles. Math Mag 72: 202–213
Fujisawa () (ed) Sampō Shinsho (). Tohoku University Digital Collection
Fukagawa H, Pedoe D (1989) Japanese Temple Geometry Problems. Charles Babbage Research Centre Winnipeg Canada
Fukushimaken Wasan Kenkyū Hozonkai () (ed) (1989) The sangaku in Fukushima (). Sōju Shuppan () Fukushima
Furuya () (ed) (1854) Sampō Tsūsho (). Tohoku University Digital Collection
Gunma Wasan Kenkyūkai () (ed) (1987) The sangaku in Gunma (). Gunma Wasan Kenkyūkai
Hanai () (ed) (1873) Taiyō Reki Zokkai (). In Wasansho Shūsei () (Wasan collection) Okumura H (ed) (2001) Iwanami Shoten
Hayashi T () (1937) Collected papers on the old Japanese Mathematics () volumes 1 and 2. Tokyo Kaiseikan ()
Heinouchi () (ed) (1848) Shouka Kujutsu Shinsho (). Tohoku University Digital Collection (non-online)
Heinouchi () (ed) (1833) Shouka Kujutsu Youkai (). Kotenseki Sogo Database in Waseda University
Hirayama (), Matsuoka () (1966) The sangaku in Yamagata ().
Ishiguro () (1835) Tokai Hyōteki (). In Wasansho Shūsei () (Wasan collection) Okumura H (ed) (2001) Iwanami Shoten
Iwai () (no date) Keihoukei Kou (). Tohoku University Digital Collection
Iwata S () (1993) The unabridged dictionary of geometry () supplement, vol 2. Maki Shoten ()
Iwata S () (1988) The unabridged dictionary of geometry () supplement, vol 1. Maki Shoten ()
Iwata S () (1982) The unabridged dictionary of geometry (), vol 6. Maki Shoten ()
Iwata S () (1980) The unabridged dictionary of geometry (), vol 5. Maki Shoten ()
Iwata S () (1978) The unabridged dictionary of geometry (), vol 4. Maki Shoten ()
Iwata S () (1976) The unabridged dictionary of geometry (), vol 3. Maki Shoten ()
Iwata S () (1974) The unabridged dictionary of geometry (), vol 2. Maki Shoten ()
Iwata S () (1971) The unabridged dictionary of geometry (), vol 1. Maki Shoten ()
Jobbings A K (2011) Two semicircles fill half a circle. Math Gaz 95:538–540
Kanai Y, Okumura H (2017) A three tangent congruent circle problem. Sangaku J Math 1:16–20
Kinki Sūgakushigakukai () (ed) (1992) The sangaku in Kinki (). Osaka Kyōiku Tosho ()
Kimura () (1828) Onchi Sansō (). Tohoku University Digital Collection
Kuroda M, Michiwaki H, Saitoh S, and Yamane M (2014) New meanings of the division by zero and interpretations on 100∕0 = 0 and on 0∕0 = 0. Int J Appl Math 27:191–198
Matsuura T, Okumura H, Saitoh S (2019) Division by zero calculus and Pompe’s theorem. Sangaku J Math 3:36–40
Mikami Y (1913-2) The parabola and hyperbola in Japanese mathematics. Tohoku Mathematical Journal, First Series 3:29–37
Mikami Y (1913) The development of Mathematics in China and Japan. Chelsea, New York
Morishima () (ed) (no date) Kansai Shachū Sansou (). Tohoku University Digital Collection
Nakasone () (1864) Ryōchi Senmen Hyō (). In Wasansho Shūsei () (Wasan collection) Okumura H (ed) (2001) Iwanami Shoten
Nipkow T, Paulson L C, Wenzel M (2015) A Proof Assistant for Higher-Order Logic. Springer-Verlag
Okayu () (ed) (1855) Hōnōcho Sekisensei Sandai Kujō (). Tohoku University Digital Collection
Okumura () (1836) Ryōchi Kodo Sampō (). In Wasansho Shūsei () (Wasan collection) Okumura H (ed) (2001) Iwanami Shoten
Okumura H (1995) Two similar triangles. Math Gaz 79:569–571
Okumura H (1999) Geometries in the East and the West in the 19th century. Symmetry: Culture and Science : 9:189–197
Okumura H (2013) Archimedean circles of the collinear arbelos and the skewed arbelos. J Geom Graph 17:31–52
Okumura H (2014) The arbelos with overhang. KoG 18:19–27
Okumura H (2017a) Configurations of congruent circles on a line. Sangaku J Math 1:24–34
Okumura H (2017b) Theorems on two congruent circles on a line. Sangaku J Math 1:35–38
