The Mathematics of Origami

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Shanna Hawkins
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1 Puish y th Appi Proiity Trust Appi Proiity Trust Th Mthmtis of Origmi SUDHARAKA PALAMAKUMBURA Origmi is wispr rt form gining popurity mong mthmtiins for its rmrk iity to prform gomtri onstrutions. This rti provis rif introution to th mthmti spts of origmi n shows how origmi n us to prform two wknown onstrutions, ng tristion n ouing th u, whih r impossi to sov using ony th ompss n th stright g. Introution Origmi is th rt of rting supturs through ppr foing. Th wor origmi is riv from two Jpns wors: Oru, whih mns fo, n Kmi, whih mns ppr. As th nm impis, in origmi th foing of ppr is us to rt signs. Cutting n guing pprs r not onsir origmi thniqus. It is iv tht origmi strt roun 105 AD with th invntion of ppr. Athough th xt p whr origmi ws orn is sujt to muh t, it is iv to hv gun in Kor, Chin, or Jpn (s rfrn 1). Origmi gn s form of rt n inspirs oth young n o u to th vst numr of sign possiiitis tht it opns (for som xmps, s In som ountris, origmi is prt of th utur; in Jpn nwy w oups n nworn is r prsnt with thousn origmi ppr rns to symoiz ong if n hppinss, whi in Chin th urning of prtiur origmi mos is trition t funrs (s rfrn 2). In th t nintnth ntury thr ws spur of intrst in stuying th mthmti spts of origmi. In 1893 T. Sunr Row puish ook nm Gomtri Exriss in Ppr Foing (rfrn 3) whih iustrts th gomtri spts of origmi. Sin thn, onsir fforts hv n m in orr to mthmtiz origmi. In 1985 Jpns mthmtiin nm Humiki Huzit prsnt st of six xioms whih trmin wht is onstruti through origmi fos. Ltr, in 2001, Koshiro Htori on mor xiom to th otion (s rfrn 1). Th svn xioms r wiy known s th Huzit Htori xioms. On of th pionrs of morn origmi, Rort J. Lng, rnty prov tht this systm of xioms is ompt (rfrn 4), i.. vry sing fo in origmi orrspons to on of th svn xioms. Huzit Htori xioms It is trition prti to us squr ppr in origmi. Hn, in this rti ppr rfrs to squr ppr hrftr. A in in origmi is rs m y ppr fo or th ounry of th ppr. Initiy, ppr onsists of four ins: its ounris. A point is th intrstion of two ins. Th Huzit Htori xioms fin wht n onstrut vi sing fo in trms of ins n points n r iustrt in figur 1. (A1) (A2) Givn two points, on n fo rs in through thm. Givn two points, on n fo rs ong thir prpniur istor, foing on point on top of th othr.

This rti provis rif introution to th mthmti spts of origmi n shows how origmi n us to prform two wknown onstrutions, ng tristion n ouing th u, whih r impossi to sov using ony th ompss n th stright g.

2 13 ( ) A1 ( A2 ) ( A3) ( A4 ) ( A5 ) ( A6) ( ) A7 Figur 1 Huzit Htori xioms, sh ins rprsnt rss n soi ins rprsnt xisting ins. (A3) (A4) (A5) (A6) (A7) Givn two ins, on n fo thir istor rs, foing on in on top of th othr. Givn point n in, on n rs through th point prpniur to th in, foing th in onto itsf. Givn two points n in, on n fo rs through on point tht mps th othr point onto th in. Givn two points n two ins, on n fo rs tht simutnousy mps on point to on in n th othr point to th othr in. Givn on point n two ins, on n fo rs prpniur to on in so tht th point mps to th othr in. Th powr of origmi On of th intrsting spts of origmi is its iity to go yon trition ompss n strightg (unmrk rur) onstrutions. Ang tristion n ouing th u r two mjor proms whih hunt mthmtiins for ovr 2000 yrs. In 1837 th Frnh mthmtiin Pirr Lurnt Wntz prov tht oth of ths onstrutions r impossi using ony th strightg n ompss (rfrn 5). Howvr, oth onstrutions n rri out using origmi fos. Tristing n ng Tristing n ng is th prom of iviing givn ng into thr qu prts using th ompss n strightg. In 1980 Hishshi A isovr n gnt mtho for tristing givn ng using origmi fos (rfrn 1). Figur 2 iustrts A s tristion mtho. Th givn ng is mrk in th figur s. Not tht th fos in this onstrution gr with th Huzit Htori xioms. 1. Th first fo (figur 2()) fins th istor rs orrsponing to th two ounris of th ppr. This is in orn with (A3).

Givn two points n in, on n fo rs through on point tht mps th othr point onto th in. Givn two points n two ins, on n fo rs tht simutnousy mps on point to on in n th othr point to th othr in.

3 14 f () () /3 () () () Figur 2 A s mtho to trist n ng. Dsh ins rprsnt rss n soi ins rprsnt xisting ins. 2. Simiry, th son fo (figur 2()) fins th istor rs orrsponing to th owr ounry n. This is so in orn with (A3). 3. Th thir fo (figur 2() n ()) is y (A6) n fos onto n onto f ; mps to. 4. Th st two fos (figur 2()) r y (A1) n fin th ins n. To s how A s mtho trists n ng, w n to onsir th rs pttrn of th mtho (figur 3). Th rs hg mps th thr points,, n to,, n, rsptivy. Thrfor, hi h i = =, (1) =, (2) h α i g Figur 3 Crs pttrn of A s onstrution for tristing n ng.

