{"id":15535,"date":"2016-06-01T15:24:37","date_gmt":"2016-06-01T12:24:37","guid":{"rendered":"http:\/\/109.232.216.219\/~bilimvegelecek\/?p=15535"},"modified":"2018-02-07T19:17:26","modified_gmt":"2018-02-07T16:17:26","slug":"arsimet-ron-graham-persi-diaconis-ve-bir-oyun","status":"publish","type":"post","link":"https:\/\/bilimvegelecek.com.tr\/index.php\/2016\/06\/01\/arsimet-ron-graham-persi-diaconis-ve-bir-oyun","title":{"rendered":"Ar\u015fimet, Ron Graham, Persi Diaconis ve bir oyun"},"content":{"rendered":"

Antik \u00e7a\u011f\u0131n en bilinen isimlerinden birisini g\u00fcn\u00fcm\u00fcz d\u00fcnyas\u0131ndan iki isimle ayn\u0131 ba\u015fl\u0131kta bulu\u015fturan \u015fey tabii ki matematik. Matemati\u011fin b\u00f6yle bir g\u00fcc\u00fc var. Zaman ve mek\u00e2n tan\u0131madan binlerce y\u0131l sonras\u0131na ta\u015f\u0131nabilir olu\u015fu ve evrenselli\u011fi\u2026<\/p>\n

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Ar\u015fimet\u2019in elyazmas\u0131ndan bir sayfan\u0131n okunur hale getirildikten sonras\u0131 ve \u00f6ncesi.<\/figcaption><\/figure>\n

Persi Diaconis, Fun Chungand ve Ron Graham\u2019\u0131n Ar\u015fimet\u2019i \u201coyun oynarken\u201d yakalamalar\u0131n\u0131n hik\u00e2yesi, New York\u2019taki bir m\u00fczayede salonunda ba\u015flar. 29 Ekim 1998. Hazine de\u011ferindeki t\u0131p ve bilim kitaplar\u0131 a\u00e7\u0131k art\u0131rmadad\u0131r. Sabah oturumunda, Marie Curie\u2019nin atomun n\u00fckleer yap\u0131s\u0131n\u0131 ke\u015ffeden Ernest Rutherford i\u00e7in imzalad\u0131\u011f\u0131 doktora tezi, Darwin\u2019in T\u00fcrlerin \u00c7e\u015fitlili\u011fi \u00dczerine <\/em>adl\u0131 eserinin ilk bask\u0131lar\u0131ndan biri, \u00d6klid-d\u0131\u015f\u0131 geometrinin ilk kitab\u0131 olarak kabul edilen Nicolai Lobachevskii\u2019nin Geometrinin Prensipleri <\/em>adl\u0131 eseri sat\u0131\u015fa \u00e7\u0131kar\u0131l\u0131r. \u00d6\u011fleden \u00f6nceki son oturumda tek bir kitap, daha do\u011frusu bir el yazmas\u0131 i\u00e7in tek oturumluk bir a\u00e7\u0131k art\u0131rmaya gidilir. Baz\u0131 b\u00f6l\u00fcmleri yanm\u0131\u015f, k\u00fcflenmi\u015f, neredeyse okunmaz haldeki bu el yazmas\u0131na 800.000 dolarl\u0131k bir ba\u015flang\u0131\u00e7 fiyat\u0131 belirlenir. M\u00fczayede evinin y\u00f6neticisi bu eserin en az ba\u015flang\u0131\u00e7 fiyat\u0131 \u00fczerinden al\u0131c\u0131 bulaca\u011f\u0131ndan emindir; \u00e7\u00fcnk\u00fc bu el yazmas\u0131 fazlas\u0131yla h\u0131rpalanm\u0131\u015f, silinip \u00fczerine ba\u015fka \u015feyler yaz\u0131lm\u0131\u015f olsa da b\u00fcy\u00fck bir matematik\u00e7iye, Sirak\u00fczal\u0131 Ar\u015fimet\u2019e ait mevcut ilk kitapt\u0131r. Bu el yazmas\u0131nda, Ar\u015fimet\u2019in \u00fcnl\u00fc eserleri Y\u00fczen Cisimler \u00dczerine<\/em> ve Y\u00f6ntem \u00dczerine<\/em>\u2019nin yan\u0131 s\u0131ra bir de Ar\u015fimet yapbozu olarak bilinen Stomachion<\/em> adl\u0131 oyun vard\u0131r.<\/p>\n

