OUR FIRST INSIGHT IN SANGAKU PROBLEMS

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1 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG 0, VOLUME 0, OUR FIRST INSIGHT IN SANGAKU PROBLEMS Ivanka Stipančić-Klaić, Josipa Matotek Univesity of Osijek, Faculty of Civil Engineeing, Dinska 6a, 3000 Osijek, Coatia, Univesity of Osijek, Faculty of Civil Engineeing, Dinska 6a, 3000 Osijek, Coatia, Abstact. Duing self-imposed isolation fom the West in Japanese mathematicians built up oiginal math wold. We can conside sangaku poblems as an unique cultual ceation with etenal beauty whee at and eligious aspect meet within mathematics. The advantage of sangaku is that it can be simple fo younge students, while some poblems ae still a challenge fo othes because they ae still unsolved. Keywods: sangaku poblem, Kowa Seki, eni, sooban, Japanese Theoem, Soddy s Hexlet. Intoduction and Histoical Review Fom the 4 th centuy onwad, the dak ages of Japanese science began. Duing the Japanese "Edo Peiod", which began in 603 and lasted until 867, the Tokugawa shogunate isolated Japan fom the est of the wold fo nealy 300 yeas, until the 9 th centuy. But, this was also a peiod of peace and significant pogess in Japanese cultue and at. Although thee wee no colleges o univesities in Japan, teaching was caied out in pivate schools juku and moe often in Shinto shines and Buddhist temples. The samuai wee tansfomed within a few geneations into a highly educated class. At the tun of the 7 th and the beginning of the 8 th centuy, math in Japan also expeienced significant advancement unde the leadeship of Kowa Seki (64-78), the most impotant Japanese mathematician of his time. He is often called the "Newton" o "Leibnitz" of Japan. In 683, Seki was the fist mathematician who studied deteminants, ten yeas befoe Leibnitz; futhemoe Seki's vesion was the moe geneal. In 685, Seki solved the cubic equation x 3 5x 4x 30 0, applying the same method as Hone would a hunded yeas late. One of the most impotant Seki's contibutions to Japanese mathematic wasan was a method named eni. Eni divided a cicle into n ectangles to appoximate a cicle, against Euope's method of exhaustion which used n-sides polygons. The given eni method is ough vesion of integal calculus. Fukagawa and Rothman in his book [] claim that it was nealy impossible fo anyone in Japan to know about integal calculus by Newton and Leibnitz duing this peiod. Sangaku appeaed duing this Edo peiod. Sangaku, o moe pecisely San Gaku liteally means mathematical tablet, and those wee coloful mathematical puzzles, i.e. geometical poblems witten and dawn on wooden tablets. They wee usually hung unde the eaves in Shinto shines and Buddhist temples as offeings to the gods, but also as Fig. : Tablet [9] challenges to all. Sangaku look like the woks of at. Each wooden tablet often includes many illustated poblems o theoems which wee fequently based on Euclidian geomety. In Fig. is pesented one of the few poblems fom the tablet in Fukushima pefectue, dealing with cicles and ellipses. It is inteesting that most

2 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG 0, VOLUME 0, sangaku contain only poblems, without the solutions. The majoity of these poblems ae not so difficult to solve, they ae at the high school o college level. But, some of them ae extemely difficult, because the poof of these theoems equies, fo example, integal calculus which Japanese did not know duing the peiod when the tablets wee ceated. Some theoems emain unpoven even by the application of the moden mathematics. Sangaku poblems at that time on aeas and volumes wee solved by expansions in infinite seies and tem-by-tem calculation. Evey tablet contains the name of ceato and the date. Hence, today we know that sangaku was done by devotees of mathematics, fom people in the poo castes up to highly educated samuai, women and childen (aged -4). Sangaku wee pobably made by goups of students in schools. Sangaku wee acts of homage, thanks to a guiding spiit, o may have been challenges to othes: Solve this if you can! []. Fig. : Sangaku in Japanese [3] The most of sangaku was witten on Kambun, an achaic Japanese dialect elated to Chinese. Kambun was language like Latin in Euope, leaned and used by educated castes. It can be concluding that authos of tablets witten on Kambun ae samuais o some highly educated pesons. In Edo peiod taveling was vey popula; so many mathematicians taveled and made sangaku all ove the county. Kazu Yamaguchi taveled acoss Japan six times between 87. and 88. He collected data about 87 sangaku and thei ceatos anged them equally in ual and uban disticts. Unfotunately, only two of these tablets ae peseved. Though seveal centuies thee wee made thousands of tablets, today still exist 880 sangaku. Clealy, sangaku had done thei job, successfully popagated knowledge and inteest fo mathematics, [], [9].. Some Typical Sangaku Poblems In sangaku poblems we can usually find multitudes of cicles o of sphees within othes, ellipses, cylindes and othe figues... Cicles in a Squae In a squae thee ae two cicles touching the bottom side of the squae. At the same time, thee ae two small cicles tangent to the same side of the squae and the incicle of the squae. Let two tangent of smalle cicles meet each othe at midpoint of the top side of squae, they make an isosceles tiangle in squae. Find the adius of the incicle of that tiangle if we know the adius of incicle of the squae. Solution: The medium cicle is half the size of the lagest cicle. Fig. 3: Cicles [3]

