Japanese Temple Geometry Problems and Inversion

Transcription

1 hp://dxdoiog/ /kekyu Noe Japaese Temple Geomey Poblems ad Ivesio Keji Hiaoka* ad Aljosa Maulic** Ioducio I acie Japa mahemaicias used o cay hei bes heoems o a shie o emple ad hag hem somewhee o he wall Such heoems wee usually paied o piece of woode boad If hee was a geomeic cosucio he quie ofe he picue was vey coloful Someimes such a picue coaied decoaios wih flowes plas mouais ec Some of hem wee eal pieces of a I was pobably a way o hak he Gods fo he mome of eligheme while solvig he poblem They wee simulaeously woks of a eligious offeigs ad woks of mahemaics They ae called sagaku which simply meas mahemaical able These wee hug i Buddhis emples ad Shio shies houghou Japa ad fo ha easo he eie collecio of sagaku poblems has come o be kow as emple geomey The yea 6 is he ime whe we should sa lookig fo he oigis of wasa Japaese mahemaics of he Edo peiod A ha ime Japa was coolled by he daimyo o i wese ems walods who wee sill fighig fo domiacy Some of hem wee vey poweful ad he couy was i a cosa sae of ues Figue Sagaku of he Takemizuke shie Nagao pefecue * Pofesso Hioshima Uivesiy of Ecoomics Hioshima Japa ** Mahemaics ad Compue Sciece Teache Juio High School of Svei Maej Viskovo Coaia

2 I 6 duig he famous Sekigahaa bale he daimyo wee defeaed by Tokugawa Ieyasu Thee yeas lae Tokugawa Ieyasu became he shogu of Japa This was he saig poi of a ew peiod i he hisoy of Japa a peiod of almos 5 yeas wihou wa kow as Gea Peace Afe he bale Tokugawa Ieyasu moved o a small a ha ime povicial ow of Edo oday s Tokyo Fo his easo he ule of he Tokugawa is also kow as he Edo peiod The couy was uied ad may chages saed akig place This was also he ime whe he Spaish Pouguese ad Duch ied o sele dow i Japa ad seghe hei ade A he same ime missioaies fom hese couies waed o cove as may souls as possible The ade wih foeiges was o cosideed poblemaic howeve covesio of people o Chisiaiy was o vey welcomed by he wo mai eligios i Japa Shio ad Buddhism This was i fac he mai souce of esios i he couy I ode o keep people calm Tokugawa Ieyasu issued a edic odeig he Pouguese ad Spaish o leave Japa emoval of missioaies he desucio of all Chisia chuches ad fobiddig Chisiaiy i Japa Tokugawa Ieyasu died a few yeas lae bu his gadso Tokugawa Iemisu fiished he ask of emovig he foeiges I 64 hee wee pacically o foeiges lef i Japa All of hese chages wee he caalys fo ew peiod i Japa someimes called sakoku o a Closed couy Closig he couy did o have exclusively egaive effecs Mos impoaly i sopped boh ieal ad exeal coflics I also foced ad i fac helped he Japaese o develop hei ow foms of a ad sciece The local a sciece ad culue saed developig apidly This was also he case wih mahemaics I his peiod Japaese mahemaics (wasa) was bo ad developed I is difficul o say i wha yea exacly he adiio of sagaku bega bu he oldes suvivig sagaku daes fom 68 ad was foud i Togachi pefecue Yamaguchi Kaza ieeeh ceuy mahemaicia meios i his avel diay a eve olde able fom 668 bu ha oe is ow los Ove he ex wo ceuies he ables spead ad appeaed all ove Japa i Shio shies ad Buddhis emples wo hids of hem i Shio shies May of he sagaku meioed i coempoay mahemaics exs wee los bu we ca guess ha hee wee oigially housads moe ha he 9 ables which exis oday This pacice of hagig ables gadually died ou afe he fall of he Tokugawa shoguae bu some examples dae as lae as 98 The laes sagaku wee discoveed i 5 Five ables wee foud i he Toyama pefecue Ealie ables wee geeally abou 5 cm by cm bu lae ables wee someimes as lage as 8 cm by 9 cm each displayig seveal geomey poblems Some of he Japaese emple geomey (sagaku) o poblems of Japaese mahemaics befoe Meiji peiod (wasa) ca be solved by usig eally useful mehod of ivesio Thee ae may poblems wih muliple cicles wih a coac wih oe aohe The mai example was a poblem poposed by Hoa Jisuke ad hug i 788 a he Yaagijima Myōkedō emple of Tokyo Yoshida Tameyuki solved his poblem wih adiioal mehods his soluio has bee foud i a upublished mauscip Soluios

