{"id":61,"date":"2020-04-24T01:11:50","date_gmt":"2020-04-23T22:11:50","guid":{"rendered":"https:\/?p=61"},"modified":"2026-08-19T19:19:42","modified_gmt":"2026-08-19T16:19:42","slug":"fraktal-geometri","status":"publish","type":"post","link":"https:\/\/leyhatlari.com\/index.php\/kutsal-geometri-2\/dogadaki-geometri\/fraktal-geometri\/","title":{"rendered":"Fraktal Geometri: Do\u011fan\u0131n ve Evrenin Matematiksel Dili"},"content":{"rendered":"

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Geometri\u2019de 20. y\u00fczy\u0131la dek \u00d6klid geometrisi kullan\u0131ld\u0131; do\u011frular, d\u00fczlemler, \u00fc\u00e7genler\u2026 Ancak zamanla do\u011fay\u0131 daha iyi anlamak ve modellemek i\u00e7in yeni bir geometriye gereksinim duyuldu. \u0130\u015fte bu da fraktal geometriydi. \u00c7\u00fcnk\u00fc do\u011fadaki kutsal geometri fraktaller halinde kendisini g\u00f6sterir. Fraktal; genellikle birbirine benzeyen de\u011fi\u015fik geometrik \u015fekillerin genel ad\u0131d\u0131r. Sonsuza dek i\u00e7 i\u00e7e ge\u00e7mi\u015f ve birbirini tekrarlayan \u015fekillerdir. Bu tan\u0131ma g\u00f6re fraktal ana \u015fekle benzeyen gitgide k\u00fc\u00e7\u00fclen alan\u0131 sonsuz olan bir \u015fekildir. Fraktalleri ikiye ay\u0131rabiliriz :<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1- Kendinin t\u0131pat\u0131p ayn\u0131s\u0131 olan fraktaller: \u00a0<\/strong><\/span><\/p>\r\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Bu tip fraktallerin herhangi bir par\u00e7as\u0131yla, b\u00fcy\u00fck olan par\u00e7a t\u0131pat\u0131p ayn\u0131d\u0131r.<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0a-Yaprak Fraktali :<\/strong><\/span><\/p>\r\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u015eekilde g\u00f6r\u00fclen en k\u00fc\u00e7\u00fck yaprak en b\u00fcy\u00fck yaprak nas\u0131l in\u015fa edildiyse o \u015fekilde in\u015fa edilmi\u015ftir. (\u015eekil 9)<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0b-Akci\u011fer Fraktali:<\/strong><\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Akci\u011ferlerimiz fraktale \u00e7ok iyi birer \u00f6rnektirler. Dikkat edin soluk borusu ikiye ayr\u0131l\u0131yor, sonra tekrar ayr\u0131lan damarlar ikiye ayr\u0131l\u0131yor, hep ikiye ayr\u0131larak ve daha da k\u00fc\u00e7\u00fclerek devam ediyor.\u00a0 Bu da akci\u011ferlerin fazla hava almas\u0131n\u0131 sa\u011fl\u0131yor. (\u015eekil 10)<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 c-A\u011fa\u00e7 Dal\u0131 Fraktali:<\/strong><\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Ayn\u0131 Akci\u011fer fraktali mant\u0131\u011f\u0131 ile olu\u015fturulan A\u011fa\u00e7 Fraktali: K\u00f6kten ba\u015flayarak her zaman ikiye ayr\u0131larak ve k\u00fc\u00e7\u00fclerek devam eder.<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u015eekil 9 Yaprak Fraktali\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u015eekil 10 Akci\u011fer Fraktali<\/em><\/span><\/figcaption>\r\n<\/figure>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0d-Sierpinski \u00dc\u00e7gen Fraktali:<\/strong>\u00a0<\/span><\/p>\r\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Polonyal\u0131 matematik\u00e7i Waclaw Sierpinski\u2019nin ad\u0131n\u0131 ta\u015f\u0131r. E\u015fkenar \u00fc\u00e7genin i\u00e7inden k\u00fc\u00e7\u00fck \u00fc\u00e7genler \u00e7\u0131kart\u0131larak elde edilir. Bu i\u015flemi tekrarlamaya devam edersek Sierpinski \u00fc\u00e7genini<\/strong>\u00a0elde ederiz. (\u015eekil 11)<\/span><\/p>\r\n

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\u015eekil 11: Sierpinski \u00dc\u00e7gen Fraktali<\/em><\/span><\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 2- K\u0131smen benzer \/ par\u00e7alar\u0131 \u2013 b\u00f6l\u00fcmleri fraktal olan \u015fekiller<\/strong>: <\/strong>Bu tip fraktallerin belirli par\u00e7alar\u0131 fraktaldir. Komple, b\u00fcy\u00fck \u015fekle k\u0131smen, par\u00e7a par\u00e7a benzerlik g\u00f6sterir.<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 a-Koch Kar Tanesi Fraktali\u00a0<\/strong><\/span><\/p>\r\n