M\u00fczayede evinin arka koltuklar\u0131 el yazmas\u0131n\u0131 almaya g\u00fcc\u00fc yetmeyen, Ar\u015fimet\u2019in \u00e7al\u0131\u015fmalar\u0131n\u0131 tutkuyla takip eden matematik\u00e7i ve bilim insanlar\u0131 taraf\u0131ndan tamamen doldurulmu\u015f, fakat as\u0131l al\u0131c\u0131lar\u0131n beklendi\u011fi \u00f6n s\u0131ralar bo\u015f g\u00f6r\u00fcn\u00fcyordur.<\/p>\n

Yunan K\u00fclt\u00fcr Bakanl\u0131\u011f\u0131 kitab\u0131 sat\u0131n alabilmek i\u00e7in m\u00fczayedeye New York ba\u015fkonsolosunu g\u00f6ndermi\u015f ve bas\u0131n a\u00e7\u0131klamas\u0131nda Yunanistan\u2019\u0131n bilimsel tarihi i\u00e7in bu el yazmas\u0131na sahip olma y\u00fck\u00fcml\u00fcl\u00fcklerinin bulundu\u011funu belirtmi\u015ftir. Ayr\u0131ca, Londra\u2019n\u0131n \u00fcnl\u00fc sahaflar\u0131ndan Simon Finch de salondad\u0131r. Kitab\u0131 bir m\u00fc\u015fterisi ad\u0131na sat\u0131n almak i\u00e7in gelmi\u015ftir, ama m\u00fc\u015fterisinin kim oldu\u011fu ve ne kadar \u00f6deyece\u011fi bilinmiyordur.<\/p>\n

Ve \u201cd\u00fcello\u201d ba\u015flar\u2026 Ba\u015flang\u0131\u00e7 fiyat\u0131 olan 800.000 dolar h\u0131zla a\u015f\u0131larak k\u0131sa s\u00fcrede 1 milyon dolara ula\u015f\u0131l\u0131r. Yunanl\u0131lar telefonla ald\u0131klar\u0131 talimatlar do\u011frultusunda fiyat\u0131 art\u0131r\u0131rlar ama her defas\u0131nda Finch kar\u015f\u0131l\u0131k verir. Ba\u015fkonsolos 1.900.000 dolarl\u0131k fiyat\u0131 onaylad\u0131\u011f\u0131nda Finch hemen 2 milyon dolara \u00e7\u0131km\u0131\u015ft\u0131r. Yunanl\u0131lar kaybeder, kitap 2 milyon dolara Finch\u2019in me\u00e7hul m\u00fc\u015fterisine sat\u0131l\u0131r. Salonda \u015fa\u015fk\u0131nl\u0131k ve \u00f6fke hakimdir, \u00e7\u00fcnk\u00fc herkes kitab\u0131n gelece\u011finden endi\u015fe duymaktad\u0131r.\u00a0 2300 y\u0131l \u00f6ncesinden gelen ve bir dehan\u0131n en \u00f6nemli \u00e7al\u0131\u015fmalar\u0131n\u0131n bulundu\u011fu el yazmas\u0131 bir eser, ulusal bir h\u00fck\u00fcmet yerine, kim oldu\u011fu bilinmeyen bir ki\u015finin eline ge\u00e7mi\u015ftir.<\/p>\n

\u00dcnl\u00fc sahaf Simon Finch, kitab\u0131 sat\u0131n alan \u015fahs\u0131n ismini a\u00e7\u0131klayamayaca\u011f\u0131n\u0131, ama bu ki\u015finin Bill Gates olmad\u0131\u011f\u0131n\u0131 belirtir.<\/p>\n