3 Japaese Temple Geomey Poblems ad Ivesio o Shipeki Sapō Poblems Yoshida s oigial soluio of Hoa s poblem was solved by usig a Japaese equivale of Descaes heoem bu his poblem ad may simila oes ca be solved moe easily by echique kow as ivesio Ivesio was discoveed by wese mahemaicias bewee 84 ad 845 This mehod was ukow o Japaese adiioal mahemaicias Hoa s poblem ad is adiioal soluio As show i Figue a lage cicle of adius coais wo cicles ad each of adius = which ae boh age o each ohe ad ouch cicle ieally The boom cicle also ouches a chai of cilces as illusaed Fuhe a chai of cicles wih adius is placed bewee he cicles ad such ha each ouches as well as cicles ad Fid ad i ems of Figue If hee cicles of adii ad ouch each ohe ouch a small cicle of adius exeally ad ouch a cicle of adius ieally as show i a Figue ha he followig elaio hold: = = Figue The Descaes cicle heoem gives he elaioship bewee he adii of fou muually age o kissig cicles Yoshida uses Descaes cicle heoem o successive iples of cicles o iducively esablish a ecusio elaioship fo he ad Fo simpliciy of he calculaio we will ake = a ad = p which we will be used i fuhe calculaio Le s fid : = p = a Le s fid : Usig Descaes cicle heoem fo { } we ge = o ( ) = ( ) 4a 4a p a a a p a

4 Fom above we ge p = a o Le s fid : Usig Descaes cicle heoem fo { } we ge = o ( ) = ( ) 4a 9a p a a a p a Fom above we ge p = 6a o 6 Le s fid 4 : Usig Descaes cicle heoem fo { }we ge 4 = o 4 8a p = 7a p Fom above we ge p4 = a o 4 Le s fid : Usig Descaes cicle heoem fo { } we ge = p p p a p p p a o p a p p a p = a p Regadig his as a quadaic equaio i x he wo soluios ae x p x x = p p = a p = p = ad x = p The o p p p = a which is he desied ecusio elaioship The geeal soluio: p = a p = a= a a p = 6a= a 4a p4 = a= a 9a p5 = 8a= a 6a ad p = a a which yields = To fid Yoshida was usig Descaes { } cicle heoem fo we will ake q o be q = Fo simpliciy ( ) = p p q p p p q p Leig p = a a fom above we ge quadaic equaio i q ( ) a = q a 7 q { } q a { q a}= The we have soluios q ad q such ha q a q = a = We discad secod soluio ad we ge he fial esul = o = 4 4 which was wie o he able Befoe we show a soluio we ge by usig a mehod of ivesio we have o ioduce ivesio Ivesio Defiiio of Ivesio Ivesio is he pocess of asfomig pois P o a coespodig se of pois P kow as hei ivese pois Two pois P ad P ae said o be iveses wih espec o a ivesio cicle havig ivesio cee T = ( x y) ad ivesio adius k if TP is he pepedicula foo of he aliude of TQP whee Q is a poi o he cicle such ha TQ is pepedicual o PQ

5 Japaese Temple Geomey Poblems ad Ivesio If P ad P ae ivese pois he he lie L hough P ad pepedicula o OP is someimes called a pola wih espec o poi P kow as he ivesio pole I addiio he cuve o which a give cuve is asfomed ude ivesio is called is ivese cuve o is ivese Fom simila iagles i immediaely follows ha he ivese pois P ad P obey TP k k = TP o k = TP TP whee he quaiy k is kow as he cicle powe The geeal equaio fo he ivese of he poi ( x y) elaive o he ivesio cicle wih he cee of ivesio ( x y) ad ivesio adius k is give by x = x y = y k ( x x ) x x y y ( ) ( ) k ( y y ) ( x x ) ( y y ) Figue 4 Popeies of Ivesios I his secio we ae goig o ioduce a few Ivesio heoems some of which ae goig o be used i a poof of Haa s heoem i chape 4 We will give oly he saemes of heoems wihou poofig hem The heoems will ell us how lie ad cicles ae goig o be cove elaios bewee adius of oigial ad coveed cicle ad some ohe impoa elaios Theoem A saigh lie passig hough he cee of ivesio ives io iself A saigh lie o passig hough he cee of ivesio ives io a cicle ha passes hough he cee of ivesio (Figue 5) Theoem If cicle C does o pass hough he cee of ivesio T he C ives io aohe cicle C (Figue 6) Theoem Figue 5 Figue 6 If cicle C does pass hough he cee of