Okumura H (2018a) A note on a problem involving a square in a curvilinear triangle. Sangaku J Math 2:3–5
Okumura H (2018b) A note on an isosceles triangle containing a square and three congruent circles. Sangaku J Math 2:8–10
Okumura H (2018c) Wasan geometry with the division by 0. Int J Geom 8:17–20
Okumura H (2018d) Solution to 2017-3 Problem 5. Sangaku J Math 2:17–21
Okumura H (2018e) Solution to the problem proposed in “Solution to 2017-3 Problem 5”. Sangaku J Math 2:24–26
Okumura H (2018f) Solution to 2017-1 Problem 4 with division by zero. Sangaku J Math 2: 27–30
Okumura H (2018g) Solution to Problem 2018-3-2. Sangaku J Math 2:54–56
Okumura H (2019a) A note on the arbelos in Wasan geometry, a problem in Sampō Tenzan Tebikigusa Furoku. Sangaku J Math 3:12–14
Okumura H (2019b) A note on the arbelos in Wasan geometry, Satoh’s problem. Sangaku J Math 3:15–16
Okumura H (2019c) A note on the arbelos in Wasan geometry, Matsuda’s problem. Sangaku J Math 3:19–23
Okumura H (2019d) The arbelos in Wasan geometry, problems of Izumiya and Naitō. J Class Geom 4:1–5
Okumura H (2019e) The arbelos in Wasan geometry, Tamura’s problem. Glob J Adv Res Class Mod Geom 8:33–36
Okumura H (2019f) A note on the arbelos in the Wasan geometry: Satoh’s problem and a circle pattern. Mathematics and Informatics 62:301–304
Okumura H (2019g) Arbeloi determined by a chord and solutions to problems 2017-3-8 and 2019-3-4. Sangaku J Math 3:41–50
Okumura H (2019h) A characterization of the golden arbelos involving an Archimedean circle. Sangaku J Math 3:67–71
Okumura H (2019i) The arbelos in Wasan geometry: Chiba’s problem. Glob J Adv Res Class Mod Geom 8:98–104
Okumura H (2019j) The arbelos in Wasan geometry: Ootoba’s problem and Archimedean circles. Sangaku J Math 3:91–97
Okumura H, Saitoh S (2018a) Applications of the division by zero calculus to Wasan geometry. Glob J Adv Res Class Mod Geom 7:44–49
Okumura H, Saitoh S (2018b) Wasan geometry and division by zero calculus. Sangaku J Math 2:57–73
Okumura H, Sodeyama C (1998) A surprising property of successively touching circles. Mathematics Plus: 6:17–18
Ono () (ed) (1855) Keiteki Sampō Shinan Taisei (). Tohoku University Digital Collection.
Saitama Prefectural Library () (ed) (1969) The sangaku in Saitama (). Saitama Prefectural Library ()
Shimura () et al (ed) (no date) Kiōshū (). Tohoku University Digital Collection
Smith D E, Mikami Y (1914) A history of Japanese mathematics. Open Court Chicago
Suzuki () (ed) (1721) Daiku Tekagami (). Tohoku University Digital Collection
Takeda () (1825) Sampō Benran (). Tohoku University Digital Collection
Takigawa () et al (ed) Mishō Sampō (), vol 9. Tohoku University Digital Collection
Toyoyoshi () (ed) (no date) Chikusaku (). Digital Library Department of Mathematics Kyoto University
Yamamoto () (ed) (1841) Sampō Jojutsu (). In Wasansho Shūsei () (Wasan collection) Okumura H (ed) (2001) Iwanami Shoten
Yasutomi Y () (ed) (1987) The extant sangaku in Iwate (). Seijisha (), Tokyo
Yoshida Y () (1632) Jinkōki (). Tohoku University Digital Collection
no author’s name (no datea) Kururisha Sandaishū (). Tohoku University Digital Collection
no author’s name (no dateb) Sendai Kokuuzoudou Sandai (). Tohoku University Digital Collection
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this entry
Cite this entry
Okumura, H. (2020). Wasan Geometry. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-70658-0_122-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-70658-0_122-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-70658-0
Online ISBN: 978-3-319-70658-0
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering