Origami: Bhoumitic Rachana (Marathi Language)

7/30/2019 Origami: Bhoumitic Rachana (Marathi Language)

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&& BXZ __ &&

m. {_qbX Am|H$ mZr `mdoir Amo[aJm_rMr ~arM nwVHo$ AmUy Z

{Xbr V|hm Iyn H$m_ AmbohUyZ _r Iwe hmoD$Z W_V: Vr MmiyZ _JHw$Ry>Z gwadmV H$am`Mr VoR>a{dbo. `mM doir S>m. bJ `mMo Origami andGeometric Constructions. ho_amR>rV ^mfmVa H$mam`Mo Z$s Ho$boVerZm|X nU _aU dhrV Ho$br. H$mhr H$mimZVa hUOoM gw_mao XmoZ VrZ_{h`mZr _aU dhr MmiVmZm hm {ZM` AmT>ibm Am{U VmH$miH$m_mbm bmJbmo. `mMr n[aUrVr AmO hmoVmZm g_mYmZ hmoV Amho.

honwVH$ H$moU dmMUma? hm Z hmoVmM Am{U `mbm ~mOma _y `ey`. _J H$embm hm IQ>mQ>mon Ago dmQo>b. nU nwVH$ dmMUo Am{U^mfmVarV H$aUo m _`o Amn`mbm Vo OmV H$iob. Ho$di dmMZmZo{VVH$m H$me nS>Uma Zmhr hon> hmoVo. `m_wiohm m{dS>r mUm`m_.

_wimV J{UVmMm A`mg H$_r Am{U AmdS A{YH$,Amo[aJm_r WmVWmM

Am{U Sm. bJ gma`m {dd {d`mVmMonwVH$ nU Ho$boYmSg !

^maVmV Amo[aJm_r gw hmoD$Z Odi Odi 40 df}Pmbr nU dV:Mr aMZm H$aUmao\$maM H$_r . `m_wiodVMr aMZm H$a`mgmR>r _mJXeZhmoBb hr ^mdZm. OoAmnUmgr R>mdo, VogdmZm gmJmdohr H$nZm.

m nwVH$mV _mPo`moJXmZ ZJ`. nwVH$ {XboAm|H$ `mZr , _amR>r

Q>H$ boIZ Ho$bo{eH$dboAm|H$ mZr , _mJXeZ bm^boVogwX lr.nmR>mao,lr. adr AmnQo>, {demIm AmnQo>, lr.H$_boe JmYr, lr. aZmH$a _hmOZ,lr. lrnmX ZmS>H$Uu `mMo. _yVdn Xo`mMoH$m_ AmnybH$sZoHo$boVoS>m.H$me gmJwS>}H$a, {dbmg gmJwS>}H$a Am{U `mMoghH$mar `mZr. gOmdQ>dJaogd H$mhr `mMoM. `mMm G$Ur Amho.

dmMH$mZm gmJmdogo dmQ>Vo H$s AmnU J{UV {eH$Vmo Bpbe_YyZ`m_wio J{UVmgmaIm {df` Am{U _amR>r ^mfm `m_wio pbQ>Vm dmT>Vo.Var nU Ho$di dmMZ Z H$aVm `mMm dmna `jmV Ho$bm Va Y` hmoBZ.


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g_Wm`m CVr_mUo "AmYr Ho$boAm{U _J gm{JVbo' `m_mUo _mPr""H$mhr'' Tesselations_bnR>mda nmhVm oVrb.

Vohm `m MwH$m AgVrb `m H$a`mMoH$m_ _r Ho$bo Amho. ""dmV

gwIm`'' Ho$boboho`Z H$moUm`m Var Cn`moJr AmboVar ^$Z nmdbmoAgo_r hUoZ

An_Vr Ag`mZoA{YH$ {b{hUog w{$H$ hmoUma Zmhr, `m_wioBWoM aOm `mdr hoCm_ .

- O. dm. nmJo


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AZwH$_{UH$m

1. am~Q bJ 72. Amo[aJm_r = ^y {_VrVrb aMZm 8

3. {_mZ AH$ nVrZo H$mJXmMo ^mJ H$aUo 12

4. n[a_o Any UmH$ (Rational fractions) 21

5. \w$Or_moQ>mo aMZm(Fujimoto Construction) 266. h JmMr aMZm Umbr 32

7. An[a_o` JwUmoma(Irrational Proportions) 35nwZamdVuV hmoUmao d AZVmn`V OmUmao AnyUmH$

8. Quadratic surds 41

9. H$moZmMr {d^mJUr 47

10. _yi VdmqH$V Amo[aJm_r 55

11. mW{_H$ Amd`H$Vm 59

12. JhrVH$ ghm Am{U KZ-g_rH$aUmMo JmobmH$ma 68

13. gJUH$mZo AMyH$ e`Vm 74

14. gX^ 81


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am~Q> bJ

S>m. bJ ho Jobr gw_mao Mmirg df} Amo[aJm_r H$aV AgyZ, OJmVrb EH$

{d`mV Amo[aJm_r aMZmH$ma AmhoV.

Amo[aJm_r Am{U J{UV `mMr gmJS Kmb m`m {H$òVrb EH$ _wI dAmJhr `$s_d AgyZ 2006 gmbt Origami Science , Mathematics andEducationm AmVaamr` n[afXoMo {Z_H$ hmoVo d H$m`_Mo _wI gX` AmhoV.

`mZr AVamimVrb Xw~uU, H$ma AnKmVmV Ord dmMdUmarAir - bagsMr

aMZmH$Vr, Amf{YH$ narjUmgmR>rMr CnH$aUo, `mgmR>r Amo[aJm_rMm dmna H$gmH$amdm hUyZ OmJ{VH$ Vamda gmJma hUyZ Cm_ H$m_ Ho$bo Amho.

`mZr dV: d g`w$nUo Amo[aJm_r aMZmH$VtMr AZoH$ nwVHo$ d nwH$iboI {b{hbo AgyZ, Amo[aJm_r, J{UV, {dkmZ, d VkmZ m {df`mda OJâAZoH$ `m`mZo {Xbr AmhoV.

`mZr {bhbo`m AmoarJm{_`m nwVH$mVyZ AZoH$ AdKS> d Zg{JH$ gmY`VVmoVV nmiUm`m aMZmMr {Z{_Vr {dfX Ho$br Amho. {deofV: ""H$sQ>H$,{H$So,Am{U mUr.'' `mZr `mgmR>r gJUH$mMm Cn`moJ Ho$bm AgyZ A`m aMmZmdbr

~Z{d`mgmRr gJUH$mMr Amkmdbr - Tree maker ,Referance point.~Z{dbrAmho. Vr gdmZm CnbY nU Ho$br Amho.

Amo[aJm_r _ò gJUH$mMm Cn`moJ AmUyZ `mZr VkmZmVrb J{Ver_oi gmYbm Amho. `mZr ~amM H$mi NASA Mo gmJma hUyZ nU nwH$iH$m_ Ho$bo Amho.

A{YH$ _m{hVrgmR>r nhm;-http://www.langorigami.com .m do~ da{_iob.


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Amo[aJm_r = y{_VrVrb aMZm

{df` AmoiI :

embo` OrdZmV H$nmg Am{U gai aofo`m y{_VrVrb aMZm m`mer K{Z>g~Y òVmo. Va AmO`m OrdZebrV embo` dmVmdaUmVrb EH$ Hw$Vyhb AerM^mdZm ^m{_{VH$ aMZo~m~V ~hVmMr Amho. naVw JrH$ d BOrpe`ZZm _m hr EH$

am oO`m OrdZmVrb AZoH$ H$m`nVr nH$s EH$, ^m{_{VH$ Am{U ZH$memH$mT>`mgmR>rMo gmYZ dmQ>Vo.

n>r d H$nmg `mMr _XV KoCZ gai aofm H$mT>Uo d H$nmgm`m ghm`mZodH$mH$ma H$g H$mTyZ aofoda g_mZ AVa Xe{dUo dm ZwgVrM gai aofm n>r`mghm`mZo H$mT>Uo ghO H$aVm òVo. `m Xmohr Jmo>tMm EH$mMdoir Cn`moJ hmoBb AgoZmhr. (Ogo {H$ EH$m _moR>m aofoda H$nmgm`m _XVrZo R>am{dH$ AVa Xe{dUo)

m{edm` EH$ doJir _ZmoaOH$ {H$`m hUOo H$mJXmbm KS>m KmbyZ aMZm

H$aUo. H$nmg d n>rnojm {h, {H$`m _OoXma d `jmV A{YH$ Cn`moJr Amho{deofV: Amo[aJm_r H$aVmZm. Amo[aJm_r hUOoM H$mJXmMm EH$ VwH$S>m KodyZ, ZH$mnVm d {MH$Q>drVm AZoH$ gwaoI {VH$Vr ~Z{d`mMr H$bm. EH$m H$mJXmVyZAZoH$ Ae` dmQ>Um`m {VH$Vr. Zg{JH$ gmY` d gwaoIVm gmYyZ ~Z{dUo eÀmho. ho AmYw{ZH$ Amo[aJm_rZo XmI{dbo Amho. Amo[aJm_r H$bmH$VrVrb gmX`mZo

gm_m` _mUyg VgoM AZoH$ Vk d J{UVkmZm AmH${fV Ho$bo Amho. EH$m_mJmo_mJ

EH$ gm`m Am{U ghOVoZo òUm`m KS>m KmbyZ gm` Ho$bobr {VH$Vr; `mVrb{d{dYVm; {dMmamMr Imobr, ho m AmH$fUmMo H$maU Amho. ~hVmer gd Amo[aJm_raMZm H$_dma KS>m KmbyZ V`ma hmoVmV d `mgmR>r òH$ gX^ q~Xy AWdm aofoMrAmIUr Ho$bobr AgVo. {H$òH$ doim gmonr qH$dm pb>. Ogo {H$ bm~r dm XrMo

g_^mJ H$aUo, g_Om 3,8,12 dJao. A`m aMZm _mnZ H$$Z e` AmhoV nUKS>m KmbyZ ho q~Xy{_i{dUo AmZXXm`r Amho. {deofV J{UV d ^y{_VrMm dmna

H$$Z H$_dma nVrZo ho gX^ q~Xy emoYyZ H$mT>`mMm AmZX g_mYmZXm`H$ Amho.hUyZM EH$ {d{e> _mU~ mJ ~Z{dVmZm EImm ~mOyMm qH$dm aofodarb

q~XwnmgyZ H$moZmMm ^mJ H$mT>`mMr AmoarJm_rVrb nV V`ma H$aUo B> hmoD$Z m


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KS>m`m H$_dma nVrZo, J{UVr, d ^m{_{VH$ nVrZo ~m gmYZo dmn$ZH$mT>md`m`m nVrda _mV Ho$br. m boIm_ò Amo[aJm_rMm dmna H$$Z, AZoH$^m{_{VH$ g_^mJ H$go H$amdm`Mo Vo {dfX Ho$bo Amho. m gd Wm AZoH$ d {d{dY

AgyZ `m J{UVmVrb AZoH$ emImer AmYma^yV AmhoV. ho H$m`jo {demb Am{U{d{dY Amho BVHo$M Zho Va `mMo J{UVm_Y`m emImer Agbobo ZmVo AM{~VH$aUmao Amho.

W_V: AmoarJm{_Vrb H$mhr H$VtMr ~mYUr {_mZ g`m nVrer(BinarySystem)H$er {ZJS>rV Amho ho XedyZ _J nwTo H$mhr {R>H$mUr `mMm {dH$mgArbitrary rational fractiondmn$Z H$gm Ho$bm Amho ho gm{JVbo OmBb. H$mhr

{R>H$mUr Va An[a_o` AH$ (irrational)gwm AmoarJm{__ò dmnabo AmhoV. `mMrCXmhaUo nU AmnU nmh. `m{edm` AmnU AXmOmZo KmVbo`m KS>mMm nU{dMma H$$`m. `mMm J{UVr _mUmer g~Y Zgob. nU `jmV `mMr ghOVmd Cn`mo{JVm A{YH$ _hdmMr Agob. `mM~amo~a _mU^yV Vdmer g~{YV A`mWmMm Chmnmoh hmoBb. Ogo H$s Ho$hm AmnU KS>mZr e` Agbobr _mU~Vm

H$$ eH$Vmo d `mgmR>r H$moU`m _yb^yV Vdm`m ghm`mZo Am{U gJUH$mMr_XV KoD$Z KS>mMm H$_dma AmamIS>m V`ma H$aVm òBb `mMr A{YH$ _m{hVrAmo[aJm_r aMZm {df`r`m nwVH$mVyZ {_iob.

mW{_H$, mV{dH$ ^mJ d `m`m

^m{_{VH$ aMZm _mUoM AmoarJm{__ò AZoH$ {d{dYVm Amho. gdW__w`V: EH$ MmagmH$ma H$mJXmMm VwH$S>m KodyZ gwadmV hmoVo. (${MV gJrAm`VmH$ma, nMH$moZmH$V qH$dm BVa H$mhr bm~r, XrMo _mU Agbobo H$mJX

dmnabo OmVmV) _w` hoVy Mmagm`m H$S>mda qH$dm _ o Hw$Rohr R>am{dH$ gw{ZpMV_mUmZwgma, gX^ q~XyMr {Z{_Vr H$a`mgmR>r KS>m KmbUo hm AgVmo. m gX^q~XyMm Cn`moJ H$$Z KS>mMr H$_dmar {ZpMV Ho$br OmVo {O`m moJo gwXa {Z{_Vr_yV $n KoVo. hr H$_dma aMZm H$aVmZmgwm H$mhr gX^ q~XybmJVmV Oo H$mhr aofmMo

NoX q~XyAgVmV qH$dm H$SoMm NoX gwm AgyeH$Vmo. AmXe AmoarJm{_`m H$_dmanVrV EH$m KS>r _mJmo_mJ EH$ Ago {ZX}e {Xbobo AgVmV Am{U òH$ KS>rgmR>rgX^ q~Xyn> Ho$bobo AgVmV Ogo qH$, NoX q~Xy, H$mJXmMr H$S>m, EImXr KS>rqH$dm XmoZ KS>mMm NoX q~Xy.


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darb Xmohr H$Vr nyU Ho$`mda KS>m CKS>`mZo AmnU EH$ Zdr aofmH$VrV`ma Ho$br. VgoM AmnU XmoZ gX^ q~XynU V`ma Ho$bo. Oo ~mOy`m aofmMo _`q~XyAmhoV. `oH$ gX^ q~Xyn>nUo {ZX}{eV Ho$bm AgyZ Vmo H$S>m d KS>r m XmoZ

aofmMm NoXq~XyAmho.`m Xmohr {H$`mnVr, AmoarJm_rZo Ho$bo`m ^m{_{VH$ aMZmMr {Z`_mdbr

Xe{dVmV.

1) gd aofm m Mmagm`m H$S>m qH$dm H$mJXmdarb KS>rZo Xe{d`m OmVmV.

2) gd gX^ q~Xy ho XmoZ aofmMo NoX q~Xy AgVmV.

3) gd KS>mMr {Z{_Vr q~Xy Am{U aofmMr EH${Vm gmYyZ qH$dm H$mhr aofmMrEH${Vm H$$Z {ZpMV Ho$br OmVo.

4) ${MV gJr EH$m KS>rbm g_^mJmV gnmQ> H$$Z(SQUASH){ZpMV Ho$`mOmVmV.


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{_mZ KS>mMo JhrVH$ (Binary Folding Algorithm)

W_ hm {df` ~Q>ZZr {dfX Ho$bm d bJ `mZr `mMm {dVma Ho$bm. m

nVrZo {_mZ AH$ nVrMm Cn`moJ H$$Z Hw$R>`mhr {_mZ AnyUmH$m`m _mUmMr{Z_mU Umbr V`ma H$aVm oVo. {_mZ _mnZ nVrV Ho$di XmoZM AH$ AmhoV 1 d0. BVa gd AH$ ho m XmoZ AH$mMm YmJm dmn$Z V`ma hmoVmV. Imbrb H$mo>H$mV

Vo 1 AH$ 10 {_mZ nVrV XmI{dbo AmhoV .

Xe_mZ {_mZ

1 12 10

3 11

4 100

5 101

6 1107 111

8 1000

9 1001

10 1010

VVm H$. 1 1 Vo 10 AH$mMo {_mZ nVrV nmVa

Z

_

2 Agm Hw$R>bmhr {_mZ AnyUmH$ AmnU Z KS>m KmbyZ XmIdy eH$Vmo.Am{U Vr nV hUOoM `m AnyUmH$mMo {_mZ AnyUmH$mMo CXmhaU.

Amn`m Xe_mZ nVrVrb AmH$So {b{h`m`m arVrMm Cn`moJ H$$Z {_mZAnyUmH$ H$mT>Vm `oVmo. Xe_mZ nVrV {b{hbobm AH$ 10 m AH$m`m KmVmMr

_mS>Ur H$Z {bhrbm OmVmo. Ogo

1043=103+0x102+4x101+3x100

= 1000 +0+40+3 -------------- (1)


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hrM nV \$$ (2) Mo KmV dmn$Z hUOoM AH$ "1' d "0' dmnZ {_mZAH$ {b{hbo OmVmV. Ogo

1011 = 1x23+0x22+1x21+1x20

= 8+0+2+1= 11 ---------------------------- (2)

m_mUo Hw$R>bmhr AH$ AmnU 1 d 0 ho XmoZ AH$ dmn$Z {_mZ nVrV{bheH$Vmo. `mM nVrZo AmnU {_mZ AnyUmH$ {h Xe_mZ nVrMr {H$`m dmn$Z{bheH$Vmo.

\$aH$ EdT>mM qH$ AnyUmH$ hm Xe_mZ nVrV qQ>~ XoD$Z {b{hbm OmVmo.`mbm AmnU Xeme {Mh hUVmo BWo AmnU `m q~Xwbm {_mZ Ae hUy eHy$.CXmhaUmW 0.753 hUOo

0.753 = 7x10-1 + 5x10-2+3x10-3 = 753/1000 ---- (3)

VgoM AmnU {_mZ AnyUmH$Mo Cma

0.111 = 1x2-1 + 1x2-2 +1x2-3 = 7/8 ------------ (4)BVa H$mhr CXmhaUo nmh

Xe_mZ {_mZ

1/2 0.1

1/4 0.01

3/4 0.11

hUOoM Hw$R>bmhr AnyUmH$ `mMm NoX hm 2 m AH$mMm KmV Agob Va{_mZ AnyUmH$ nVrV q~XyZVa gw{ZpMV AH$mV {b{hVm `oBb.

Hw$R>`mhr AnyUmH$mMr {_mZ AnyUmH$ H$mT>`mMr nV :-

(1) W_ {_mZ q~Xy{bhm.(2) Amn`m AnyUmH$mg 2 Zo JwUm.

(3) m JwUmH$mamVyZ Ambobm nyUmH$mMm ^mJ (1 qH$dm 0) q~XynwTo {bhm.


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(4)n wTo nwhm H$_mH$ (2) d H$_mH$ (3) m {H$`m, òH$ doir òUmao AmH$S>oCO`m ~mOyg dmT>dV eof ey` òB n`V H$am.

mMmM AW `mdoirZ

_

2 hm AnyUmH$, {_mZ nVrV ~XbVmo V|hm {_mZq~XyZVa `mo` Vo ey` n[a_mU KoD$Z "_' hm AH$ {_mZ nVrV ~XbVmo.

naVw Oa AnyUmH$mMm NoX AH$ hm 2 AH$mMm KmV Zgob Va H$m` Agm Z

C^m amhVmo. CXmhaUmW Oa AmnU1

3hm AnyUmH$ {_mZ nVrV ~Xbbm Va

`m_ò ey` ~mH$s H$YrM òV Zmhr. Va AmnUmg EH$ AmH$S>mMr A_`mX

gmIir {Xgob.1

3 = 0.010101................Oa AH$ n[a_o` (rational)Agob Va Vmo XmoZ nyUmH$mMo JwUmoma AgVmo Am{U hUyZM AH$mMr A_`m{XV

gmIir {XgyZ oVo.

