b. de smit and h. w. lenstra jr. - artful mathematics: the heritage of m. c. escher

8/3/2019 B. de Smit and H. W. Lenstra Jr. - Artful Mathematics: The Heritage of M. C. Escher

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Lb zkocgbl}lcb ca }dk :00? Eh}dkeh}los Hwhzkbkss

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acz czghblvlbg ncohn oknkjzh}lcbs ca Eh}dkeh}los Hwhzkbkss Ecb}d"

]dk Eh}dkeh}lohn_}z{o}{zk ca Ksodkz“sPzlb} Ghnnkz|J" mk _el} hbm D" W" Nkbs}zh Fz"

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]dk jks} ktpnhbh}lcb ca dcw Pzkb}kb}kb}ccbs}knnlbg

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J" mk _el} hbm D" W" Nkbs}zh Fz" hzk h} }dk Eh}dkeh}lsod

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pzlb} ghnnkz| j{} zkm{okm j| h aho}cz ca :;1" Lb czmkz}c mc f{s}lok }c }dk qhz|lbg hec{b} ca mk}hln }dh}dk bkkmkm. Ksodkz ho}{hnn| ehmk ac{z s}{mlks lb%s}khm ca h slbgnk cbk $skk Y?U,. cbk acz khod ca }dkac{z oczbkzs ca }dk nl}dcgzhpd" Alg{zk ; sdcws }dks}{m| acz }dk ncwkz zlgd} oczbkz" Khod ca }dksks}{mlks sdcws h pcz}lcb ca }dk pzkqlc{s cbk $ecm%{nc =, j{} jncwb {p j| h aho}cz ca =" Eh}dkeh}l%ohnn| wk eh| hs wknn qlkw Ksodkz“s ac{z s}{mlks hsh slbgnk mzhwlbg }dh} ls lbqhzlhb} {bmkz sohnlbg j|h aho}cz ca :;1" _x{hzk j| sx{hzk. Ksodkz }dkbal}}km }dk s}zhlgd} sx{hzk gzlm ca dls ac{z s}{mlkscb}c }dk o{zqkm gzlm. hbm lb }dls wh| dk cj}hlbkmPzkb}kb}kb}ccbs}knnlbg " ]dls ls lnn{s}zh}km lbAlg{zk 1"

Alg{zk :" H o|onlo ktphbslcb ktpzksskm {slbgs}zhlgd} nlbks"

Alg{zk ?" H o|onlo ktphbslcb ktpzksskm {slbgo{zqkm nlbks"

Alg{zk =" Ksodkz“s gzlm"

Alg{zk ;" Cbk ca Ksodkz“s s}{mlks"

Alg{zk 1" Al}}lbg }dk s}zhlgd} sx{hzks cb}c }dko{zqkm gzlm"



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Bkw Ksodkz E{sk{e

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Hnn|b Fhoiscb

Alg{zk 7" ]dk sx{hzk HJOM }zhbsaczekm }c }dks}zhlgd} wcznm"

Alg{zk 4" H ; t ; sx{hzk }zhbsaczekm }c }dko{zqkm wcznm"

Alg{zk <" H 7 t 7 sx{hzk }zhbsaczekm }c }dko{zqkm wcznm"

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HPZLN :00? BC]LOK_ CA ]DK HE_ ==<

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Knnlp}lo O{zqks

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a $:;1v, 5 a $v,. g$οw , 5 g$w ,"

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lsceczpdlse. lb c}dkz wczms. hb lsceczpdlse ca cbk%mlekbslcbhn ocepnkt hbhn|}lo qhzlk}lks"

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czmlbh}ks pzcpkzn|. wk eh| hss{ek }dh} d$8 ,58"

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ktp√↚ O∗#ο √↚ 0

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8= N:;1 5 Vπl#: ) V ncg=" P{nnlbg }dls plo}{zk

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= Nο 5 Vπl#: ) V$πl ncg=,#$πl ) :ncg=, " _lbok}dk nh}}kz nh}}lok ocb}hlbs :π l. }dk plo}{zk ohb bcwjk p{sdkm aczwhzm j| }dk ktpcbkb}lhn ehp" ]dlspzcm{oks }dk plo}{zk cb }dk ncwkz zlgd}. wdlod lslbqhzlhb} {bmkz e{n}lpnloh}lcb j| hnn ac{z}dzcc}s ca ο"]dk jc}}ce dczlvcb}hn hzzcw zkpzkskb}s }dke{n}lqhn{km ehp w d$w , 5 w θ 5 ktp$θ ncg w , 9 l} lscbn| ecm{nc }dk sohnlbg s|eek}z| }dh} l} ls wknn%mkalbkm"