6 ivesio T he C ives io a saigh lie ha does o pass hough he cee of he ivesio (Figue 7) Theoem 4 If is he adius of C ad is he adius of C he ad ae elaed by = d k whee d is he disace bewee T ad he cee of C Theoem 5 If L is he legh of he age fom T o he ivese cicle C he L Figue 7 = k (Figue 8) Theoem 9 If wo cicles ae age o each ohe a T hey ive io paallel lies If wo cicles ae age o each ohe a a poi P ha is o he cee of ivesio he he ivese cicles mus be age o each ohe a some poi P Poi of agecy ae peseved Theoem A cicle i s ivese ad he cee of ivesio ae colliea Theoem By he pope choice of he cee of ivesio T wo cicles ha ae o i coac ca be iveed io wo coceic cilces Theoem If fou cicles ca be iveed io fou cicles of equal adii whose cees fom he veices of a ecagle he = 4 whee 4 ae he adii of he oigial cicles 4 Soluio o Haa s poblem by usig ivesio Figue 8 Theoem 6 Poi o he cicle of ivesio ae ivaia Theoem 7 Coceic cicles whose cee is he cee of ivesio ive io coceic cicles Theoem 8 The cee of he ivese cicle is o he ivese of he cee of he oigial cicle Figue 9

7 Japaese Temple Geomey Poblems ad Ivesio Radius of he oue cicle α is adii of wo lages iscibed cicles β ad γ ae We eed o fid ad poof wha is he adius of he h cicle i oue o ie coac chais i ems of ( ad ae used o desigae he adii of he h cicle i he oue ad ie chais also o desigae he cicles hemselves) Pyhagoea heoem gives ( ) = ( ) which leads o Similaly by usig Pyhagoea heoem o small cicle (Figue ) oe ges 5 Figue o o cicles α ad β so i mus ive io a cicle ha lies bewee α ad β as show i a Figue Figue Figue Now we ll sa employig ivesio we ll ive figue wih espec of he poi T chose as show i a Figue Because hey pass hough he cee of ivesio T α ad β mus ive io saigh lies (Theoem ) ad hoizoal lie because we have chose T o lie diecly below O Fo simpliciy we will ake he adius of ivesio cicle o be k By defiiio we have TO TO The TO = so TO = Similaly fo poi B TB TB = TB = so TB = Nex we have o coside uppe cicle which does o pass ough T so i mus ive io aohe cicle (Theoem ) This cicle is age Similaly cicle is age o α β ad γ = so i mus ive io he cicle show i above picue The same is ue fo all he cicles i oue chai We ge a esul ha all he ivese cicles i oue chai have he same adius = = = = = I he same way we ge ha all he cicles of ie chai ive io cicles of equal adius = = = = = Le s elae ad o cosideig The disace fom T o a cee of cicle γ is d (by defiiio i Theoem 4) d = Theoem 4 saes ha ( d ) = which yields = = 8 = 4 Similaly = 6 Now whe we have ad i ems of

8 we ca ge ad Le L be a age fom T o as show i a picue below Fom he Figue we ca see ha x The disace L bewee T ad = ca be calculaed by Pyhagoea heoem M = L M = x By Theoem 5 L = By iseig ad L i above equaio x oe ges Fo ie chai (Figue 4) pocedue is simila L = M M = x I his case Theoem 5 gives L = By iseig ad L i above equaio oe ges x = ( ) 4 Figue Figue 4 Ackowledgme Aljosa Maulic oe of he auhos has also leaed why ad how Japaese mahemaics Wasa was developed ) We have leaed abou bih life ad deah of sagaku Sagaku is a he same ime a piece of a ad a wok of mahemaics bu also a eligious offeig which was haged i shies ad emples ad ha is why i is called emple geomey A Maulic has foud ieesig he fac ha Japaese mahemaicias i ha peiod of closed couy had simila poblems which hey ve yig o poof ad fid soluios fo as hei Wese colleagues Some aeas of hei ieess wee almos he same hey have jus used diffee mehods fo solvig he poblems Some of he sagaku poblems ae ice examples of ha May of hose poblems wee solved by usig adiioal Japaese mehods 6) If we y o solve some of hese poblems oday by usig wese heoems oe fids ha soluio ca be much simple bu he fial esul is he same Oe of he mehods which is eally useful ad acie Japaese mahemaicias did kow abou if is mehod we ioduced i his pape mehod of ivesio This mehod makes a lo of poblems udesadable ad easie o solve eve o a high school sudes Alhough is simple ad useful his mehod is o well kow oday because i is augh i schools o uivesiies ay moe (a leas o i Coaia ad may Euopea schools ad uivesiies) Refeeces ) Fukagawa H ad Rohma T (8) Scaed Mahemaics Japaese Temple Geomey Piceo New Jesey USA; Woodsock Oxfodshie Uied Kigdom; Piceo Uivesiy Pess ) Majewski M Je-Chug Chua Nishizawa H The New Temple Geomey Poblems i Hioaka s Ebisui Files hp://acmmahadechog/ep/ivied/ 5_88pdf ) Vice J ad Vice C Japaese emple geomey uivesiy of Melboue hp://fileseicedgov/fullex/ej74pdf 4) Muaa T Idigeous Japaese Mahemaics Wasa hp://fomalhaupsasakuaejp/sciece/muaa/ Idigeouspdf 5) Wolfam Mah Wold hp://mahwoldwolfamcom/ivesiohml 6) Piees hps://wwwpieescom/pi/ /