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A\u015fa\u011f\u0131da Koch kar tanesi fraktalinin ad\u0131mlar\u0131n\u0131 inceleyiniz. Bu fraktalin di\u011ferlerinden fark\u0131, k\u0131smen par\u00e7al\u0131 fraktal olu\u015fudur. \u00a0\u00a01904 y\u0131l\u0131nda Alman matematik\u00e7i\u00a0Helge Van Koch<\/strong>\u00a0taraf\u0131ndan a\u00e7\u0131klanan bu ilgin\u00e7 \u015fekli elde etmek i\u00e7in bir e\u015fkenar \u00fc\u00e7gen al\u0131n\u0131r, her kenar\u0131 \u00fc\u00e7 e\u015fit aral\u0131kla i\u015faretlenir ve ortadaki b\u00f6l\u00fcmler \u00e7\u0131kart\u0131l\u0131r ve bu buralara kenarlar\u0131 \u00e7\u0131kart\u0131lan par\u00e7alar kadar olan yeni e\u015fkenar \u00fc\u00e7genler konulur. Bu durumda yeni \u015feklimizin \u00e7evre uzunlu\u011fu \u00f6ncekinin 4\/3 kat\u0131 olmu\u015ftur. (\u015eekil 12)<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u015eekil 12 Koch Kar Tanesi Fraktali<\/em>\u00a0<\/span><\/figcaption>\r\n<\/figure>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Bu \u015fekilde her yeni ad\u0131mda, bir \u00f6nceki ad\u0131mda elde edilen do\u011fru par\u00e7alar\u0131na ayn\u0131 i\u015flem uygulan\u0131nca sonu\u00e7ta fraktal bir \u015fekil ortaya \u00e7\u0131kar. \u0130\u015fleme bu \u015fekilde devam edilip n<\/em>. ad\u0131ma gelinirse e\u011frinin toplam uzunlu\u011fu (4\/3)\u00a0n<\/sup><\/em>\u00a0olacakt\u0131r. E\u011fer\u00a0n<\/em>\u00a0yeterince b\u00fcy\u00fck al\u0131n\u0131rsa e\u011frinin uzunlu\u011fu da sonsuza gidecektir. Di\u011fer bir deyi\u015fle Koch e\u011frisinde iki nokta aras\u0131ndaki uzakl\u0131k sonsuzdur. E\u011fer bu e\u011fri yak\u0131ndan incelenirse \u015feklin tamam\u0131 ile onu olu\u015fturan alt par\u00e7alar\u0131n bir birine benzer oldu\u011fu g\u00f6r\u00fcl\u00fcr. \u00d6rne\u011fin \u015feklin tamam\u0131n\u0131 3 kat k\u00fc\u00e7\u00fclt\u00fcrseniz bir alt par\u00e7as\u0131n\u0131 elde edersiniz. Bu k\u00fc\u00e7\u00fcltme i\u015flemine sonsuza kadar devam edebilirsiniz.<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Koch kar taneci\u011fi d\u00fczenli denilen, yani \u00e7izim kurallar\u0131 belli olan fraktallere bir \u00f6rnektir. Bu d\u00fczenli fraktallerin ortak \u00f6zellikleri, e\u011frinin bir b\u00f6l\u00fcm\u00fcn\u00fc ne kadar b\u00fcy\u00fct\u00fcrseniz b\u00fcy\u00fct\u00fcn e\u011fri tam olarak ayn\u0131 \u015fekli s\u00fcrd\u00fcrmeye devam etmektedir.<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Koch e\u011frisinin ilgin\u00e7 bir \u00f6zelli\u011fi alan\u0131n\u0131n sonlu olmas\u0131d\u0131r, \u00e7\u00fcnk\u00fc onu bir daire i\u00e7ine s\u0131\u011fd\u0131rmak m\u00fcmk\u00fcnd\u00fcr. Bununla birlikte her ad\u0131mda uzunlu\u011fu biraz daha artar. Yani alan\u0131 sonlu ama \u00e7evresi sonsuzdur!<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u015fekil 12b fraktal alt\u0131gen kristaller<\/span><\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0D\u00fcz bir al\u00fcminyum folyo 2 boyutludur ancak onu buru\u015fturursan\u0131z iki art\u0131 bir \u015fey boyutunda bir fraktale d\u00f6n\u00fc\u015ft\u00fcr\u00fcrs\u00fcn\u00fcz. Bu yeni boyutun kesin de\u011feri folyonun ne kadar k\u0131r\u0131\u015f\u0131k oldu\u011funa ba\u011fl\u0131d\u0131r ve fraktal kar taneci\u011fi i\u00e7in kullan\u0131lan y\u00f6ntemle yakla\u015f\u0131k olarak hesaplayabiliriz. Aradaki tek fark, cetvelle uzunluk \u00f6l\u00e7mek yerine bu sefer s\u00fcrekli k\u00fc\u00e7\u00fclen \u00f6l\u00e7\u00fc plakalar\u0131 ile alan \u00f6l\u00e7\u00fcyor olmam\u0131zd\u0131r.<\/span><\/p>\r\n