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bcw glqkb j| Nο 5 θ√8N:;1. hbm azce ~ο~ 2 8 wkmkm{ok

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nhzgkz }dhb }dk cbk ekhs{zkm lb Ksodkz“s gzlm" ]dkzkhmkz eh| bc}lok scek c}dkz mlaakzkboks. hnn ca wdlod lbmloh}k }dh} Ksodkz mlm bc} pkzako}n| hodlkqkdls s}h}km p{zpcsk ca mzhwlbg h ocbaczehn gzlm. j{}l} ls zkehzihjnk dcw oncsk dk gc} j| dls cwbdkhmhodk%oh{slbg pzcokss"

Alnnlbg lb }dk Dcnk

Lb czmkz }c alnn lb }dk dcnk lb Pzkb}kb}kb}ccbs}knnlbg .wk alzs} zkocbs}z{o}km Ksodkz“s s}{mlks azce }dkgzlm hbm }dk nl}dcgzhpd j| zkqkzslbg dls cwb pzc%okm{zk" Acz }dls p{zpcsk wk {skm sca}whzk spkolhnn|wzl}}kb j| Fccs} Jh}kbj{zg. h eh}dkeh}los s}{mkb}

h} Nklmkb" Hs cbk ohb skk lb Alg{zk 8:. }dk jnhbispc} lb }dk elmmnk ghqk zlsk }c hb kep}| splzhn lb}dk zkocbs}z{o}km s}{mlks. hbm }dkzk wkzk c}dkz le%pkzako}lcbs hs wknn" Bkt}. }dk M{}od hz}ls}s DhbsZlod}kz hbm Fhox{knlbk Dcas}zh ocepnk}km hbm hm%

f{s}km }dk plo}{zks cj}hlbkm9 skk Alg{zk 8?"

Wdkb l} ohek }c hmmlbg }dk bkoksshz| gzh|sohnk.wk zhb lb}c pzcjnkes ca mlsocb}lb{c{s zkscn{}lcbhbm odhbglbg nlbk wlm}ds" Wk mkolmkm }dh} }dkbh}{zhn wh| ca cqkzocelbg }dksk pzcjnkes whs j|zkx{lzlbg }dk pltkn mkbsl}| cb c{z knnlp}lo o{zqk }c

jk {blacze lb }dk Dhhz ekhs{zk" Lb pzho}lohn }kzes.wk p{nnkm }dk sik}odks jhoi j| }dk ktpcbkb}lhna{bo}lcb. cj}hlblbg h mc{jn| pkzlcmlo plo}{zk cbO9l} whs lb }dh} plo}{zk }dh} }dk gzh|sohnk whs hmmkm.

j| Fhox{knlbk Dcas}zh. wdlod zks{n}km lb }dk mc{% jn| pkzlcmlo plo}{zk lb Alg{zk 8=" ]dk ocepnk}km qkz%slcb ca Ksodkz“s nl}dcgzhpd. sdcwb lb Alg{zk 8;. whs}dkb khs| }c pzcm{ok"

C}dkz ocepnkt hbhn|}lo ehps d6O∗#ν ↚

O∗#:;1 acz qhzlc{s ν glqk zlsk }c lb}kzks}lbgqhzlhb}s ca Pzkb}kb}kb}ccbs}knnlbg " ]c skk }dksk

Alg{zk 88" ]dk pkzako}n| ocbaczehn gzlm"

Alg{zk 8:" Ksodkz“s nl}dcgzhpd zko}lalkm. j|ekhbs ca dls cwb gzlm"

Alg{zk 8?" H mk}hln ca }dk mzhwlbg ehmk j|Dhbs Zlod}kz hbm Fhox{knlbk Dcas}zh"



HPZLN :00? BC]LOK_ CA ]DK HE_ =;8

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]dk hssls}hbok ca Fccs} Jh}kbj{zg. Oczmcb Hz}.Jz{bc Kzbs}. Zlodhzm Gzckbkwkgkb. Fhox{knlbkDcas}zh. hbm Dhbs Zlod}kz ls gzh}ka{nn| hoibcwn%kmgkm" Oczmcb Hz} dcnms }dk ocp|zlgd} }c hnn ca E" O" Ksodkz“s wczis" ]dk pzcfko} whs s{ppcz}km