Geli\u015fmelerin bundan sonras\u0131 daha ilgin\u00e7tir\u2026 O g\u00fcn m\u00fczayede salonunda bulunan sanat m\u00fczesi k\u00fcrat\u00f6r\u00fc William Noel, \u00e7al\u0131\u015ft\u0131\u011f\u0131 m\u00fczede kitab\u0131n sergilenmesinin m\u00fcmk\u00fcn olup olmad\u0131\u011f\u0131n\u0131 sahaf Finch\u2019e bir e-postayla sorar. Yan\u0131t olumludur. Noel, bir ka\u00e7 ay sonra kimli\u011fini a\u00e7\u0131klamayan, kendini Bay B.\u00a0 olarak tan\u0131tan kitab\u0131n sahibiyle bir \u00f6\u011fle yeme\u011finde bulu\u015fur. Noel, 2300 y\u0131l \u00f6nce bir d\u00e2hinin kaleme ald\u0131\u011f\u0131 bir esere ula\u015facak olman\u0131n heyecan\u0131yla oturmu\u015ftur yemek masas\u0131na. Bay B. yemekte, kitab\u0131 Noel\u2019in m\u00fczedeki \u00e7al\u0131\u015fma odas\u0131na b\u0131rakt\u0131\u011f\u0131n\u0131 a\u00e7\u0131klad\u0131\u011f\u0131nda Noel yerinde duramaz olmu\u015ftur. Bu g\u00f6r\u00fc\u015fmeden birka\u00e7 ay sonra Bay B. Noel\u2019e y\u00fckl\u00fc miktardaki paran\u0131n kar\u015f\u0131l\u0131\u011f\u0131 olan bir \u00e7ek g\u00f6nderir. Bu \u00e7ek, kitab\u0131n okunur hale getirilebilmesi i\u00e7in yap\u0131lacak teknik \u00e7al\u0131\u015fmalardaki harcamalar\u0131n kar\u015f\u0131l\u0131\u011f\u0131d\u0131r.<\/p>\n

William Noel ve Ar\u015fimet \u00fczerine \u00f6nemli bir otorite olan Reviel Netz, el yazmas\u0131n\u0131n okunup, \u00e7\u00f6z\u00fcmlenmesi i\u00e7in y\u0131llarca s\u00fcrecek hummal\u0131 bir \u00e7al\u0131\u015fma i\u00e7ine girerler. Kitap \u00e7\u0131plak g\u00f6zle okunamayacak kadar y\u0131pranm\u0131\u015ft\u0131r, \u00fcstelik orta\u00e7a\u011fda sayfalar\u0131 \u00fczerine Ar\u015fimet\u2019in el yazmalar\u0131 silinerek dualar yaz\u0131lm\u0131\u015ft\u0131r. Bu dualar\u0131n alt\u0131nda yer alan Ar\u015fimet\u2019in el yazmalar\u0131na dijital ortamda \u00e7oklu dalga boyunun kullan\u0131ld\u0131\u011f\u0131 \u00f6zel g\u00f6r\u00fcnt\u00fcleme teknikleriyle ula\u015f\u0131l\u0131r. Bilim ve matematik tarihi i\u00e7in \u00e7ok \u00f6nemli, hatta sars\u0131c\u0131 bulgulara rastlan\u0131r.<\/p>\n

Ar\u015fimet\u2019in el yazmas\u0131ndan bir sayfan\u0131n okunur hale getirildikten sonras\u0131 ve \u00f6ncesi.<\/p>\n

Ba\u015flang\u0131\u00e7ta, bu kitab\u0131 Bay B\u2019nin ki\u015fisel m\u00fclkiyetine almas\u0131 \u00fczerine olu\u015fan tepkilerin yersizli\u011fi anla\u015f\u0131l\u0131r, bilim tarih\u00e7ileri Bay B\u2019ye defalarca te\u015fekk\u00fcr ederler.<\/p>\n

\"\"Ar\u015fimet, kitab\u0131n son sayfalar\u0131nda Antik \u00e7a\u011fda Stomachion<\/em> olarak adland\u0131r\u0131lan bir oyunla u\u011fra\u015fm\u0131\u015ft\u0131r. Klasik metinlerde ge\u00e7en bu Yunanca kelimenin hem bu oyunu hem de kar\u0131n a\u011fr\u0131s\u0131n\u0131 ifade etmek \u00fczere kullan\u0131ld\u0131\u011f\u0131 belirtilmektedir. Oyun ad\u0131n\u0131 kar\u0131n a\u011fr\u0131s\u0131na neden olacak kadar zor olmas\u0131ndan alm\u0131\u015ft\u0131r. Oyunun amac\u0131 14 ayr\u0131 par\u00e7an\u0131n bir kare olu\u015fturacak \u015fekilde bir araya getirilmesidir. A\u015fa\u011f\u0131da bu 14 par\u00e7adan olu\u015fan bir kare g\u00f6r\u00fclmektedir.<\/p>\n

Bu oyunun bir yapboz olarak ele al\u0131nmas\u0131n\u0131n \u00f6tesinde her matematikseverin hemen akl\u0131na gelecek bir soru vard\u0131r:\u00a0 Bu par\u00e7alar\u0131n t\u00fcm\u00fcn\u00fc kullanarak ka\u00e7 farkl\u0131 \u015fekilde kare elde edilebilir?\u00a0 Bu yapbozun ba\u015fka bir versiyonu a\u015fa\u011f\u0131daki \u015fekilde g\u00f6r\u00fclmektedir.<\/p>\n