AnyUmH$mMm ~Xb {_mZ nVrV Ho$`mda {_iUmam AH${h Vo JwUmoma{_id`mgmR>r Kmbmd`m`m KS>mMr nV AgVo. W_ AmnU H$`m KS>m

Kmbmd`m`m Vr gkm nmhy.(1) {_mZ AnyUmH$m EdT>o AVa KS>m KmbyZ {ZpMV H$a`mgmR>r `mMm {_mZ

AnyUmH$ {bhm.

(2) _J AnyUmH$m m CO`m ~mOyg gwadmV H$$Z (gdmV H$_r _hdmMm AmH$Sm) n{h`m AmH$S>mgmR>r (Omo Zoh_rM 1 Agob H$maU `m`m nwT>bo eyÀmnU AJmoXaM H$mTyZ H$mT>bobo AgVrb) MmagmMr daMr ~mOy nm`mbm

{_idyZ H$S>obm (Pinch) KS>r KmbyZ CKS>m.

(3) `mZVa Cabo`m AnyUmH$mVrb nwT>`m òH$ AH$mgmR>r Oo 1 qH$dm 0AgVrb H$Vr -

(A )AH$ 1 Agob Va H$mJXmMr daMr ~mOy AJmoXa Ho$bo`m IyUog {_idyZ Z{dZIyU H$Z CKS>m Am{U

(~) Oa AH$ ey` Agob Va H$mJXmMm Vi AJmoXa Ho$bo`m IwUoer OwidyZ ZdrZIyU H$Z CKS>m. ho JhrVH$ {_mZ AnyUmH$m`m {dVmam~amo~a VmSyZ nm{hboVa Vo H$go H$m_ H$aVo ho ghO g_Oob.


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Figure 4. (Top) Folding the bottom edge up to a crease rgives a new crease (r/2) from thebottom. (Bottom) Folding the top edge down to a crease rgives a new crease ((1+r)/2) from thebottom.


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m CXmhaUmd$Z AmnU Ago hUy eH$Vmo. qH$ H$g nVrZo H$_dmamZoAnyUmH$ {ZpMV H$$Z XmoZ H$mao AmnUmbm H$moU`mhr {_mZ AnyUmH$mMr KS>r

KmbVm òBb.

m d$Z Ago {g hmoVo qH$ n{h`m nVrnojm {nX AnyUmH$m`m ghm`mZo

KS>m Kmb`mMr nV A{YH$ Cn`moJr Amho. Z_

2 m AnyUmH$mgmR>r n{h`m

nVrV 2Z-1 BV`m KS>m Kmbm`m bmJVrV Va Xwg`m nVrV \$$ Z.

{_mZ g`m nVrZo AMyH$VoH$So OmUo. (Binary Approximations)

`m AnyUmH$mMm NoX hm 2 AH$mMm KmV AgVmo VohmM {Z[V AmH$S>m_ò{_mZ AnyUmH$ gn hmoVmo. BVa ~`mM AnyUmH$mMm {_mZ nVrVrb ~Xb hmAZV AgyeH$Vmo Am{U _J Vmo AnyUmH$ H$mhr _`mXo ZVa gw{Z[V Ho$bm OmVmo.

Omo OdiVrb Odi Agy eH$Vmo. (Close Adoroximation) Am{U ho Hw$R>`mhrAH$ Umbr_Yo hmoUmaM Ogo {_mZ, {_mZ, MVwl_mZ dm Xe_mZ nV CXmhaUmW

Xe_mZ nVrV

1

3 =0.333(AZVmn`V) hUyZ Oa Xeme {MhmZVa EH$M AH$

KoD$Z Wm~bmo Va =1 3

3 10hm OdiMm AH$ R>aob, Oa XmoZ AH$mZVa Wm~bmo Va

=1 33

3 100Omo

3

10nojm A{YH$ Odi Agob Am{U 3 AH$ KoVbo Va (0.333),

=1 333

3 1000Oo `mhZ A{YH$ Odi Agob. hrM dVwpWVr {_mZ nVrV nU

nhmd`mg {_iVo. g_Om AmnU 13

hm AnyUmH$, {_mZ nVrV q~Xy ZVa 2 AH$m

ZVa Wm~bmo Va1

4= 0.01 ho

1

3AH$mMo AJXrM {d{M d Z nQ>Umao H$` hmoBb.

nU AmnU 0.0101 bm Wm~bmo Va5

16hUOo

1

3`m ~aoM Odi nmohMy. Am{U

0.010101=21

64

KoVbo Va hm AnyUmH$ 1% nojm H$_tOdi Agob. hUOoM

AmnU Hw$R>bmhr AnyUmH$, {_mZ AnyUmH$ nVrV A`m arVrZo I{S>V H$$ eH$Vmo.AmnUmg {H$Vr AMyH$Vm gmYbr Amho ho _m{hV Agob. `m_wio AmnUmbm KS>m

KmbyZ gw{ZpMV _mUm`m ~aoM Odi OmVm òBb Am{U `mgmR>r AnyUmH$mMo


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dmadma KSr KmbyZ h>rM AMyH$Vm {_id`mMr nV AmnU BVa AnyUmH$mgmRrhr dmn$ eH$Vmo. g_Om Amnbm AnyUmH$ 1

5Amho. `mMo {_mZ nmVa 0.0011

`m gmRr KSmMr H$m` Umbr gwm VerM hUOo daMr ~mOyXmoZXm KSr Kmbm.

_J nm mH$SyZ XmoZXm KSr Kmbm Ago AmnUmg B doim H$aV amhm. gd {_mZ ZgUmaon[a_o` AnyUmH$ (non binary rational fractions) ho AZVmn`V VoM AmH$So dmadmaòUmao AgVmV Am{U `mgmRr hr nV Cn`moJmV AmUVm òBb.

mMrM CbQ> {V{H$`m Oa AmnU {dMmamV KoVbr, hUOoM W_ KS>mKmbm`Mo R>a{dbo. g_Om qH$ da`m ~mOyZo XmoZ KS>m d nm`mH$SyZ 3 KS>m Agodmadma Ho$bo Va AmnU H$moUVm AnyUmH$ AMyH$ emoYy? A`m {H$òMm {_mZAnyUmH$ 0.11000 Agob. Va _J `mMo nmVa H$a`mMr gd_m` H$m` UmbrAmho. dmadma VmoM AH$ òUmam n[a_o` AnyUmH$ ~XbVmZm, naV naV òUmam AH$

hm AnyUmH$m `m NoX aofo`m da {bhm`Mm Am{U aofoImbr NoX {b{hVmZm {VVHo$MAmH$So NoXm_ò {bhm`Mo. \$$ Or AH$ nV hdr Agob `m AH$mVyZ 1 dOm

H$am`Mm. AmnUmg {_mZ nVrV ~Xb hdm hUyZ 2-1=1 hm AH$ {OVHo$

AmH$So Aem_ò AmhoV {VV`m doim NoX _ò {bhm`Mm. hUyZ :0.11000=

11000

11111{_mZ =

24

31= Xe_mZ ------------(7)

AmnU Or iterative (dmadma )nV1

3m AnyUmH$ H$[aVm dmnabr.

(AmH$Vr 5) `m_ò1

3 d2

3 m XmohrMr EH$$nVm Pmbr. `mMo H$maU Amn`m

AMyH$Vm emoY`m`m nVrV AmnU 01 {h Umbr dmnabr Vr XmoZ AnyUmH$mgmR>rEH$M Amho Vo Ago : 0.01 Am{U 0.10 `mMo dmadma KS>r Kmb`mMm H$maEH$gmaImM Amho. (CYCLIC)m d$Z Ago {XgyZ òVo {H$ dmadma KS>`m`mnVrV dH$s` pW`Va(cyclic permutation) Agob `mMm g~Y Va AmnUmg

EH$ n[a{_V gMmer òBb(rational set.)CXmhaUmW (011)(d$Z Imbr, d$ZImbr, d$Z Imbr) {h nV dmnabr Va

=001 1

221 7 =

010 2

121 7Am{U =

100 4

111 7--------------(8)


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AmnU Hw$R>bmhr AH$ _J Vmo n[a_o`(rational) qH$dm An[a_o`(irrational)Agob {_mZ AH$ UmbrV ~Xby eH$Vmo. `m_wio `m AH$mer OdimV Odi

OmUmar KS>mMr H$m` Umbr V`ma hmoVo .

{_mZ AnyUmH$m`m AMyH$Vo`m g{hVoMm(power of binary approximationalgorithm)^mdrnUm dm gm_` ho nwH$i H$_r KSm KmbyZ, C XOmMr AMyH$Vm

gmYVm òVo. Amn`mbm nm{hOo Agbo`m AMyH$VogmR>r {H$Vr KS>m bmJVrb VoAmnUmg J{UVmZo gmJVm òBb. Oa AmnUmbm a hm AnyUmH$ (B) AMyH$VoZo

H$mT>m`Mm Agob Va {_mZ AnyUmH$ nVrZo H$mT>`mg bmJUmar KS>mMr g`mm_ò

[ (Log2

1

) - 1 ] ------------------------------------------------------(9)

m_ò {h g`m Odi`m nyUmH$m`m CV_ _`mXoer {ZJS>rV Agob.(function) `mhZhr H$_r AMyH$Vm AOyZ H$_r KS>m KmbyZ gmYVm òBb . OaAmnU {_mZ AnyUmH$ _`m{XV n[adVZmZVa {ZX}{eV Ho$bm g_Omm/p Va `mMr

H$_dmar {h {_mZ KS>r UmbrZo. AmnUbin(m/p) Yabr Va Vr

Rank (bin (m/p)) =log2

p [2 is base of log]------------------------(10)

Ho$di VmpdH$ J{UVr >rH$moZmVyZ {dMma Ho$bm Va J{UVr dmnamZo AMyH$Ho$bobr aMZm AmZXXm`r Agob nU `mdhm[aH$ >rH$moZmVyZ `jmer OdiAgUmar aMZm H$_r XOmMr Agbr Var gmonr Ag`m_wio A{YH$ Cn`moJr Agob.