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Zkakzkboks

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Y?U K" ]DÌ $mkslgb,. ]dk Ehglo ca E" O" Ksodkz . Dhzz| B"

Hjzhes. Bkw \czi hbm Ncbmcb. :000"

Alg{zk 8=" ]dk s}zhlgd} mzhwlbg p{nnkm jhoi. j| }dk ocepnkt ktpcbkb}lhn a{bo}lcb. }c h mc{jn|pkzlcmlo plo}{zk. wl}d gzh|sohnk hmmkm" ]dk dczlvcb}hn pkzlcm ls ncg :;1. }dk qkz}lohn pkzlcm ls :πl"

Alg{zk 8;" ]dk ocepnk}km qkzslcb ca Ksodkz“s nl}dcgzhpd wl}d ehgblaloh}lcbs ca }dk okb}kz j| aho}czsca = hbm 81"



=;: BC]LOK_ CA ]DK HE_ QCN[EK ;0. B[EJKZ =

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Acz eczk }dhb h d{bmzkm |khzs. eh}dkeh}lolhbsdhqk mzhwb }zlhbgnk }kssknnh}lcbs lb }dk Pclbohzìmlsi ecmkn ca }dk d|pkzjcnlo pnhbk. hbm scek hz}ls}sdhqk ac{bm lbsplzh}lcb lb }dksk ph}}kzbs" ]dk M{}odgzhpdlo hz}ls} E" O" Ksodkz $84<4‟8<7:, whs ecs}nlikn| }dk alzs} cbk }c jk sc lbsplzkm" Dk {skm „onhs%slohn― s}zhlgd}kmgk hbm ocephss ocbs}z{o}lcbs }cozkh}k dls d|pkzjcnlo ph}}kzbs" Eczk zkokb}n| c}dkzsdhqk {skm ocep{}kz gzhpdlos }c ozkh}k d|pkzjcnloph}}kzbs" Wk wlnn alzs} ktpnhlb dcw Ksodkz mzkw }dk

{bmkzn|lbg }kssknnh}lcbs aczdls ph}}kzbs9 }dkb wk wlnnmksozljk qkzslcbs ca }dkocep{}kz pzcgzhe }dh}gkbkzh}km }dk mkslgb acz }dk:00? Eh}dkeh}los Hwhzk%bkss Ecb}d pcs}kz"

Ksodkz“s Ek}dcms

Lb 8<;4 D" _" E" Octk}kzskb} Ksodkz h ocp| ca dlsphpkz „Oz|s}hn s|eek}z|hbm l}s gkbkzhnlvh}lcbs―

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Ksodkz whs sdcoikm jk%oh{sk }dh} alg{zk sdcwkm dle }dk ncbg%mkslzkm sc%n{}lcb }c dls pzcjnke ca mkslgblbg zkpkh}lbg ph}%}kzbs lb wdlod }dk ec}las jkocek kqkz sehnnkz}cwhzm h olzo{nhz nlel}" Octk}kz“s Alg{zk 7 ocb%}hlbkm }dk ph}}kzb ca o{zqlnlbkhz }zlhbgnks sdcwb lbAlg{zk 8 $j{} wl}dc{} }dk sohaacnmlbg," Ca oc{zsk.}dh} ph}}kzb ohb jk lb}kzpzk}km hs h }zlhbgnk }ks%

sknnh}lcb lb }dk Pclbohzì mlsi ecmkn ca }dk d|pkz% jcnlo pnhbk. hn}dc{gd Ksodkz whs pzcjhjn| eczklb}kzks}km lb l}s ph}}kzb%ehilbg lepnloh}lcbs"

]dk pclb}s ca }dk Pclbohzì mlsi ecmkn hzk lb}k%zlcz pclb}s ca h jc{bmlbg olzonk lb }dk K{onlmkhb

pnhbk" D|pkzjcnlo nlbks hzk zkpzkskb}km j| mlhek%}kzs hbm olzo{nhz hzos }dh} hzk cz}dcgcbhn }c }dk

jc{bmlbg olzonk YGzkkbjkzgU" ]dk Pclbohzì mlsiecmkn ls h}}zho}lqk }c hz}ls}s jkoh{sk l} ls ocbacz%ehn $hbgnks dhqk }dklz K{onlmkhb ekhs{zk, hbm l}ls mlspnh|km lb h jc{bmkm zkglcb ca }dk K{onlmkhbpnhbk sc }dh} l} ohb jk qlkwkm lb l}s kb}lzk}|" Ksodkz

