Pythgors theorem revised : + =? Kees Lemmes 0th Jue 2003, v.2 My 2004 Itrodutio I most sietifi mthemtil pulitios yet usolved prolem (or t lest oe tht is thought to e usolved :) is desried i few lies d the the rest of the rtile is out wys to solve the prolem usig erti ew d stoishig pproh. I this rtile however I d like to disuss prolem tht is thought to e solved lredy more th 2000 yers go : the Pythgors theorem out retgulr trigles tht gives the legth of the diste etwee the edpoits of 2 perpediulr lies tht strt i the sme poit. I ll try to show tht the ommoly epted solutio for the legth of the hypoteuse s give y equtio () is ot i ll ses the most ovious solutio. 2 + 2 = 2 () lthough mesuremet of retgulr trigle immeditely shows you tht the formul give y Pythgors yields ideed orret swer, I do hve resos to elieve tht i some ses the swer evertheless seems to e give y equtio (2) : + = (2) Ok, I her the lughter ow d you my thik wrily out those silly omthemtiis tht wt to iterfere with mteril they hve o kowledge out, ut efore you disrd this rtile s solute osese, plese red t lest the ext setio efore doig so! 2 Prolem desriptio If we look t figure we see the stdrd trigle s used i most textooks to disuss Pythgors theorem. If we proeed from over to the swer is give y equtio
Figure : The si Pythgors prolem (2), ut if we move diretly from to the swer is give y equtio (). So fr o prolems whtsoever. Let us ow try to follow the stirse pth s show i figure 2. If we ompute the totl diste l from to over this pth it is esy to see tht the swer is give y : l = 2 + 2 + 2 + 2 = + 2 2 2 2 Figure 2: The diste from to usig 2 steps I similr wy we ompute the totl diste l from to for figure 3 : l = 4 + 4 = + d for the geerl se i figure 4 : l = + = + (3) Eve i this lst se it seems ovious tht the diste l lwys equls +, o mtter how lrge we hoose, so it looks s if we my write : 2
Figure 3: The diste from to usig 4 steps l = lim { + } = + (4) Eh step of the stirse is defied y trigle with sides d d i the limitig se for eh retgulr side of the trigle must eessrily hve legth of zero, so oviously ll poits of the trigle iludig the lowerright orer must e loted o the lie -! (s is lso idited y the rrow) However, the diste from to is eve i tht se equl to + ordig to equtio (4), whih is still quite differet from the stdrd swer s give y the Pythgors theorem. Figure 4: The diste from to usig steps We lso look t the prolem from slightly differet poit of view : osider stirse-shped lie goig from to tht osists of ifiite umer of stirse steps with ifiite short sides d d d. I tht se the legth of the lie is give y : l = Z Z d + d = + whih is gi the sme result s we otied erlier. 3
3 Possile Expltios lthough the resoig ove seems plusile eough eve simple exmple shows tht the swer is ot wht we expet from prtie. Suppose =2 d =6 : the ordig to Pythgors the hypoteuse =20, while ordig to the method from ove the diste l=28. Mesuremet shows tht the diste is ideed 20 d ot 28, so wht s wrog with the theory? I my opiio the questio should NOT e why the 2 methods give differet swer for the sme (pperet) legth, ut should e whether the 2 distes d l re or should e ideed the sme thig d if ot, wht is the the differee? s my yer old so otied whe I ofroted him with this prolem : i the first se the lie hs oly 2 fixed poits whih re the strtig poit d the edpoit. I the seod se however the lie osists of ot oly the strtig d edig poit ut hs my extr poits etwee. Of ourse he ws solutely right (I m little preised ;-), ut why should tht mke y differee if ll poits re extly o the lie -? possile expltio my e foud from prolem tke from hos theory : osider ostlie, let s sy the diste from Hoek of Holld to De Helder mesured over the Northse ost. If we hve very simple mp this diste wo t e y loger th the legth of stright lie goig from HvH to DH, ut the more detiled the mp, the more urves we ll oserve i the ostlie d thus the loger the mesured diste will e. d suppose tht we relly wlk the diste over the eh d mesure every tiy urve log the wy? Or tht we ll use mirosope to exmie eh sigle gri of sd? Oviously the diste will irese with the mout of detil we tke ito out d there is o reso to elieve tht there is y limit to this umer. So, the orret swer is y umer etwee the shortest diste over the stright lie d ifiity! However, lthough i our prolem from ove we do fid loger diste th the shortest oe possile, we defiitely do ot fid oe tht goes to ifiity. Furthermore, if we use ifiite umer of steps - therey forig ll poits from these steps to e effetively o the sme lie s - d so they should t hve y ifluee t ll - we still fid lrger diste th simply, whih t lest otrdits our ituitio. 4 Future reserh I hve t ee le to solve this prolem yet, ut of ourse I m oly physiist, without my mthemtil skills :) yoe with more skills who feels him/herself 4
tempted to give stisftory expltio for this prolem? Or mye the prolem ws lredy solved y someoe else log go, ut I simply ws t le to fid the result? I do hve some possile ides where (prt of) the expltio my e foud, ut hve t ivestigted them yet. It my however e useful to metio some of them here : Priiple of the lest tio? lultio of vritios? Theory of lie itegrls? Frtl dimesios? Referees [] De iteresstse ewijze voor de stellig v Pythgors, ruo Erst, (Epsilo, 2002) [2] Jmes Gleik, hos, (Pegui, reprit editio Deemer 998) [3] Hughes e reht, Viieuze irkels, (ert kker, tweede druk 988) 5