Japanese temple geometry Jill Vincent & Claire Vincent University of Melbourne <[email protected]> <[email protected]> Sangaku etween the 17th and 19th centuries, the Japanese government closed its Bborders to the outside world in an attempt to become more powerful. Foreign books were banned, people could not travel, and foreigners were not allowed to enter the country. One result of this isolation was the flourishing of sangaku — wooden tablets inscribed with intricately decorated geometry problems that were hung beneath the eaves of Shinto shrines and Buddhist temples all over Japan (Fukagawa & Pedoe, 1989). The hanging of tablets in Japanese shrines is a centuries-old custom, but earlier tablets generally depicted animals. It is believed that the sangaku arose in the second half of the 17th century, with the oldest surviving sangaku tablet dating from 1683. About 900 tablets have survived, as well as several collections of sangaku prob- lems in early 19th century hand-written books or books produced from wooden blocks. Although their exact purpose is unclear, it would appear that they are simply a joyful expression of the beauty of geometric forms, designed perhaps to please the gods. Many of the problems were three-dimensional, involving spheres within spheres. Only in a few cases were proofs included. Earlier tablets ) 1 were generally about 50 cm by ( 8 1 30 cm but later tablets were some- l a n times as large as 180 cm by 90 cm, r u o each displaying several geometry J s c i problems. The 1875 tablet at the t a m Kaizu Tenman shrine (see Figure 1) e h t has twenty-three problems. a M r The tablets were usually beauti- o i n e fully painted and included the S n name and social rank of the presen- a i l a r ter. Although many of the problems t s u A Figure 1. The Kaizu Tenman shrine in Siga Prefecture. are believed to have been created 8 Japanese temple geometry Australian Senior Mathematics Journal 18 (1) 9 - emain Japanese Temple n methods. oblems for which ester . Solutions for Examples 1, ove by W also includes many pr ficult to pr . emple Geometry (Fukagawa & Problems ovided. Each of the following examples has y dif e pr oblems ed in the West. Pedoe suggests that the Japanese ed in the West. y Pr Figure 2. Temple geometrytablet from Osaka. Figure 2. Temple egarding an ellipse as a plane section of a cylinder, rather an ellipse as a plane section of a cylinder, egarding (now out of print), dedicating it to ‘those who love Cabri Geometry II Japanese Temple Geometry Problems Japanese Temple ovided by the authors of this article. All diagrams have been eface to Japanese T e based on those given by Fukagawa and Pedoe, with the r emple Geometr ems that would be ver In his pr Sangaku problems have been made known through the work of Hidetosi have been made known through Sangaku problems In some of the examples given by Fukagawa and Pedoe original solutions estern and the Soddy Hexlet such as the Malfatti Theorem geometry, Geometry everywhere’. Japanese temple geometry was also described in a Scientific American article by Fukagawa and Rothman (1998). Fukagawa — a Japanese high school teacher who is recognised as a sangaku Fukagawa — a Japanese high school teacher who is recognised expert. Emeritus of In 1989 Fukagawa and Dan Pedoe — a Professor the University of Minnesota — published Mathematics from 1. Problems involving tangent circles 1. relationship. on the Pythagorean The solutions to Examples 1 to 3 rely of a segment of a circle. calculation of the area Example 4 requires Japanese T by highly educated members of the samurai class, some bear the names of village teachers and farmers.women and children, 2 shows a tablet Figure a temple in the Osaka Prefecture. from answers, but not solutions, ar been taken from 5, 7 and 8 ar constructed using ing solutions pr Geometry Problems than of a cone, or as generated by the focus-directrix method, has given rise than of a cone, or as generated by the focus-directrix to theor modern included while in other cases their solutions are are suggestions. method of always r Pedoe, 1989, p. xii), Pedoe comments that Japanese temple geometry the discoveryappears to have pre-dated of of several well-known theorems W and ellipses involving circles as well as including many theorems Theorem, which have never appear t n e Example 1. Miyagai Prefecture, 1892 c n i V A circle with centre O and radius r touches another circle, centre O and 1 1 2 & t radius r , externally. The two circles touch the line l at points A and B respec- n 2 e c tively. n i V 2 Show that AB = 4r1r2 Figure 3. Two tangent circles. 