\"\"Bu 14 par\u00e7a kullan\u0131larak yukar\u0131daki gibi ka\u00e7 farkl\u0131 \u015fekilde kare olu\u015fturulabilir? Matemati\u011fin kombinatorik alan\u0131nda kar\u015f\u0131l\u0131\u011f\u0131 olan bu soruyu yan\u0131tlamak hi\u00e7 de kolay de\u011fil. Ne yaz\u0131k ki el yazmalar\u0131n\u0131n son \u00fc\u00e7 yapra\u011f\u0131 y\u00fczy\u0131llar \u00f6nce kopar\u0131ld\u0131\u011f\u0131ndan Ar\u015fimet\u2019in bu soruyu nas\u0131l yan\u0131tlad\u0131\u011f\u0131n\u0131 bilemiyoruz. \u00c7\u00fcnk\u00fc Stomachion<\/em>\u2019la ilgili sadece bir yaprak kalm\u0131\u015f. El yazmalar\u0131n\u0131n tamamen g\u00fcn \u0131\u015f\u0131\u011f\u0131na \u00e7\u0131kar\u0131ld\u0131\u011f\u0131 2003 y\u0131l\u0131na kadar Ar\u015fimet\u2019in bu oyunu sadece bir yapboz olarak ele ald\u0131\u011f\u0131 d\u00fc\u015f\u00fcn\u00fclm\u00fc\u015ft\u00fcr. Oysa ki Reviel Netz tam olarak okunamayan elindeki o tek yapraktaki baz\u0131 ifadelere bakarak, bir oyun olarak ilgilenmenin \u00f6tesinde Ar\u015fimet\u2019in Stomachion<\/em>\u2019a matematiksel bir yakla\u015f\u0131m getirmi\u015f olabilece\u011fini d\u00fc\u015f\u00fcnmektedir.<\/p>\n

Netz, Stanford \u00dcniversitesi Matematik B\u00f6l\u00fcm\u00fc\u2019ndeki meslekta\u015f\u0131 Persi Diaconis\u2019e ve matematik\u00e7i arkada\u015flar\u0131na problemin \u00e7\u00f6z\u00fcm\u00fcn\u00fc sorar: Stomachion<\/em> karesini yerle\u015ftirmenin ka\u00e7 yolu var? G\u00fcn\u00fcm\u00fcz matematik d\u00fcnyas\u0131n\u0131n \u00f6nemli kombinatorik\u00e7ilerinden olan Diaconis problemi \u00f6\u011frencileriyle de payla\u015f\u0131r, ama bir haftadan fazla bir s\u00fcre ge\u00e7mesine kar\u015f\u0131n matematik\u00e7ilerden herhangi bir yan\u0131t gelmez. \u015e\u00f6yle bir d\u00fc\u015f\u00fcnenler vard\u0131r: E\u011fer g\u00fcn\u00fcm\u00fcz matematik\u00e7ileri bu problemi an\u0131nda \u00e7\u00f6zemiyorsa Ar\u015fimet \u00e7\u00f6zm\u00fc\u015f olamaz. Ama bu sonuca ula\u015fanlar, Ar\u015fimet\u2019in binlerce y\u0131l sonras\u0131na \u0131\u015f\u0131k tutan \u00e7al\u0131\u015fmalar\u0131n\u0131 ve dehas\u0131n\u0131 hesaba katmad\u0131klar\u0131ndan yan\u0131lm\u0131\u015f olabilirlerdi. Nitekim Netz, aylarca s\u00fcren u\u011fra\u015f sonunda elindeki metni ileri g\u00f6r\u00fcnt\u00fcleme teknikleriyle okunur hale getirdi\u011finde Ar\u015fimet\u2019in Stomachion<\/em>\u2019u bir matematiksel oyun olarak \u00e7\u00f6z\u00fcmlemi\u015f olabilece\u011fi sonucuna ula\u015fm\u0131\u015ft\u0131r.<\/p>\n