Oa 11000

AMyH$Vm (Or ~`mM Amo[aJm_r H$m` UmbrV _mU _mZVmV) `mgmR>r

g_rH$aU 9 _ò XmI{d`m_mUo 9 KS>m bmJVrb. nU `jmV ho Cma `mnojm H$_r KS>mZr gmYbo OmVo {VWo AMyH$Voer WmoS>r VS>OmoS> Ho$br OmVo.

VgoM m nVrMm Xwgam \$m`mMm _wm Agm {H$ òH$ doir gnyU KSrKmb`mMr Oar Zmhr Va H$mJXm m H$Soda NmoQmgm {M_Qm KoD$Z gX^ q~Xy{_i{dVmòVmo. `m_wio dmadma KSm KmbyZ H$mJX MwaJiyZ QmH$ mMr Oar Zmhr.

{OWo AmnUmbm nm{hOo Agbobo AVa AMyH$ nm{hOo {VWo {_mZ UmbrMmCm_ Cn`moJ hmoVmo. {_mZ Umbr {h AnyUmH$mMm NoX hm AH$mMm KmV AgVmo{VWo eV{VeV AMyH$ AgVo. BVa AnyUmH$mgmRr A{YH$ AMyH$Vm {_i{d mgmRrBVa nVr nU dmnaVm òVmV. `mMm {dMma AmnU nwTo nmh.


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m nVrV AmnUmg Ago {Xgob {H$ nm mMr XmoZ AnyUmH$mV Ram{dH$ JwUmomamV{d^mJUr gmYbr Amho.

, ----------------------(14)BWow Am{Ux ho AH$ {_mZ nVrVrb AnyUmH$ {ZdS>bo AmhoV.

, ---------------------------(15)

`md$Z Ago {g hmoVo qH$

Y = mm+n

, z= nm+n

-----------------------------------------------(16)

gmh{OH$Ma

bho JwUmoma V`ma H$a`mgmRr KSmMr H$mÙmbr Aer amhrb.

(1) phm Agm {ZdS>m qH$ `mMm 2 Mm bKwm_ KmV hma Am{Ub-a m nojmA{YH$ Agob.

(2) VgoMm=a, n=b-a Aer `mMr J{UVr gòV qH$_V Agob.(3) {_mZ AH$ Umbr dmn$Zw=

m

p, x=

n

p, ho AnyUmH$, Mmagm`m S>m`m d

CO`m ~mOyda {ZpMV H$am.

(4) 'w d'x ho nm`m`m H$mon`m {d {XeoZo OmoS>m.

(5) m XmoZ aofmMm NoX q~Xy,a

bhm AnyUmH$ Xe{dVmo d {h `m q~XyMr CMr VgoM

S>m`m ~mOynmgyZMo AVa AgVo.Y=a/b Z= 1-y W X Rank

1/3 2/3 1/2 1 3

1/5 4/5 1/4 1 4

1/6 5/6 1/8 5/8 8

1/7 6/7 1/8 3/4 7

2/7 5/7 1/4 5/8 7

3/7 4/7 3/4 1 4

1/9 8/9 1/8 1 5

2/9 7/9 1/4 7/8 7

4/9 5/9 1/2 5/8 6

1/10 9/10 1/16 9/16 10

3/10 7/10 3/8 7/8 8

VVm H$_mH$ 3 _ò VVm 2 MoM AnyUmH$ KoD$Z `mMm XOm Xe{dbm Amho.


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34

y=a/b 1-y m/p a/p Rank

1/3 2/3 1/2 1/2 4

1/5 4/5 1/4 1/4 6

1/6 5/6 1/2 1/4 5

1/7 6/7 3/4 1/4 6

2/7 5/7 3/4 1/2 5

3/7 4/7 3/4 3/4 6

1/9 8/9 1/8 1/8 8

2/9 7/9 1/8 1/4 7

4/9 5/9 1/8 1/2 6

1/10 9/10 1/4 1/8 7

3/10 7/10 1/4 3/8 7

VVm 6 :- hJm`m _o`mZwgma, {_mZ Umbr AnyUmH$, `m`maMZogmR>r bmJUmao AnyUmH$ d XOm

hr aMZm nVr Zmo_m`m aMZonojm gmYr gai Amho Am{U g_moarb ^wOmOmoS>Um`m H$UmMr KS>r Xm~ Z XoVm Ho$di gX^ hUyZ dmnabr Va H$mJXmda oHw$Rohr IyU amhV Zmhr.


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22

(32)/2

21

(21)/2

11/2

(1+2)/2

2

1/2

2(1+2)/2

(32)/2

1/2

22

21

11/2

(21)/2

21 2+1

2


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y Am{Ux AVa AgUmao XmoZ H$U Mmagm`m XmoZ wOmZm NoX XoV AgVrbAm{U `m`m NoX q~XyMr CMr AmnUw KoVbr, Va N>oX q~XynmgyZ Mmagm`m

nm`mda H$mT>bobm b~ hm nm`mMo XmoZ ^mJw

zd

w

ym _mUmV NoX KoVmo d

nm`mMr bm~r {h Imbrb JwUmomamZo {_iVo.

1 1

)y z+w( ----------------------------------------------------------(36)

AmVm AmnU mMr VwbZm AmH$Vr H$_mH$ 20 _Yrb Mmagm`m ~mOy`mbm~rer hUOoMx(4 + 2 )er H$$`m. Oa Xmohr g_rH$aUmMr VwbZm Ho$br Vaw, x, y, z m AVamMr Cmao H$mT>Uo e` hmoBb. Or aMZm VwbZoZo gmonr Amho.

X (4 + 2 ) =1 1

)y z+w( ------------------------------------------(37)

m_ò AmnUx = w Aer AVao KoVbr {H$ Amnbo g_mZVoMo g_rH$aU AgohmoBb.

(4 + 2 ) =1 1

)y z+ --------------------------------------------------(38)

Oa AmnUmg(4 + 2 ) m gòMo XmoZ ^mJ H$aVm Ambo Va Cma H$mTUoghO hmoBb. m mJmH$mamMr Cmao H$mT`m`m AZoH$ nVr AmhoV nU W_V: EH$gmono Cma nmh. Am{U _J `md$Z gdgmYmaU arV H$er Agob `mH$So bj XoD$.

Amn`m g_rH$aUmMo Cma ----

(4 + 2 ) = {(2) + (2 + 2 )} =

1 1

1 1 1( ) ( )2 2

+

----------------------------(39)

hUOoM y Mr qH$_V1

2d z Mr qH$_V 11

2

hUOoM N>oX XoUm`m

H$UmMm NoX q~XyMm b~ hm Mmagm`m nm`mbm m JwUmomamV NoX XoBb.

Y =

1

2 ho AVa emoYUo gmono Amho. nU z =1

12

ho {VVHo$ ghO Zmhr.nU AmnUmg Ago gmJVm oBb qH$ Amo[aJm_r _Yrbfish base _ò mMm Cn`moJhmoVmo. nhm AmH$Vr H$_mH$.


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1/2

11/2

Figure 22. Construction of 11/2.

Figure 23. Folding sequence to find the initial division.


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H$moZmMr {d^mJUr

AmoarJm{__`o {OV`m _mUmV ~mOyMr 1/3 , 1/5, d 1/7 Mr {d^mJUr

Cn`moJmV AmUbr OmVo `m _mZmZo Aem H$maMr H$moZmMr {d^mJUr {h AJXrMVwaiH$ _mUmV dmnabr OmVo. `m_mUo gai aofm qH$dm ~mOyMo 2 `m KmVmVhUOo 1/2 ,1/4, dJao H$aUo gmono Amho VgoM H$mhrgo H$moZm`m ~m~VrV AmhoH$m ? EImXm Ago g_Oob {H$ {OVHo$ aofoMo R>am{dH$ g_^mJ H$aUo ghO e` AmhoVgoM H$moZm`m ~m~VrV nU Agob. naVw g` doJio Amho d H$moZmMr g_^mJmV{d^mJUr AdKS> Amho.

dVwV H$nmg d n>r`m ghm`mZo aofoMo Amd`H$ {VVHo$ ({H$Vrhr) EH$gmaIo^mJ H$aUo gmono Amho VgoM g_OyZ Oa AmnU H$moUmMo VrZ g_ mJ H$nmg d n>rZo

H$am`Mo hQ>bo Va Vo Ae` Amho ho gd _m` Amho. H$nmg d n>rZo g_^mJ H$aUo{h J{UVmMr emIm Amho Ago mMrZ H$mimnmgyZ hUOoM gmYmaUnUo 20 bmIdf(20 millenia)nydunmgyZ _mZbo OmVo Am{U AmOVmJm`V m gmYZmZr H$moZmMo

VrZ g_^mJ H$aUo Ae` Amho ho {g Ho$bo Jobo Amho. H$nmg d n>rZo g_^mJH$a`m`m arVtMm dmna AmoarJm{__`o Ho$bm Amho, Vmo J{UVmMm ^mJ WmoS>mgm~mOybm RoD$ `m.