2 2 2 O1P + PO2 = O1O2 r r 2 AB2 r r 2 ()1 − 2 + = ()1 + 2 2 2 2 2 2 r1 − 2r1r2 + r2 + AB = r1 + 2r1r2 + r2 AB2 4r r = 1 2 Example 2. Gumma Prefecture, 1824 A circle with centre O3 touches the line l and the two tangent circles, with centre O1, radius r1, and centre O2, radius r2. Show that ) 1 ( 8 1 l a Figure 4. Three tangent circles. n r u o J s c i t a m e h t a M r o i n e S n a i l a r t s u A 10 J a p Example 3. Aichi Prefecture, 1877 a n e ABCD is a square of side a, and M is the centre of a circle with diameter AB; s e t CA and DB are arcs of circles of radius a and centres D and C respectively. Find e m p l the radius, r1, of the circle, centre O, inscribed in the area common to these e g circles. Find also the radius, r , of the circle, centre O', which touches AB and e 2 o m touches CA and DB externally as shown. e t r y Figure 5. Inscribed circles. Example 4. Fukusima Prefecture, 1883 ABCD is a rectangle with AB = . Semicircles are drawn with AB and CD A as diameters. The circle that touches AB and CD passes through the intersec- u s t r tions of these semicircles. Find the area S of each lune in terms of AB. a l i a n S e n i o r M a t h e m a t i c s J o u r n a l 1 8 ( Figure 6. Lunes problem. 1 ) 11 12 Australian Senior Mathematics Journal 18 (1) Vincent & Vincent two par Now we will find the area ofeachlune. Now wewillfindthearea BC' ABC Ex visualisation ofacir Examples 5and6r 2. Pr of the two semicircles. Consider semicircle Considersemicircle of thetwosemicircles. r the known tohaveinvestigatedlunesandshownthatitwaspossiblesquare egion betweentwocir am The area of the lune is therefore independentof oftheluneistherefore The area First of all we will show that the circle does pass through theintersection doespassthrough First ofallwewillshowthatthecircle = is atrianglewithrightangleat ple 5.F BC oblems basedontriangles ts ofequalar . Pisthepointon r om anear ely onknowledgeofsimilartriangles.Example7requires cle andanellipseasplanesectionsofacylinder. ea. Showthat2 cles (Cooke,1997). l y 19th century bookoftemple geometry problemsy 19th Figur BC such thattheline e 7.Equalar PC' C = , and eas pr AB DHC . oblem. C' . Let is thepointon C'P divides thetriangleinto BC π . Hippocratesisalso = a . So AB AB = . such that Japanese temple geometry Australian Senior Mathematics Journal 18 (1) 13 . F at MD meets AC . AB . DCF . AB is the midpoint of to M P cle in triangle , and from from Figure 8. Inscribed circle problem. Figure 8. Inscribed circle a PH is a square of side is a square ample 6. Miyagi Prefecture,ample 1877 Find the radius of the incir Ex ABCD Draw a perpendicular 14 Australian Senior Mathematics Journal 18 (1) Vincent & Vincent example transfor anellipsemaybetransformedwhereby The ellipseinthis intoacircle. factor of A Miyagi Prefecture, 1822 7. Example Figure 9transformsFigure intotheconditionthattriangle semiminor axis Show that the area oftriangle Show thatthearea equal. are . , B Japanese templegeometerswer and C b/a are three points on an ellipse with centre pointsonanellipsewithcentre three are . Theconditionofequalar b ms intoacir such thatthear Figure 9. Ellipseproblem. cle withradius eas e conversantinaffine transformations, S 1 , S eas forthecurvilinear trianglesin 2 and b by scalingthemajoraxisa S 3 of thecur A'B'C' O , semimajoraxis is equilateral. vilinear triangles a and Japanese temple geometry Australian Senior Mathematics Journal 18 (1) 15 . a BE for n / π e 11, . a and tan n / uction is “right over π is the width , sin FC n . a / π , so that e sent thus in Japan’ (p. 132). In Figur Figure 11. Pentagon problem. of the pentagon in terms of t , probably derived from their relationship to the sides of their relationship derived from , probably n Figure 10. Pentagon formed from tying a knot in a strip of paper. [love letters] wer e the edges of the strip of paper -gons inscribed in or circumscribed about a circle of unit diameter. about a circle -gons inscribed in or circumscribed doux ar n [s] CD Fukagawa and Pedoe note that ‘the method of constr The authors of this article have used the exact value of cos 36° in the solution to Example 9. Calculate the side Billet and Construct pentagon by tying a knot in a strip of paper of width a regular Example 8: From an early 19th century book of temple geometry 8: From an earlyExample problems 19th These problems suggest that, like the Pythagoreans, some Japanese temple suggest that, like the Pythagoreans, These problems geometers found the form of the pentagon, associated with the golden ratio, a geometrically pleasing one.