Problemle u\u011fra\u015fmay\u0131 s\u00fcrd\u00fcren Diaconis, \u00e7\u00f6z\u00fcm u\u011fra\u015f\u0131na \u00fcnl\u00fc kombinatorik \u00e7iftini, UC San Diego\u2019dan Ron Graham ve Fun Chungand\u2019\u0131 da dahil eder. Bu arada akademik \u00e7evrelerde olduk\u00e7a yayg\u0131nla\u015fan problemi bilgisayar uzman\u0131 Bill Cutler bilgisayar algoritmalar\u0131 yard\u0131m\u0131yla \u00e7\u00f6zer (!). Cutler 536 temel \u00e7\u00f6z\u00fcm \u00fcretmi\u015ftir. Daha sonra her bir \u00e7\u00f6z\u00fcm i\u00e7in 32 d\u00f6nd\u00fcrme yap\u0131larak 536 x 32 =17.152 sonucu elde edilir. Bu \u00e7\u00f6z\u00fcmden k\u0131sa bir s\u00fcre sonra Ron Graham ve Fan Chungand problemin matematiksel \u00e7\u00f6z\u00fcm\u00fcn\u00fc yapt\u0131klar\u0131n\u0131 duyururlar, sonu\u00e7 17.152\u2019dir.<\/p>\n

2300 y\u0131l \u00f6nce bir d\u00e2hinin u\u011fra\u015ft\u0131\u011f\u0131n\u0131 bildi\u011fimiz ama \u00e7\u00f6z\u00fcme ula\u015f\u0131p ula\u015fmad\u0131\u011f\u0131ndan emin olamad\u0131\u011f\u0131m\u0131z bu problem g\u00fcn\u00fcm\u00fcz matematik\u00e7ileri taraf\u0131ndan \u00e7\u00f6z\u00fclm\u00fc\u015ft\u00fcr. M\u00fczayedede ortaya \u00e7\u0131kan el yazmas\u0131yla Ar\u015fimet adeta kendisinin binlerce y\u0131l \u00f6nce kombinatorik hesaplar\u0131n\u0131 yapabildi\u011fi mesaj\u0131n\u0131 g\u00f6ndermi\u015ftir. Antik\u00e7a\u011fda bir kombinatorik problemiyle u\u011fra\u015fan ilk ki\u015finin Hipparchus oldu\u011fu san\u0131l\u0131rken, William Noel ve Reviel Netz, bu alanda \u00e7al\u0131\u015fan ilk matematik\u00e7inin Ar\u015fimet oldu\u011funu g\u00f6stermi\u015ftir.<\/p>\n

Ar\u015fimet\u2019in par\u015f\u00f6mene kam\u0131\u015f kalem kullanarak yazd\u0131\u011f\u0131 \u00e7\u00f6z\u00fcm\u00fc Graham ve Chungand k\u00e2\u011f\u0131t, kalemle yapm\u0131\u015ft\u0131r. Bilimlerin en h\u0131zl\u0131 geli\u015feni ve ayn\u0131 zamanda binlerce y\u0131ll\u0131k kuramlar\u0131 koruyan tek bilim dal\u0131d\u0131r matematik. Par\u015f\u00f6men k\u00e2\u011f\u0131da d\u00f6n\u00fc\u015f\u00fcyor, ama zaman\u0131n f\u0131rt\u0131nas\u0131 g\u00fcn\u00fcm\u00fcz matematik\u00e7ilerinin bir problemde Ar\u015fimet\u2019le bulu\u015fmas\u0131n\u0131 engelleyemiyor.<\/p>\n

\"\"Stomachion\u2019<\/em>un \u00e7\u00f6z\u00fcm y\u00f6ntemi<\/strong><\/p>\n

Birbirinden farkl\u0131 Stomachion<\/em> kareleri elde etmenin en kolay yolu d\u00f6nd\u00fcrme ve par\u00e7alar\u0131n yerlerinin de\u011fi\u015ftirilmesidir. Graham ve Chungand, baz\u0131 par\u00e7alar\u0131n \u201cbirbirlerine yap\u0131\u015ft\u0131r\u0131lm\u0131\u015f\u201d gibi hareket ettiklerini g\u00f6zlemleyip, bu par\u00e7alar\u0131n hi\u00e7 bir yer de\u011fi\u015ftirme ya da d\u00f6nd\u00fcrme i\u015flemiyle birbirlerinden ayr\u0131lmayaca\u011f\u0131n\u0131n geometrik kan\u0131t\u0131n\u0131 yaparlar. A\u015fa\u011f\u0131daki \u015fekilde birbirlerinden ayr\u0131lamayacak olan par\u00e7alar i\u00e7lerine nokta koyularak g\u00f6sterilmi\u015ftir. B\u00f6ylece \u00f6nemli bir sadele\u015fme elde ederek, \u015fekli 14 par\u00e7a de\u011fil, 11 par\u00e7a olarak d\u00fczenleyebilir.<\/p>\n