whs hjnk }c zkocbs}z{o} }dk olzo{nhz hzos lb Octk}kz“salg{zk hbm }dkb {sk }dke }c ozkh}k dls alzs} olzonknlel} ph}}kzb. Olzonk Nlel} L $Alg{zk :h,. wdlod dk lb%on{mkm wl}d dls nk}}kz }c Octk}kz" Alg{zk :j sdcwsh zc{gd ocep{}kz zkbml}lcb ca }dk Olzonk Nlel} L mk%slgb" Ksodkz“s ehzilbgs cb }dk zkpzlb} }dh} Octk}kzskb} }c dle sdcw }dh} }dk hz}ls} dhm ac{bm }dkokb}kzs ca scek ca }dk olzo{nhz hzos hbm mzkw nlbks}dzc{gd ocnnlbkhz okb}kzs Y_odh}}sodbklmkz8U"

]dk okb}kz ca hb cz}dcgcbhn olzo{nhz hzo ls kt%}kzbhn }c }dk mlsi hbm ls ohnnkm l}s pcnk" ]dk nco{sca hnn pcnks ca hzos }dzc{gd h pclb} lb }dk mlsi ls hnlbk ohnnkm }dk pcnhz ca }dh} pclb} YGccmehb%

_}zh{ssU" ]dk kt}kzbhn mc}s lb Alg{zk 8 hzk }dk pcnksca }dk nhzgkz hzos. hbm }dk kt}kzbhn nlbk skgekb}socbbko}lbg }dke hzk phz}s ca pcnhzs ca }dk pclb}sca lb}kzsko}lcb ca }dcsk hzos" Alg{zk 8 sdcws cbklb}kzlcz pclb} hbm l}s pcnhz hs h nhzgk mc} hbm h }dloinlbk cb }dk nka}9 h olzo{nhz hzo hbm l}s pcnk hzk sle%lnhzn| kepdhslvkm cb }dk zlgd}"]dk kt}kzbhn wkj ca pcnks hbm pcnhz skgekb}s ls scek}leks ohnnkm }dksohaacnmlbg acz }dk }kssknnh}lcb" ]dk aho} }dh} }dk pc%nhzs hzk nlbks ohb jk {skm }c spkkm {p }dk s}zhlgd}%kmgk hbm ocephss ocbs}z{o}lcb ca }zlhbgnk }ksskn%nh}lcbs" Acz kthepnk. glqkb }wc pclb}s lb }dk mlsi.}dk okb}kz ca }dk cz}dcgcbhn hzo }dzc{gd }dke ls}dk lb}kzsko}lcb ca }dklz pcnhzs"

Octk}kz ktpnhlbkm }dk jhslos ca }dksk }kodblx{kslb dls zk}{zb nk}}kz }c Ksodkz YZccskqkn}U. hn}dc{gd

j| }dh} }lek Ksodkz dhm alg{zkm c{} ecs} ca }dls. hs kqlmkbokm j| Olzonk Nlel} L " Nlik Ksodkz.eh}dkeh}lolhbs dhqk }zhml}lcbhnn| mzhwb }zlhbgnk}kssknnh}lcbs lb }dk Pclbohzì mlsi ecmkn {slbgs}zhlgd}kmgk hbm ocephss }kodblx{ks. coohslcbhnn|sdcwlbg }dk sohaacnmlbg" ]dls }kodblx{k whs scek%}dlbg ca h gkcek}zlo „acni hz}― {b}ln }dk zkokb}phpkz j| Odhle Gccmehb%_}zh{ss YGccmehb%_}zh{ssU. lb wdlod }dk ocbs}z{o}lcb ek}dcms wkzkalbhnn| wzl}}kb mcwb"

Acz pcsl}lqk lb}kgkzs p hb m x . wl}d 8#p)8#x > 8#:. }dkzk ktls} }kssknnh}lcbs ca }dk d|pkzjcnlopnhbk j| zlgd} }zlhbgnks wl}d ho{}k hbgnks π #p hbmπ#x " H zkg{nhz p%slmkm pcn|gcb. cz p%gcb. ohb jkaczekm azce }dk :p }zlhbgnks hjc{} khodp%acnm zc}h}lcb pclb} lb }dk }kssknnh}lcb" ]dkskp%gcbs acze }dk zkg{nhz }kssknnh}lcb yp. xu j|p%slmkm pcn|gcbs. wl}d x ca }dke ekk}lbg h} khodqkz}kt" Alg{zk = sdcws }dk }kssknnh}lcb y1. =u $wl}d hokb}zhn gzc{p ca alsd cb }cp ca l}," Hs ohb jk skkb.Ksodkz ksskb}lhnn| {skm }dk y1. =u }kssknnh}lcb lbOlzonk Nlel} L "