Th st two fos (figur 2()) r y (A1) n fin th ins n. To s how A s mtho trists n ng, w n to onsir th rs pttrn of th mtho (figur 3).

4 15 n By (1), (2), n (3), is ommon to oth n. (3). (4) Sin is th prpniur istor of, thn = n, from (4), =. Sin is th mipoint of n is th mipoint of, it foows tht = = α. (5) Not tht = = 90 α = g = α. Sin g = g, g = g = α. Thrfor, y (5), = = g = α n th ins n trist th ng. Douing th u Douing th u is n nint prom tht ws prov unsov using th ompss n strightg. Givn u of voum V n si ngth, th prom is whthr u with voum 2V n si ngth 3 2 n onstrut. Th prom n ru to iviing givn in sgmnt into two prts, n, suh tht : = 1: 3 2. In 1985, Ptr Mssr foun mtho to ou th u through origmi (rfrn 1; s figur 4). 1. Th first fo (figur 4()) fins th istor rs. This is in orn with (A3). 2. Th point is rought onto (figur 4() n ()). This is n (A2) fo. Hr w fin th nw point whih is tristion point of th si pq. Th rmining tristion point n otin y mking th istor rs twn p. Hn, th squr is ivi into thr qu prts (figur 4()). p () () () p g q () () f q (f) Figur 4 Mssr s mtho of ouing th u.

Givn u of voum V n si ngth, th prom is whthr u with voum 2V n si ngth 3 2 n onstrut. Th prom n ru to iviing givn in sgmnt into two prts, n, suh tht : = 1: 3 2.

5 16 h j Figur 5 Tristing th si of th squr. 3. Finy, th points n f r mpp onto th two ins n pq, rsptivy (figur 4() n (f)). This fo is y (A6). Th nw point g is foun, whih ivis th si pq y th rtio, gq : pg = 1: 3 2. In th first hf of this onstrution, th ppr is ivi into thr qu prts, s iustrt in figur 5. Lt th ngth of on si of th squr. Thn = h = /2. Consir hj. By th Pythgorn rtion h 2 + hj 2 = j 2, w hv (/2) 2 + hj 2 = j 2. Howvr, hj + j = ; thrfor, ( ) 2 + ( j) 2 = j 2 = j = 5 3, hj = Hn, th rtio twn th sis of hj is, hj : h: j = 3: 4: 5( hj turns out to sr mutip of 3: 4: 5 right-ng tring!). Sin hj n r simir trings, = h hj = /2 = 4 3 = = 2 3. Hn, is tristion point. Aftr iviing th ppr into thr qu prts th rminr of th onstrution (s figur 4() n (f)) ivis pq in th rtio 1: 3 2. Figur 6 pits th unrying gomtry. Lt pg = y n gq = x. Consiring gqr, w hv = x tn + x s = = x(tn tn 2 ) ( sin = = x + 1 sin 2 = = x 1 + sin2 1 sin sin 1 sin. (6) ) Consiring gst, w hv sin = y /3. /3

Thn = h = /2. Consir hj. By th Pythgorn rtion h 2 + hj 2 = j 2, w hv (/2) 2 + hj 2 = j 2. Howvr, hj + j = ; thrfor, ( ) 2 + ( j) 2 = j 2 = j = 5 3, hj = 2 8 8.

6 17 y x 3 t p 3 g 3 s 3 q r Figur 6 Diviing th si y th rtio 1: 3 2. Sustituting this rsut into (6) yis 3y = x 2 3y. Sin = x + y, 3y x + y = x 2(x + y) 3y. Simpifition of this xprssion s to (x/y) 3 = 1 2 ; thrfor, qg gp = Aknowgmnts Th uthor wou ik to thnk n it this rti to th mmrs of th mzing mthmti forum Mth Hp Bors ( forum/). Thir ontriutions m m ook t mthmtis from iffrnt prsptiv. Rfrns 1 E. D. Dmin n J. O Rourk, Gomtri Foing Agorithms: Linkgs, Origmi, Poyhr (Cmrig Univrsity Prss, 2007). 2 F. Tmko, Ppr Pns n Jumping Frogs (Chin Books, Sn Frniso, 1998). 3 T. S. Row, Gomtri Exriss in Ppr Foing (Th Opn Court Puishing Compny, Chigo, 1917). 4 R. J. Lng, Origmi n gomtri onstrutions, hj.php (2010). 5 D. M. Burton, Th History of Mthmtis: An Introution (MGrw Hi, Nw York, 2006). Suhrk Pmkumur is n unrgrut stunt t th Univrsity of Prniy, Sri Lnk. His mthmti intrsts inu ryptogrphy n oing thory. In his spr tim, h iks to ri his iy n istn to musi.

om/ forum/). Thir ontriutions m m ook t mthmtis from iffrnt prsptiv. Rfrns 1 E. D. Dmin n J. O Rourk, Gomtri Foing Agorithms: Linkgs, Origmi, Poyhr (Cmrig Univrsity Prss, 2007). 2 F.

Schedule C. Notice in terms of Rule 5(10) of the Capital Gains Rules, 1993

Schedule C. Notice in terms of Rule 5(10) of the Capital Gains Rules, 1993 (Rul 5(10)) Shul C Noti in trms o Rul 5(10) o th Cpitl Gins Ruls, 1993 Sttmnt to sumitt y trnsror o shrs whr thr is trnsr o ontrolling intrst Prt 1 - Dtils o Trnsror Nm Arss ROC No (ompnis only) Inom Tx