_m`{_H$ embo` ^y{_VrV H$nmg d nrMm dmna H$$Z J{UV gmoS{dUo hmAZw d gdmZm AmhoM. H$nmg d nrZo gmoS{dbobo Z AmoarJm{_er gmY` Xe{dVmV.

m XmoZhr H$mamV AmnU ^m{_{VH$ aMZm H$aVmo Am{U `mgmRr dmnabobo {Z`_ {h

gmaIoM AmhoV. H$nmg d nrMm Cn`moJ; H$nmgmZo dVwi d nrZo gai aofm gmYbrOmVo. {deofV: {Xbo`m gai aofog g_mVa aofm H$mTUo, H$moZmMm Xw mOH$ H$mTUo qH$dm^m{_{VH$ aMZm Ogo {H$ g_ yO {H$moU, g_{^yO H$mQH$moZ {H$moU ,d Mmag m

gmRr Ho$bm OmVmo. Oa B{Vhmg n{hbm Va 300B.C. dfmnydu wbrS(Euclid)`mElementsm nwVH$mV A`m aMZmMr _m{hVr Ag`mMo {XgVo.

Amn`m nydOmZr H$nmg d n>rMm AmYma KoD$Z AZoH$ dfmnydu ^m{_{VH$aMZmMm emoY bmdbm hmoVm naVw `m H$mimV hUOoM AWog _Yrb JrH$gm_m`m`m B{VhmgmZwgma gw_mao 400 B.C._Yo m_w`mZo VrZ Z gVm{dV hmoVo.`mMr aMZm e` ZhVr. `m aMZm _m Amo[aJm_r _ o {deof _hd nyU AmhoV.


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48

g`m _m{hV Agbo`m gX^mZwgma Vo Ago 1) dVwim`m jom\$imBVH$m MmagV`ma H$aUo. 2) (Cube) KZm`m jo\$im`m XwnQ> jo\$imMm KZ ~Z{dUo(doubling cube) hUOoM EH$ gX aofm KoD$Z, Xwgar gX^ aofm Aem V`ma H$aUo

qH$ {OMo AVa _yi aofo`m3

2 Agob Am{U 3) {Vgam Z Agm qH$ EH$m {Xbo`mH$moZmMo VrZ g_mZ ^mJ H$aUo. `mMm Chmnmoh Amn`m Ammm`m {df`mer Amho.

JrH$mMo J{UV em m VrZ ZmZr ^S>mdbo hmoVo. (AmO{_Vrg J{UVmMo~aoM Vk m Zmer {ZJS>rV AmhoV). AZoH$ J{UV VkmZr m ZmMm ~amMH$mi g$ nmR>nwamdm Ho$bm Amho. nU `W eodQ>r AmO 2200 dfmZr Agm {ZH$f

H$mT>bm {H$ Ho$di H$nmg d n>rMm Cn`moJ H$$Z Cma {_i{dUo Ae` Amho.H$nmg Am{U n>r`m ghm`mZo gai aofm d dVwi H$mT>Vm òVo nU Amo[aJm_r _ò_m gaiaofoVrb KS>`mM KmbVm òVmV. naVw AmM`mMr Jmo> ht qH$ {Xbo`mH$moZmMo VrZ g_^mJ ~Z{dUo d KZm`m XwnQ> jo\$imMm KZ ~Z{dUo m aMZmAmo[aJm_rMm dmna H$$Z {g H$aVm òVmV.

H$nmg, n>r d Amo[aJm_r m`mV VwbZm Ho$br AgVm Xmohr H$mamZr V`ma

H$aVm oUm`m aMZm_ò Amo[aJm_r daMT> Amho. H$nmg Am{U n>rZo H$aVm oUm`maMZm m XmoZ KmVm`m g_rH$aUm nwa`m (Quadratic Equations) _`m{XV amhVmV.

Oa AmnUmg Ram{dH$ aofmMm g_yh {Xbm Agob Va H$nmg Am{U nr`m ghm`mZo`mMo H$moU`mhr JwUH$m_Yrb EH$rH$aUmMr (linear combination) aMZm H$$eH$Vmo. `mV AVamMr JwUH$ {d{dYVm dm AVamMo dJ_yi Ag`mMr e`Vm Amho.

`mMmM {ZH$f Agm {H$ H$nmg d nr`m gh`mZo Amn`mbm XmoZ KmVmMo {H$dmA{YH$ KmV Ag`mg `mMo XmoZ KmVm m _mUmV(convertibility in to quadraticequation) namdVZ H$aUo e` Agbo nm{hOo Va A`m aMZm AmnU H$$ eHy$.

naVw 2 `m AH$mMo KZ_yi Agbobo AVa qH$dm H$moZMo VrZ g_^mJ H$aUo`mgmR>r VVr` KmVmMo g_rH$aU gmoS>dmdo bmJob, Va dVwim`m jo\$imMm MmagV`ma H$a`mgmR>r> m AVamMr ~mOy H$mT>mdr bmJob. ( hmtranscendental

AH$ Amho) `mMo Cma H$moU`mhr KmVmMo g_rH$aU KoVbo Var {_iUo Ae`H$maU `m_ò AZVmn`V OmUmao dYm{ZH$ g_rH$aU Agob. `m_wio m VrZAZwmarV g_`mMo Cma AmnUmg \$$ 200 dfmnydu {_iy eH$bo Amho.


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q

q

q


50/84

A

X

A

Y

Z

A

X

A

Y

Z

A

X

Y

A

Z

X

Z


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51

Jacques JustinMr ~{hdH$ H$moZmMo VrZ g_^mJ H$a`mMr aMZm nV --

1) ZOX hm ~{hdH$ H$moZ AgyZ `mMo VrZ g_^mJ H$ad`mMo AmhoV.

2) ZO Am{UXO m aofm dmT>dm. (AVa bm~ H$am )

3) O m q~Xwd$ZA d X OmoS>Umar KS>r Kmbm. Vr aofobm b~ Agob.

4) A dA ho O m _`mnmgyZMo g_ AVamdarb gX^ q~Xy Z$s H$am.

5) A Am{UA ho EH$mM doirXOd YO m aofmda RoD$Z KS>r Kmbm.

6) O m q~Xy _YyZ OmUm`m darb KS>rbm b~ H$mT>m. Or b~ aofm VrZg_^mJmMm EH$ ^mJ Agob.

`md$Z bjmV òBb qH$ H$moU`mhr dVwimbm d `m`m ^mJmbm Oa doJ m_mUmV g_mZ NoX m`Mo AgVrb Va AmnU H$moZmMo VrZ d XmoZ g_mZ ^mJ mnVrMm g`w$ dmna H$$Z hm Z gmoSdyeH$Vmo qH$dm Xwg m ^mfoV Ago hUVmòBb qH$ Oa AmnUmg dN H$moZ d wOm AgUmar ~h ^yOmH${VMr aMZm H$aVm oBb

{OWo dN hmprime AH$ Amho hUOoM Vmo 2 3 1n m +x . (BWo d m d dn ho

H$moUVohr nyUmH$ AmhoV ). mMmM AW Agm qH$ AmoarJm_rZo KSm KmbyZ AmnUH$moZmMo 2,3,4,6,8,9,12 B. g_^mJ H$$ eHy$.

gnyU dVwimMo dN g_^mJ H$a`mMr arVAustrian ZmJ[aH$ d J{UVk am~QJoaoQbJa(Robert Geretschlager) 21-24 mZr m{_{VH$ nV 1890 gmbremoYbr hmoVr. `mZrN m AH$mMr ~h- ^yOmH$Vr H$mT`mMr EH$ gdgmYmaU arV

emoYbr hmoVr. (2n.3m+1) A`m H$maMo AH$ hUOo (3,5,7,13,17.....) m

nVrMm dmna g`w$nUo H$moZmMo 2 qH$dm 3 `m KmVm`m gòV H$aUo qH$dm

~h wOmH$VrMr aMZm H$a`mgmRr hUOoM2J.3k(2n.3m+1){OWo òUmam AH$ hm

A{d^m` AH$ (prime no.) Agob. {h nV g{dVanUo XoUo {df`mbm Y$Z Zmhr

naVwm mJmMo eodQr H$mhr Oar, {deof d gdgmYmaUnUo Cn w$ AH$mMr ~h wOmH$Vr

nV XmI{dbr Amho. mMm dmna H$$Z AmnUmg Ho$diN = 11 hm AH$ gmoSyZ BVa

gd AH$m`m g_mZ ~h wOm AgUm m ~h wOmH$VtMr aMZm H$aVm oBb.

H$moZmMo g_^mJ H$a`mMr AMyH$ nV tours de force A{Ve` {H$MH$Q>

AgyZ AmoarJm{_gmR>r Ag^dZr` hUVm òBb. H$maU `m nVrV nyU H$mJXâ


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(1/2)(0+r)r


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_yi VdmqH$V Amo[aJm_r

AmVmn`V {dfX Ho$bo`m KS>m`m doJdoJ`m H$m`nVr `m mW{_H$

AmhoV. Ogo {H$ EH$m {~Xwbm, Xwgam q~Xy {_idyZ KS>r KmbUo qH$dm EH$m aofodaXwgar aofm RoD$Z KS>r KmbUo. (H$moZ Xw^mOH$) qH$dm XmoZ gX^ q~XyKS>rZo OmoS>Uo.B. g. 1970 ZVa H$mhr Amo[aJm_r H$aUm`m `$sZr Aem doJdoJ`m KS>mMrg{dVa Zm|X H$$Z, {H$VrH$mao Aem KS>m KmbVm oVrb d `mVrb gX^ q~XyMoEH$_oH$mer Agbobo AVa d bm~rMr JwUmomao VnmgyZ `mMm A`mg gw Ho$bm.

m gX^mV W_Humiki Huzita [27-29]`mZr 6 doJdoJ`m nVtMm EH$ gMV`ma Ho$bm. {OWo EH$ q~Xyd aofm KoD$Z {H$Vr H$ma`m doJdoJ`m KS>m KmbVm`oVrb? Agm.