3 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG 0, VOLUME 0, solution.. Sangaku with Pentagons: Golden Ratio Fig. 4: Pentagon [3] t a. This is sangaku fom Miyage pefectue, 9. Six conguent ight tiangles fan out along the sides of a egula pentagon of side a. Find the length of the hypotenuse t of these tiangles in tems of a. Solution: The diagonal with a side of pentagon in golden atio, using simila tiangles and the fact cos 36 leads to Two conguent egula pentagons with a common side ae inscibed in a lage cicle. A cicle of adius touches the lage cicle and the sides of pentagons, and the incicle of the tiangle ABH has adius t. Show that t. Solution: Simila tiangles EPR and ABH. Fig. 5: Two pentagons [4].3. Balls Poblem Restoed by H. Fukagawa Fig. 6: Sangaku with Balls [] Solutions: 05, 3, 5; 075, 56, 55. Thee ae N balls. Fist stack them with 9 balls on the top and m balls on the bottom, as in the figue left. Then stack them with 6 balls on the top and n on the bottom, as in the figue ight. Find N, m, and n..4. Sangaku Poposed by Okuda Tsume In cicle of AB R, daw acs of adius R with centes A and B espectively and ten inscibed cicles, two of diamete R (light), 4 dak of adius t, and 4 lighte of R adius t'. Show that t t. Fo solution apply 6 Pythagoean Theoem and the law of cosine. This is a ae poblem poposed by woman. Fig. 7: Cicles and Acs [] Fig. 8: Sangaku in Secto of Annulus [].5. Sangaku Poblems upon a Secto of Annulus Hee is example of sangaku that could be dawn upon folding fan such as wee popula in late Edo peiod. This is also an example of how in Japanese poblems the paametes wee sometimes stange, but the aim was to have nice esult o to make calculation easie. 3

4 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG 0, VOLUME 0, In a secto of adius R two cicles of adius ae tangent to each othe and touch the secto intenally. At the same time a small cicle of adius t touches both the secto and a chod of length d. If d and t 0. 34, find. Often they offe wong answe on tablets. Hee, the coect answe is 3. 05, but the answe on the tablet states that Fig. 9: Fan Sangaku [].6. Poblem Maxime In a secto AOB of adius daw a small cicle of adius x with cente O. Daw the tangent to the small cicle fom the vetex B as shown. As x is vaied, the aea S (x) of the black pat of the figue will also vay. Shaw that S (x) is a 93 maximum when x Sangaku fom Yamaguchi Kazan s Diay Hung by Seki Teutoshi in Zenkoji temple in 804 this sangaku is known as Japanese theoem. Yamaguchi Kanzan did not ecod the answe in his Tavel Diay because it was too complicated. In a egula n-sided polygon, with length side a, we daw the diagonals fom the vetex A, so we get ( n ) tiangles. Find the sum of adii of the inscibed cicles in tems of a and pove that it does not depend on the way we tiangulate the polygon. Fig. 0: Japanese Theoem [].8. Still Unsolved Poblem with Ellipse by Sawa Masayoshi Fig. : Unsolved Poblem [] 3. Solids in Sangaku Tablets 3.. A Special Case Tangent Sphees In the ight tiangle an ellipse is inscibed with its majo axes paallel to the hypotenuse. Two cicles of adius ae inscibed within ellipse and the thid cicle with the same adius touches the ellipse and two shote sides of the tiangle, a and b. Find in tems of a and b. This poblem is fom an 8 tablet in Kanagawa Pefectue, which pedates by moe than a centuy a theoem of F. Soddy, the Bitish chemist [], [3], [5]. Fig. : Soddy s Hexlet [3] 4