Graham ve Chungand, dahice bir hesaplamayla i\u00e7inde sadece d\u00f6nd\u00fcrme veya sadece simetriyi kullanarak 268 temel \u00e7\u00f6z\u00fcm\u00fc i\u00e7eren bir liste haz\u0131rlarlar. Daha sonra bu temel \u00e7\u00f6z\u00fcmlerin her birine farkl\u0131 d\u00f6nd\u00fcrmeler ve simetriler uygulayarak 64 \u00e7\u00f6z\u00fcm \u00fcretilir. B\u00f6ylece t\u00fcm \u00e7\u00f6z\u00fcmlerin say\u0131s\u0131 268 x 64 =17.152\u00a0olur. Bu e\u015fitlikteki\u00a0 say\u0131s\u0131 4 x 2 x 8\u00a0olarak yaz\u0131l\u0131rsa, \u00a0\u00e7arpan\u0131 yukar\u0131daki \u015feklin alt yar\u0131s\u0131n\u0131n sa\u011f \u00fcst k\u00f6\u015fesindeki \u00fc\u00e7genlerin ve \u00fcst yar\u0131s\u0131n\u0131n sol alt k\u00f6\u015fesindeki \u00fc\u00e7genlerin ikizli\u011finden gelir. \u00a0\u00e7arpan\u0131 \u00fcst yar\u0131n\u0131n sa\u011f k\u00f6\u015fesindeki \u00fc\u00e7genlerle alt yar\u0131n\u0131n sa\u011f alt k\u00f6\u015fesindeki \u00fc\u00e7genin yer de\u011fi\u015ftirmesinden dolay\u0131 belirir. \u00a0\u00e7arpan\u0131 ise simetri ve d\u00f6nd\u00fcrmelerden elde edilir.<\/p>\n

Okurun \u00e7\u00f6z\u00fcm hakk\u0131nda fikir edinebilmesi i\u00e7in verdi\u011fimiz bu k\u0131sa bilgilerden daha fazlas\u0131 ve \u00e7\u00f6z\u00fcm i\u00e7in Kaynak 2\u2019ye ba\u015fvurulabilir. Bu kaynakta, bir \u00e7\u00f6z\u00fcmden farkl\u0131 bir \u00e7\u00f6z\u00fcme \u00e7izgeler kuram\u0131n\u0131n y\u00f6ntemleriyle ge\u00e7ildi\u011fi, ayr\u0131ca t\u00fcm \u00e7\u00f6z\u00fcmler aras\u0131nda Hamilton yoluyla<\/em> dola\u015f\u0131labilece\u011fi a\u00e7\u0131klan\u0131yor.<\/p>\n

Kaynaklar<\/strong><\/p>\n

1) Netz, R, Noel, W, Ar\u015fimet\u2019in Elyazmalar\u0131<\/em>, \u00c7ev. Zennur Anbarc\u0131o\u011flu, Alfa yay\u0131nc\u0131l\u0131k, 2012.<\/p>\n

2) http:\/\/www.math.ucsd.edu\/~fan\/stomach\/tour\/sym.html<\/p>\n","protected":false},"excerpt":{"rendered":"

Antik \u00e7a\u011f\u0131n en bilinen isimlerinden birisini g\u00fcn\u00fcm\u00fcz d\u00fcnyas\u0131ndan iki isimle ayn\u0131 ba\u015fl\u0131kta bulu\u015fturan \u015fey tabii ki matematik. Matemati\u011fin b\u00f6yle bir g\u00fcc\u00fc var. Zaman ve mek\u00e2n tan\u0131madan binlerce y\u0131l sonras\u0131na ta\u015f\u0131nabilir olu\u015fu ve evrenselli\u011fi\u2026 Persi Diaconis, Fun Chungand ve Ron Graham\u2019\u0131n Ar\u015fimet\u2019i \u201coyun oynarken\u201d yakalamalar\u0131n\u0131n hik\u00e2yesi, New York\u2019taki bir m\u00fczayede salonunda ba\u015flar. 29 Ekim 1998. Hazine […]<\/p>\n","protected":false},"author":375,"featured_media":15536,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[185,25,514],"tags":[1116,675,208],"class_list":["post-15535","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-148-sayi","category-matematik","category-matematik-sohbetleri","tag-arsimet","tag-geometri","tag-matematik"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t