{OWo q~Xy, aofm mao {H$VrH$mao KSm H$aVm oVrb Vo g_OVo. Vo H$ma"HuzitaAxioms" (HA) `m ZmdmZo gwn[a{MV AmhoV. {OWo q~Xy d aofmMm gX^ KoD$Z

Kmbmd`m m KS>mMm Chmnmoh Amho. {Xbobo q~Xy, Am{U aofm`m gMmMm dmna

H$$Z"Huzita" Vdm`m Cn`moJmZo AZoH$ ZdrZ aofm V`ma H$aVm oVmV. {OWo Ow`md Z`m aofm`m NoX q~XyMm gX^ KoCZ A{YH$ gX^ q~XyMr aMZm Ho$br OmVo. mZ`m q~XyMm dmna H$Z nU EH$ doJio aofmMo Omio AmnU {dUyeH$Vmo.

A`m H$mao EImm gMmVrb q~Xy m gX mZoHAMm dmna H$$Z AmnU ZdrZgX q~XyMm doJim gM V ma H$$ eH$Vmo, {OWo MmagmMo H$monao, ~mOy mMm dmna W_VHo$bm OmVmo Am{U `mMm `jmV d A`mgmgmRr nwH$i Cn moJ hmoVmo.

ej{UH$ \$m`Xm hUyZ nm{h`mg AmnU VVr` KmV Agbobo g_rH$aUgmoS>dy eH$Vmo. {OWo \$$ H$_dma KS>mMm AmamIS>m dmnabm OmVmo. {deofV:mMrZ H$mir H$nmg d n>rZo V`ma Z H$aVm oUm`m g_`m gmoS>{dVm `oVmV. OgoAmnU nydu n{hbo Amho qH$ H$moZmMo VrZ g_^mJ H$aUo-VgoM KZmMo VVr` _yi

H$mT>`mMr nV {h AmnU nmhUma AmhmoV (10) naVw `jmV `m VdmMm

mW{_H$ Cn`moJ AMyH$VoZo H$_r KmVmMr g_rH$aUo gmoS>{d`mg _XV hmoVmo.

m gdH.A.H$m`nVtMo g{dVa {ddaU (31) nwTo {Xbo Amho. OarmgAxiomshUOoM _ybVdo hQ>bo Agbo Var m aMZm nVrZo q~Xyd aofmMm


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m {ddaUmMr AmH$Vr 33 _ o H$mT>br Amho. d U hm q~XyhUOo Ama q~Xyd$Z aofoda Q>mH$bobm b~ Omo H$_rV H$_r AVa Xe{dVmo. hUOoM tU-1 ()hm ~Xbd U () m aofodarb'I' m AVamda AgUma Amho d Omo +ve, d -ve

AgyeH$Vmo.mVrb `oH$ q~Xy(4) d (2) {h g_rH$aUo hr nyaH$ AgyZ, XmoZrhr g_rH$aUo

aofo`m gX^ `m`m Xe{dVmV.

KS>r KmbUo

KS>r KmbUo hUOo EH$ gai aofm H$mT>Uo, `mbm AmnU KS>rMr aofm hUVmo.

(Fold line, Crease) m KS>r_wio H$mJXmMo XmoZ ^mJ nS>VmV. EH$mbm pWa dXwg`mbm ApWa ^mJ hUy`m. hr pWa d ApWa Zmdo gmHo${VH$ AgyZ AmnUEH$gmaIr ~XbyeH$Vmo, Amn`m H$m_m`m OarZwgma.

O|hm KS>r KmVbr OmVo V|hm, {VMr `m`m Vr aofm ~ZVo {O`m ApWa^mJmVrb gd q~Xw`m gX^mZm AmnUL

f(d

f,

f) Ago Zmd XoD$`m.

AmnU KmVbobr KS>r EH$ aofm hmoVo d aofobm XmoZ _`m{XV gX^ AgVmV(Degree of freedom) Oo (parameters)d

fAm{U

fZo g~mo{Ybo OmVmV Am{U

hUyZM J{UVr ^mfoV AmnU `mMr `m`mUf= U(

f) Aer H$aVmo.

Oa KS>rMr aofm Lf(d

f,

f) Zo Xe{dbr Agob Va `mdarb ApWa ^mJmVrb

`oH$ q~Xy 'P' hm aofoda P' Zo Xe{dbm OmVmo.

P' = P-2[(P-dUf).Uf]Uf= P-2(d

f- P.U

f)U

f-------------------------------------------------------------(47)

AmnU'P'hm q~Xy F(P) m gkoZo XmIdy`m. hUOoM

F(P) = P +2(df- P.U

f) U

f----------------------------------------------------(48)

`m_wio XmoZrhr g_rH$aUo g_mZ ho n> {XgVo.

hUOoM'P' m H$moU`mhr aofodarb q~Xy,

P, F(F(P))= P -----------------------------------------------------------------(49)


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mMmM gai AW, KS>r KmVbr Am{U Vr CbJS>br Va `m `m q~XyMogX^ ~XbV ZmhrV .

Lf(d

f,

f) m aofodarb H$moUVmhr q~Xy hUOoM

F(P) = P --------------------------------------------------------------(50)

hUOoM `oH$ q~Xy hm ~XbV ZgyZ pWa Amho. Am{U aofoda KmVbo`mKS>rMr Vr `m`m hmoVo. L hr aofoMr `m`m, F(P) `m gd q~XyMm EH$ gM Omo(44) ho g_rH$aUmZo V`ma hmoVmo Aer Agob.

EH$rH$aU[Alignments]

EH$ KS>r KmbyZ XmoZ R>iH$ Jmo>r H$em EH$ AmU`m OmVmV Vo nmh.EH$rH$aUmMm EH$ H$ma hUOo XmoZ doJ`m Jmo>r EH$ AmUUo. Amn`mH$SoXmoZ R>iH$ Jmo>r AmhoV. EH$ q~Xyd Xwgar aofm. `m AmYr AmnU EH$rH$aUmMr`m`m nmh`m.

XmoZ q~XyMo EH$rH$aU-Point PP

P1

(x1, y

1)dP

2(x

2, y

2) m XmoZ q~XyMo EH$rH$aU Pmbo Amho mMm AW

Agm {H$ Xmohr q~XyMo j d ` AjmMo gX^ EH$ AmhoV (co-ordinates areequal) m EH$rH$aUmbm AmnUP

1P

2Zo Xe{dVmo

P1P

2hUOox

1=x

2Am{Uy

1= y

2------------------------------------(51)

hr Xmohr g_rH$aUo g_mZ Pmbr hUOo AmnU (2DOF) XmoZ dmV`o J_mdbr.

EH$m q~XyMo gai aofoer EH$rH$aU(PL)

P1

(x1, y

1) hm q~XyL(d,) m aofoer EH${V Pmbm Amho. hUOoP

1hm

aof o`m g_rH$aUmV (44) g_m{d> Pmbm Amho. m q~Xy Am{U aofo`mEH$rH$aUmbm AmnU Xmohr ~mOyg ~mU H$mTyZ : P

1 L(d,) Ago Xe{dVmo

hUOoM

P1 L(d,) ho e` Amho Ho$di P

1.U().d = 0m g_rH$aUmZo---(52)

BWo EH$M g_rH$aU gm` H$amd`mMo Ag`mZo AmnU EH$MDOF J_mdVmo.


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hr EH$rH$aUmMr {H$`m, XmoZ H$Vr EH$_oH$mg nyaH$ Ag`mZoM hmoVmV.

P1 L(d,) dL(d,)P

1Agob VaM hmoVmV. --------------------(53)

EH$m aofoMo Xwg`m aofoer EH$rH$aU(LL)

L1(d

1,

1) d L

2(d

2,

2) mMo EH$rH$aU Pmbo Amho Ago AmnU, OaL

1

aofodarb òH$ q~Xy L2

aofodarb òH$ {~Xyer OmoS>bm Jobm Agob VaM hUyeH$Vmo. Am{U hr `m`m {Mhm{H$V Xe{dVmZm AmnU Aer XmIdy.

U1= U(

1), U

2= U(

2)

_J Oa AmnUL1 m aofoMr gX^ nyU `m`m (46 m g_rH$aUm_mUo)n{hbr VaP1

hm q~Xy, L1

m aofoda Imbrb g_rH$aUmZo {_iVmo.

P1= dU

1+ t U

11 -----------------------------------------------------------------(54)

hUOoM H$mhr ( d, t) gmR>r aofmMo EH$rH$aU Xe{dVo qH$ P1q~Xy, g_rH$aU

nU 44 nyU H$aVmo. Ogo qH$ -

(d1U1 + tU11)U2 - d2 = 0 ------------------------------------------------------(55)

g_rH$aU gmoS>dy`m hUOo

(d1(U

1. U

2) -d

2) + t (U

11. U

2) = 0 -------------------------------------------(56)

S>mdr ~mOy{dMmamV KoV`mg't' `m oH$ {H$_Vr m g_rH$aUmV ~gVmVhUOoM g_rH$aUmV Agbo`m òH$ pWamH$mMr (Constant) qH$_V ey` Amho.hUyZM

d1(U

1. U

2) - d

2= 0 --------------------------------------------------------------(57)

U11. U

2= 0 ---------------------------------------------------------------------- (58)

`mMmM AW Oa XmoZ aofm EH${V Ho$`m Va, XmoZ g_rH$aUo gmoS>dmdr

bmJ`m_wio AmnbmDOFXmoZZo H$_r hmoVmo.mMm ^m{_{VH$ AW Agm qH$ g_rH$aU 58 ho XmoZ aofm g_mVa Ag`mMo

Xe{dVo. Va g_rH$aU 57 `m aofm EH$_oH$mg NoX XoVmV ho {g H$aVo.


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dVwV: Oa XmoZ aofm EH$_oH$mg NoX XoV AgVrb Va Ho$di g_rH$aU 58 honwaogo Amho ho ghO XmI{dVm oBb.