Alg{zk 86 ]dk }zlhbgnk }kssknnh}lcbca }dk Pclbohzì mlsi }dh} lbsplzkm

Ksodkz. hncbg wl}d }dk„sohaacnmlbg― acz ozkh}lbg l}"

Mc{gnhs M{bdhe ls pzcaksscz ca ocep{}kz solkbok h} }dk

[blqkzsl}| ca Elbbksc}h. M{n{}d" Dls kehln hmmzkss ls

mm{bdhe@m"{eb"km{ " M{bdhe mkslgbkm }dk Eh}dkeh}%

los Hwhzkbkss Ecb}d pcs}kz. wdlod ls hqhlnhjnk h}

d}}p6##eh}dacz{e"czg#ehe#0?#pcs}kz"d}en "

http://mathforum.org/mam/03/poster.html





HPZLN :00? BC]LOK_ CA ]DK HE_ =;?

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wl}d okb}kz C }dh} aczes }dk c{}kz kmgks ca }dksohaacnmlbg" ]dkb dk ocbs}z{o}s }dk d|pkzjcnlozlgd} }zlhbgnk wl}d h qkz}kt hbgnk ca π #p h} C

hbm l}s d|pc}kb{sk hncbg h zhml{s ca P azce C }ccbk ca }dk qkz}loks H ca P " ]dk slmk ca }dk zlgd}}zlhbgnk }dzc{gd C ls phz} ca h pkzpkbmlo{nhz

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jk mcbk j| alzs} lbqkz}lbg hnn }dk qkz}loks lb }dk olz%o{nhz hzos }dh} acze }dk slmks ca }dk okb}zhn p%gcb.aczelbg bkw p%gcbs. hbm }dkb lbqkz}lbg qkz}loks lb

}dk slmks ca }dk bkw p%gcbs l}kzh}lqkn| hs ehb|}leks hs mkslzkm" J{} Gccmehb%_}zh{ss mksozljksh eczk kaalolkb} hn}kzbh}lqk ek}dcm {slbg aho}shjc{} }dk gkcek}z| ca olzonks"

C{z Alzs} Ek}dcm

Lb 8<40 L mkolmkm }c }z| }c {sk ocep{}kz gzhpdlos}c zkozkh}k }dk mkslgbs lb khod ca Ksodkz“s ac{zOlzonk Nlel} pzlb}s $oh}hncg b{ejkzs =:<. =?:. =?=.hbm =?1 lb YJccnU," ]dk ehlb odhnnkbgk skkekm }c

jk albmlbg h zkpnloh}lcb hngczl}de }dh} wc{nm mzhwkhod ocp| ca h ec}la ktho}n| cbok" ]dkzk wkzk }wczkhscbs acz }dls" Alzs}. wk wkzk {slbg pkb%pnc}}kz}kodbcncg| }dkb. sc e{n}lpnk zkmzhwlbgs ca }dkshek ec}la oc{nm }khz }dzc{gd }dk phpkz" Kaal%olkbo| whs h skocbm zkhscb6 }dk b{ejkz ca ec}laslbozkhsks ktpcbkb}lhnn| azce }dk okb}kz. hbm lbka%alolkb} hngczl}des elgd} pzcm{ok hb ktpcbkb}lhnb{ejkz ca m{pnloh}lcbs"

Eczkcqkz. L whb}km }dk zkpnloh}lcb hngczl}de }c j{lnm }dk ph}}kzb c{}whzm kqkbn| lb „nh|kzs― sc}dh} }dkzk wc{nm jk bc fhggkm kmgks" H} }dh} }leke| ocnnkhg{k Fck Ghnnlhb dhm scek {bmkzgzhm{h}kzkskhzod s}{mkb}s wdc wkzk wczilbg cb albmlbgDheln}cblhb ph}ds lb }dk Oh|nk| gzhpds ca albl}kgzc{ps YGhnnlhbU" L }dc{gd} }dh} }dklz }kodblx{ksoc{nm hnsc jk hppnlkm }c }dk lbalbl}k s|eek}z|gzc{ps ca Ksodkz“s Olzonk Nlel} mkslgbs" ]dls }{zbkmc{} }c jk }dk ohsk. hn}dc{gd wk ac{bm }dk mkslzkmph}ds lb }wc s}kps"