5 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG 0, VOLUME 0, Two sphees touch each othe and also touch the inside of the lagest sphee. A chain of smalle, diffeent-size sphees cicles the neck between the sphees. Each small sphee touches its neighbous and given thee sphees. Find how many sphees must be thee? How ae the adii of the sphees elated? Easy way to solve this poblem is by the method of invesion. Taditional Japanese mathematicians did not know about invesion. This poblem is connected with the envelope of Soddy s hexlets is a Dupin cyclide, an invesion of the tous. 3.. Sangaku Dated fom 798 Fig. 3: Small Sphee aound a Big Sphee [3] Let a lage sphee be suounded by 30 small, identical sphees, each of which touches its fou small-sphee neighbos as well as the lage sphee. How is the adius of the lage sphee elated to that of the small sphees? 3.3. Sphee and Cylinde If a cylinde intesects a sphee so that the outside of the cylinde is tangent to the inside of the sphee, what is the suface aea of the pat of the cylinde contained inside the sphee? [5], [7] Sangaku fom Atsuta Tablet Fig. 4: Sphee Tangents Cylinde [3], [7] Two paallel planes sepaated by a distance d cut a sphee of adius. Find the suface aea A (dak) of the cut-out section in tems of d and. Solution: A d. Fig. 5: Two Paallel Planes Cut Sphee [] 3.5. Eule and Sangaku fom Daikokutendo Temple, Tokyo On a sphee of adius daw thee cicula acs AB, AC and CA. Let the staight line segments AB c, BC a and CA b. Find the aea S of the spheical tiangle ABC in tems of a, b, c and. The oiginal solution involved definite integal. We can find the same poblem in L. Eule s wok Vaie Speculations Supe Aea Tianguloum Sphaeocoum []. 5 Fig. 6: Sphee and Thee Cicula Acs []

6 PROCEEDINGS OF SYMPOSIUM ON COMPUTER GEOMETRY SCG 0, VOLUME 0, 3.6. Kyoto s Gion Shine Poblem Fig. 7: Elliptic Cylinde [] Find the aea A on an elliptic cylinde that intesects two sectos of a ight cicula cylinde, if the diamete of the cylinde is D and the depth of the secto is d. This famous poblem was solved by Ajima Nanobu at the age of 4, in 774. The oiginal solution lead to equation of 04 degees in tems of the length of the chod, and Ajima educed it to a poblem on the tenth degee by the same method Laplace used in 7 to expand deteminants Two Cylindes cut the Sphee (89, Tokyo) Two identical and paallel ight cicula cylindes pass though the cente of the sphee such that the line of contact between the cylindes passes though cente of the sphee. If the adius of the sphee is, find the volume V cut out of the sphee. Also find the aea of the suface cut out by the cylindes. V 6 3 (), A 8() 9 Fig. 8: Cylindes Cut Sphee [] 4. Summay Today sangaku poblems daws attention of students, mathematicians, atists and othe people just like befoe 50 yeas ago. Galleies and scientific aticles on this topic can be found on the net. Why? Fo mathematicians these ae complex examples, including moe challenging in positive way. We have to combine what we know; it has to look nice, and must have an elegant solution. They ae indeed wothy of ou attention. These poblems unique among the wold s cultual ceations [], ae good educational examples and should not be fogotten in pactice. Refeences [] FUKAGAWA, H., ROTHMAN, T. Saced Mathematics: Japanese Temple Geomety Pinceton Univesity, 008. [] [3] [4] RIGBY, J., Taditional Japanese Geomety, 00 [5] [6] [7] EMIKO TSUSUMI, Visual Communicaton in the Edo Ea, Bjelolasica, 006 [8] [9] 6