KS>m KmbyZ EH$rH$aU

_mJrb ^mJmV AmnU EH$rH$aUm`m VrZ {H$`m n{h`m P P1,

P L, L LAmVm AmnU EH$m KS>rZo e` Agbo`m gd H$ma`mEH$rH$aUmMm {dMma H$$ `m. Aem H$maMo EH$rH$aU H$mJXmda AmYrM CnbYAgbo`m KSmer {ZJSrV Agob. {H$dm ZdrZ KmVbo`m KSrZo V`ma Pmbo Agob.

AmnU Agm nydnj _mSy qH$ `m_ò AmYrM ApVdmV Agbo`m XmoZ

ApWa qH$dm XmoZrhr {H$`mMo pWa (KS>mMo) EH$rH$aU H$am`Mo Amho AgoEH$rH$aU KS>rZo H$aUo Ae` Amho hUyZM `mMm Cn moJ KS>rMr OmJm {ZpMVH$a`mgmR>r (g_rH$aU H$a`mgmR>r) hmoUma Zmhr. `m_wio Cabobr EH$rH$aUmMr{H$`m {h EH$ Va ApWa dm pWa ho AmYrM ApVdmV Agbo`m {H$`mMon[aUm_ d KS>r `m_ò Agob.

W_ XmoZ {H$`mMm {dMma H$$ m. `m AmYrM Agbo`m KS>`m`m gX^mVAgVrb Am{U `mnH$s EH$ pWa d Xwgar ApWa Agob. m_wio Amn`mbm nmM

H$ma`m EH$rH$aU nVr {_iVrb. `mMo {ddaU V$m H$_mH$ 9 _ò Amho.

VVm H$_mH$ 9 q~Xyd aofmMo EH$rH$aUmMo 5 doJdoJio H$ma.

AmnU m V`mVrb XmoZ eodQ>`m H$mamMr XmIb KoD$ `m. H$maU EH$KS>r Xwg`m KS>rda KmbVmZm (XmoZ g_rH$aUo 15 H$_mH$ d H$_mH$ 16) XmoZAQ>tMr nyVVm Oar Amho. _J Amn`m KS>rMr dmT> H$aUo Agmo {H$ `m aofobm NoX

XoUo Agmo. {VWo EH$mM g_rH$aUmMr Oar Amho.


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Xwg`m gMmVrb EH$rH$aU ho AmYr`m {H$`m d ZdrZ KS>rMo EH$rH$aUAgob. {OWo XmoZ H$ma g dVmV. q~Xyd KS>rMo EH$rH$aU qH$dm EH$m aofoMo `mM

KS>r`m aofoer EH$rH$aU hUOoM mVrb eodQ>Mr {H$`m ZJ` R>aob H$maU

`mMm KS>rdarb KS>rZo ZdrZ H$mhrM V`ma hmoV Zmhr. hUOoM Cabobr {H$`mVVm H$_mH$ 10 _ò {dfX Ho$br Amho.

VVm 10 q~Xyd aofoMo EH$rH$aUmMo CXmhaU

`m_wio EH$m KS>rZo Ho$bo`m `: H$pMV ~XbmMo {ddaU gnyU hmoVo.

AZoH$ EH$rH$aU

AmVm AmnU EH$rH$aU {H$òMm dmna H$$Z KSrMr `m`m H$$ m. hUOoMEH$ dm AZoH$ EH$rH$aUm m {H$`m H$$Z KS>rMo WmZ {ZpMV H$$ `m (`mMmMAW `mMo Oo XmoZ gX^ AmhoVd

f Am{UfVo emoYUo) mgmR>r AmnUmg {OVH$sg_rH$aUo bmJVrb {VV`m H$s`m H$Z _m{hV Zgbobo gX^ emoYmdo bmJVrb.Oo BWo XmoZ AmhoV.

AWmVM XmoZ g_rH$aUmgmR>r, XmoZ EH$rH$aUm`m {H$`m bmJUma `m A`mF(P

1) P

2(EH$ q~Xy Xwg`m q~Xyer OmoS>Uo. P, Am{UF(L

1L

2EH$ aofm

Xwg`m aofoer OmoS>Uo) KS>rMr {ZpMVr H$a`mgmR>r m XmoZ {H$`m nwaoem AmhoV.

(Sufficient)`mMm gX^Huzita Mo JhrVH$ (Axioms) 02 d 03 _ o VVmH$_mH$ 11 _ò XmI{dbm Amho.

F ( P 1 ) P2 02

F ( L1

) L2

03

XmoZ {H$`m `mDOF XmI{dVmV Am{U `m g~{YVHAJhrVH$.BVa Mma EH$rH$aUm`m {H$`m m EH$mM g_rH$aUm`m AmhoV. `m_wio

XmoZ {H$ m EH$ H$$Z XmoZ g_rH$aUo {_i{dVmo Oo KS>rMo gd gX^ n> H$aVmV.


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g_rH$aU 69 _òy Mm gdmV _moR>m KmV hm KZ (3) òVmo VaxMo g_rH$aUdJ KmVmV òVo Am{U `{edm` H$moUVrhr g`m KZ KmV Xe{dV Zmhr `m_wiog_rH$aU KZ Km>VmMo AgyZ dH$mH$ma Xe{dVo d Ago dH$mH$ma J{UVm`m AZoH$

emIm_ò Cn`moJmg oVmV.H$ 35 _ò {Xgbo`m dH$mH$ma AmboImMo _w` {deofImbr Xe{dbo AmhoV.

Ogm Ver dH$Vm (asymptotes) H$So PwH$Vo(dH$aofoH$S>o gVV OmUmar nU {Vbm Hw$R>`mhr q~XyV Z {_iUmar aofm)

gdgmYmaUV \$mgm m AmH$mamMr {h dH$Vm AgyZ (loop) {VMm NoX q~Xy

P2 = (a,b)

L2{h H$moUVrhr aofm OmVrV OmV VrZ doim NoX XoVo hUOoM gdm{YH$ EH$rH$aU

(A{YH$mV A{YH$) Ago VrZ {RH$mUrP2

hm q~XyL2

er H$ eH$Vmo. OaL2

dH$mH$mamg EH$mM {RH$mUr NoX XoBb Va EH$M EH$rH$aU e` Amho. Am{UOaL

2NoX XoV Zgob Va EH$sH$aUmMr gwVam_ e`Vm Zmhr.

Oa AmnU m dH$mH$Vr`m g_rH$aUmVrb ~Xb`m {H$_Vr_ o WmoS>mgm\$aH$ Ho$bm CXmhaUmW

x x + a

y y+b--------------------------------------------------(70)

Va g_rH$aUmVrb ~Xb EH$gKr` hmoVmo.

---------------------(71)

m_ò Oa AmnU b=-1

2Aer qH$_V {Xbr Va {_iUm`m dH$ AmH$mamg

Ophuride Curve (33) Am{Ua=0, b= -1

2m CmamgCissoid of Diocles

(34) g~moYVmV.

BWo dUZ H$aVmZm hoVwnydH$ Ago JhrV Yabo Amho qH$ L2 pWa Amh P2~XbVm n[adVZerb Amho, Am{U m J{hVH$mbm Y$Z òH$ KmVbobrL

2KS>r,

hrP1

dP2

ho XmoZrhr KS>r`m EH$mM ~mOybm AgVrb Aer KS>r KmVbr Amho.


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Aem H$m UmbrMo Cma XmoZ KZ g_rH$aUm`m dH$mH$ma AmH$Vr`m NoX{~XyZr {_iob. Or g_rH$aUo (26 ) Zo {dfX Ho$br AmhoV BWo 5 doJdoJir Cmao{_iVrb H$maU ho 5 NoX q~Xym XmoZ KS>mZr {_imbo AmhoV. AmH$Vr _`o Vo

doJ`m VhoZo Xe{dbo AgyZ KS>mMr nVr gai gmonr Amho. nmM Cmao Ho$diJ{UVmZo H$mT>Uo {H$dmTrial & error Zo {_i{dUo ho EH$ H$>X H$m_ AgyZ dH$mH$maAmH$VrZo H$mT>Uo ghO e` hmoVo.

Aem H$ma`m XmoZ dH$mH$ma AmH$VrnmgyZ 7 NoX q~Xy{_i{dUo gm` hmoD$eH$Vo, Oo Ago Xe{dVo qH$ {_iUmar ~h wOmH$Vr hr H$_rV H$_r 7 `m KmVm`mg_rH$aUmMr Agob naVw AmnU Ago {ZpMV {dYmZ H$$ eH$V Zmhr. {H$ gd ~h^yOmH$Vr 7 `m KmVm`m e` AgVrb d gd gmYmaUnUo Ago Ag`mMr e`Vm

\$maM H$_r Amho Am{U `m_wio A{YH$ KmV Agbo`m g_rH$aUmMr Cmao KS>mZr(Origami) {_iVrb Ago {dYmZ H$aUo YmS>gmMo R>aob. hm Z AZwmarV Amho.


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AmH$Vr H$_mH$ 39-7 _òHHA`m gd aMZm nH$s AmnUmg Ho$di Mma(4) Z`m aofm V`ma H$aVm oVmV: XmoZ H$moZ Xw mOH$ Am{U XmoZ Mmagm`m _` aofmAm{U `mgmRr EH$ KSr Kmbmdr bmJVo hUOo oH$sMm XOm (Rank) 1 Amho.

m Mma ZdrZ aofmMo EH$_oH$mer ,Am{U AmYr`m aofmer hUOoM Mmagm`mH$S>mZr {_imbobo NoX q~Xyn{hbo qH$ nmM (5) ZdrZ q~Xyn> hmoVmV.Vo Ago {H$òH$ H$SoMm _`q~Xy, Am{U MmagmMm _`q~Xy{_iyZ EHy$U nmM. Omo oH$ZdrZ q~XyMmagm`m H$S>mda Amho hUOo KS>r Z KmbVm {_imbm hUyZ 0 XOmMmAmho. H$S>mMm _`q~Xy Ho$di EH$m KS>rZo {Z_mU Pmbm Ag`_wio `mMm XOm

EH$ Amho. _`mq~XwMr `m`m AZoH$ aofmMm N>oXq~XyAer Agbr Var oH$ doirVmo H$moU`mhr XmoZ KS>m_wio {_iVmo hUOo `mMm XOm XmoZ Amho. gmame VaAmn`mH$So EHy$U 9 NoXq~Xy(Mma H$monao, Mma H$S>mMo _`q~Xyd EH$ MmamgmMm

_`q~Xy) d 12 aofm (AmH$Vr H$_mH$ 39 _Yrb 03 _ò OmS> H$$Z n>Ho$bo`m) AmhoV Am{U `mMm XOm Amho {OWor= 1Amho.