]dk alzs} s}kp lbqcnqkm albmlbg h Dheln}cblhbph}d lb }dk Oh|nk| gzhpd ca }dk s|eek}z| gzc{pca }dk }kssknnh}lcb yp. xu " ]dls whs mcbk j| MhqlmWl}}k. cbk ca Ghnnlhb“s zkskhzod s}{mkb}s" FcdbNlbmgzkb. h [blqkzsl}| ca Elbbksc}h M{n{}d s}{mkb}.lepnkekb}km }dk ocep{}kz hngczl}de. wl}d ek}zhbsnh}lbg Wl}}k“s ph}d lb}c psk{mc%ACZ]ZHBYM{bdhe8U"

]dk Oh|nk| gzhpdca h gzc{p G wl}d hsk} ca gkbkzh}czs _ lsmkalbkm hs acnncws6}dk qkz}loks hzk f{s}}dk knkekb}s ca G .hbm }dkzk ls hb kmgk

azce t }c | la | 5 stacz scek s lb _ " ]kod%blohnn|. }dls mkalbks hmlzko}km gzhpd. j{} lbc{z ocbs}z{o}lcbs }dklbqkzsk ca kqkz| knk%ekb} ca s wlnn hnsc jklb _ . sc acz slepnlol}|wk eh| hss{ek }dh}c{z Oh|nk| gzhpds hzk{bmlzko}km" Hs hb kt%hepnk. }dk s|eek}z|gzc{p ca }dk }kssknnh%}lcb yp. xu ls mkbc}km

Yp. xU" ]dh} s|eek}z|gzc{p ls }dk shek hs}dh} ca }dk }kssknnh%}lcb j| zlgd} }zlhbgnkswl}d hbgnks π#p hbmπ #x " ]dk s}hbmhzmsk} ca gkbkzh}czs acz}dk gzc{p Yp. xU lsyP. X . Zu . wdkzk P .X . hbm Z hzk zkanko%}lcbs hozcss }dk }zl%hbgnk slmks cppcsl}k}dk hbgnks π#p. π#x .

hbm π#:. zkspko%}lqkn|. lb cbk s{od }zl%hbgnk"

]dkzk ohb jk cbk%wh| cz }wc%wh|Dheln}cblhb ph}ds lb}dk Oh|nk| gzhpds ca s|eek}z| gzc{ps ca d|pkzjcnlo ph}}kzbsYM{bdhe?U" Dcwkqkz.cbk%wh| ph}ds hzks{aalolkb} acz c{z hn%gczl}des. sc lb }dlshz}lonk „Dheln}cblhb

ph}d― wlnn hnwh|s mk%bc}k h cbk%wh| Dhe%ln}cblhb ph}d"

]dkzk ls h {ska{nqls{hn zkpzkskb}h}lcbacz }dk Oh|nk| gzhpdsca }dk gzc{ps Yp. xU hbm }d{s acz }dklz Dheln}cblhbph}ds" H a{bmhekb}hn zkglcb acz }dk }kssknnh}lcbyp. xu ls h }zlhbgnk }dh} wdkb ho}km cb j| }dk s|e%ek}z| gzc{p Yp. xU dhs }dh} }kssknnh}lcb hs l}s czjl}"]dls a{bmhekb}hn zkglcb ohb jk }hikb }c jk h zlgd}

Alg{zk ?6 ]dk Oh|nk| gzhpd ca }dk gzc{pY1.=U wl}d h Dheln}cblhb ph}d"

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Nlel} L "

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zkskb}lbg }dk zkanko}lcbhozcss cbk slmk"

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jnhoi hbm gzh|. zkpzkskb}}dk Oh|nk| gzhpd9 }dk nlgd}nlbks sdcw }dk }zlhbgnk }ks%sknnh}lcb" ]dk Dheln}cb%lhb ph}d ocbsls}s ca }dk

dkhq| jnhoi nlbk skg%ekb}s" Ksskb}lhnn| }dkshek ph}d wczis acz Yp. xUlb gkbkzhn. }dc{gd snlgd}„mk}c{zs― e{s} jk }hikb la p 5 ? c z x 5 ? YM{bdhe?U"

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http://www.d.umn.edu/~jgallian/progbib.html


http://geom.math.uiuc.edu/graphics/pix/Special_Topics/Hyperbolic_Geometry/escher.html




http://www.sckans.edu/~bridges/










=;1 BC]LOK_ CA ]DK HE_ QCN[EK ;0. B[EJKZ =

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b. de smit and h. w. lenstra jr. - artful mathematics: the heritage of m. c. escher

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