AmVm AmnU AmUIr EH$ KS>r KmVbr Va H$m` hmoBb ho OmUyZ KoD$`m.

AmnUmg _yiMo 9 q~Xy_m{hV AmhoVM. nU m Z`m KS>r_wio AOyZ XmoZ q~Xy{_iVrb d Aer KS>r AmnU Mma H$mao KmbyeH$Vmo, hUOo Amn`mbm EHy$U9x8 = 72 Z`m q~Xy`m OmoS>m {_i{dVm oVrb. VgoMHHA _Yrb 01 d 02

m aMZm nVrZo XmoZ kmV (_m{hV Agbobo) q~Xy`m EH$rH$aUmZo EH$ ZdrZ q~Xy{_iVmo.`m_wio aMZoMm hm Q>nm JmR>`mZVa AmnU 9 q~XwnmgyZ 153 q~Xyn`VOmD$ eH$Vmo. VgoM 03 hr aMZm XmoZ aofmMm dmna H$aVo, Va 04 ho aMZm q~Xyd aofm

`m`m EH$rH$aUmZo hmoVo, B`mXr `m_wio oH$ Q>`mda AmYr ^m{_VrH$ loUrZo(Geometric progressive) aofm V`ma hmoVmV. (Am{U XmoZ KS>m`m NoX {~XyZrAmn`mbm A{YH$ ^m{_{VH$ NoX q~Xy{_iVmV.) `m_wio nwZ:nwhm `mM aofm d

q~Xy{_iUo d^m{dH$ Agbo Var ZdrZ q~Xyd aofm `m`m gòMr dmT> hr dJloUr`m (Exponentially) _mUmV drJV hmoVo.

Oem AmnU A{YH$ KS>m KmbyVgo oH$ KS>r ~amo~a, KS>r d q~Xy`mMmXOm {H$Vr KS>m Am{U q~Xy EH${V AmUyZ {_i{dbm OmVmo. hmda R>a{dbmOmVmo. oH$ Z{dZ {~Xyhm XmoZ KS>mMm NoX q~XyAg`m_wio `mMm XOm


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AMyH$VoZo, H$_dma KS>m KmbyZ, AmnUmg hdm Agbobm q~XyH$_dma aMZm H$$ZH$gm Z$s H$aVm òBb.

EImm JwUmomamMo Cma ({H$Vr AMyH$ ho OmUyZ) AmnU {_mZ AnyUmH$nVrZo aMZm H$$ eH$Vmo. nU Oa {Xbobm AH$ 'x' Agob d `m`m OdiMmaMZm H$aVm òUmam AnyUmH$ hmNXOmMm Agob, `mMr AMyH$Vm AmnUmg _m{hVAgob Va {VWo'x' Mm XOm nU {VVH$mM Agob.

gd gmYmaUnUo H$moUVrhr aMZm H$aVmZm AmnU HHA Vrb gd {H$`mnVr,Am{U {d{dY H$ma`m aofm d q~Xy `m`m OmJm R>adyZ Amn`m Mmagm`m

KS>m KmbVmo. nU A`mdoir Amn`mH$So Aer H$moUVrhr gmo`rH$a nV CnbYZmhr qH$ Or Amn`mbm h`m Agbo`m XOmMm (rank) q~XyAMyH$VoZo {_iobAm{U Ag`mg Ver Ho$di EH$M nV CnbY Amho Ago ImrnydH$ d R>m_nUo

gmJyeH$V Zmhr.

naVw `jmV, H$mhrer AMyH$ Zgbobr nV nU Cn`moJr nS>Vo. Oa 106

q~XyMr AMyH$Vm .001 `m Odi Agob Va 106 XOmMo qH$dm `mhZhr H$_r Agoq~Xyd aofm mMm Cn`moJ H$$Z hdm Agbobm q~Xy{_i{dVm oVmo. \$V Amn`mbm`mo` aMZmH$_ emoYVm Ambm nm{hOo. AWmVM ho Ho$di hmVmZr KS>m KmbyZgm` H$aVm òUma Zmhr `mgmR>r gJUH$mMr _XV `mdr bmJob.

Am{U mgmR>r Sm. am~Q bJ `mZrRefrence finder hmc++ _Yrb moJm_{b{hbm Amho. `mgmR>r AmnUmbm h`m Agbo`m q~XyMo gX^ gJUH$mg nwadmdo

bmJVmV. _J Vmo gd q~XyMo MmamgmVrb. WmZ Z$s H$$Z Cm_ Aer H$_rV H$_rKS>mZr V`ma hmoUmar aMZm nVr AmnUmg XoVmo.

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References

[1] James Brunton, Mathematical exercises in paper folding,Mathematics in School, Longmans for the MathematicalAssociation, vol. 2, no. 4, July 1973, p. 25.

[2] Robert J. Lang, Four Problems III, British Origami, no. 132,October, 1988, pp. 711.

[3] Robert J. Lang, Western Pond Turtle, Origami Design Secrets, AK Peters, 2003.

[4] Shuzo Fujimoto and M. Nishiwaki, Sojo Suru Origami Asobi EnoShotai (Invitation to creative origami playing), Asahi Culture Centre,

1982.

[5] Jeannine Mosely, private comm.

[6] Masamichi Noma, Origami Tanteidan Newsletter, issue 14.

[7] Koji Husimi, Origami no kikagaku (Origami and Geometry), NipponHyoronsha, Tokyo (1979).

[8] Kunihiko Kasahara and Toshie Takahama, Origami for the

Connoisseur, Japan Publications,1987, pp. 1819.

[9] Kunihiko Kasahara, Origami Omnibus, Japan Publications, 1988,pp. 7677.

[10] Aleksandr Ia. Khinchin, Continued Fractions, Dover Publications,1997.

[11] Peter Engel, Folding the Universe: Origami from Angelfish to Zen,

New York, Vintage Books, 1989.

[12] Jun Maekawa, Evolution of Origami Organisms, Symmetry: Cultureand Science, vol. 5, no. 2, 1994, pp. 167177.

[13] Kunihiko Kasahara, Viva! Origami, Tokyo, Sanrio, 1983.

[14] Robert J. Lang, Albert Joins the Fold, New Scientist, vol. 124,no. 1696/1697, December 23/30, pp. 3857, 1989.

[15] Robert J. Lang, Origami: Complexity Increasing, Engineering &Science, vol. 52, no. 2, pp. 1623, 1989.


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[16] David Auckly and John Cleveland, Totally Real Origami andImpossible Paper Folding, American Mathematical Monthly, vol.102, no. 3, pp. 215226, March 1995.

[17] Thomas Hull, A Note on Impossible Paperfolding, The American

Mathematical Monthly, vol. 103, no. 3, March, 1996.

[18] Tsune Abe, described in British Origami, no. 108, p. 9, 1984

[19] Koji Fusimi, Trisection of angle by Abe, Saiensu supplement,October, 1980, p. 8.

[20] Jacques Justin, described in British Origami, no. 107, pp. 1415,1984.

[21] Robert Geretschlger, Euclidean Constructions and the Geometryof Origami, Mathematics Magazine, vol. 68, no. 5, December,1995, pp. 357371.

[22] Robert Geretschlger, Folding the Regular Triskaidekagon,presented at AMS Joint Mathematics Meeting, Baltimore, MD.,January 9, 1998.

[23] Robert Geretschlger, Folding the Regular 19-gon, presented at

AMS Joint Mathematics Meeting, Baltimore, MD., January 9, 1998.

[24] Robert Geretschlger, Solving Quartic Equations in Origami,presented at AMS Joint Mathematics Meeting, Baltimore, MD.,January 9, 1998.

[25] James Pierpont, On an undemonstrated theorem of theDisquisitiones Aritmeticae, American Mathematical MonthlyBulletin, no. 2, 18951896, pp. 7783.

[26] Robert J. Lang, The Complete Book of Origami, New York, DoverPublications, 1988.

[27] Humiaki Huzita and Benedetto Scimemi, The Algebra of Paper-Folding (Origami), Proceedings of the First International Meetingof Origami Science and Technology, H. Huzita, ed., 1989, pp. 215222. Lang, Origami and Geometric Constructions 55

[28] Humiaki Huzita, Understanding Geometry through OrigamiAxioms, Proceedings of the First International Conference onOrigami in Education and Therapy (COET91), J. Smith ed., BritishOrigami Society, 1992, pp. 3770.


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[29] Thomas Hull, Geometric Constructions via Origami, Proceedings ofthe Second International Conference on Origami in Education andTherapy (COET95), VAnn Cornelius, ed., Origami USA, 1995, pp.3138.

[30] Peter Messer, Problem 1054, Crux Mathematigorum, vol. 12, no.10, December, 1986.

[31] Thomas Hull, http://web.merrimack.edu/hullt/geoconst.html, 2003.

[32] Koshiro Hatori, http://www.jade.dti.ne.jp/~hatori/library/conste.html,2003.

[33] http://mathworld.wolfram.com/Ophiuride.html, 2003.

[34] http://mathworld.wolfram.com/CissoidofDiocles.html, 2003.

[35] Robert J. Lang, Origami Insects II, Gallery Origami House, Tokyo,2003 [in press].

[36] John Montroll, A Plethora of Polyhedra in Origami, Dover Publications,2002.


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Origami: Bhoumitic Rachana (Marathi Language)

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