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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Fraktaller konusundaki \u00e7al\u0131\u015fmalar bilimin bir\u00e7ok dal\u0131nda yararl\u0131 fikirlere de kaynakl\u0131k yapmaktad\u0131r \u00e7\u00fcnk\u00fc bu geometri do\u011fal olaylar\u0131n matematiksel olarak modellenmesinde avantajl\u0131d\u0131r. Yine de b\u00fct\u00fcn bu giri\u015fimler yaln\u0131zca bir ba\u015flang\u0131\u00e7t\u0131r \u00e7\u00fcnk\u00fc fraktallerin boyutsal ayr\u0131cal\u0131klar\u0131 d\u0131\u015f\u0131nda temel \u00f6zellikleri hen\u00fcz tam olarak kavranm\u0131\u015f de\u011fildir. Mandelbrot\u2019un \u00e7al\u0131\u015fmas\u0131n\u0131n a\u00e7t\u0131\u011f\u0131 bu yeni kap\u0131n\u0131n ard\u0131nda ke\u015f\u00adfedilmeyi bekleyen daha nice \u015fey bulunuyor anla\u015f\u0131lan\u2026[1]<\/a><\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Do\u011fada Fraktal \u00d6rnekleri:<\/strong><\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Fraktal geometrik formlar do\u011fada pek \u00e7ok canl\u0131da ve varl\u0131kta kendisini g\u00f6stermektedir: K\u0131rm\u0131z\u0131 lahanada, e\u011frelti otunda, piramit karnabahar\u0131nda, aloe bitkisinde, mercan poliplerinde, deniz kabuklular\u0131nda, a\u011fa\u00e7 dallanmas\u0131nda, yaprak yap\u0131s\u0131nda, buz kristallerinde, kar tanesi kristallerinde, bizmut kristallerinde, kan damarlar\u0131nda, n\u00f6ronlarda\u2026 \u00a0\u00a0\u00a0(\u015eekil 13)<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u015fekil 13 do\u011fada fraktal \u00f6rnekleri<\/span><\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n

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\u015fekil 13b do\u011fada fraktal \u00f6rnekleri<\/span><\/figcaption>\r\n<\/figure>\r\n<\/div>\r\n

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[1]<\/a> Sibel \u00c7a\u011flar, Do\u011fan\u0131n Geometrisi \u2013 Fraktal Geometri<\/span><\/p>\r\n

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\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Geometri\u2019de 20. y\u00fczy\u0131la dek \u00d6klid geometrisi kullan\u0131ld\u0131; do\u011frular, d\u00fczlemler, \u00fc\u00e7genler\u2026 Ancak zamanla do\u011fay\u0131 daha iyi anlamak ve modellemek i\u00e7in yeni bir geometriye gereksinim duyuldu. \u0130\u015fte bu da fraktal geometriydi. \u00c7\u00fcnk\u00fc do\u011fadaki kutsal geometri fraktaller halinde kendisini g\u00f6sterir. Fraktal; genellikle birbirine benzeyen de\u011fi\u015fik geometrik \u015fekillerin genel ad\u0131d\u0131r. Sonsuza dek […]<\/p>\n","protected":false},"author":1,"featured_media":65,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2568],"tags":[514,513,523,525,528,517,511,524,455,519,36,533,37,518,527,510,526,520,522,531,529,458,38,532,25,515,484,516,530,39,457,521,512],"class_list":["post-61","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-dogadaki-geometri","tag-agac-dali-fraktali","tag-akciger-fraktali","tag-altigen-kristaller","tag-biyoloji-ve-fraktallar","tag-buz-kristalleri","tag-doga-ve-geometri","tag-dogadaki-fraktallar","tag-dogal-fraktal","tag-dogal-yapilar","tag-fibonacci-ve-fraktal","tag-fraktal","tag-fraktal-boyutlar","tag-fraktal-geometri","tag-fraktal-geometrik-sekiller","tag-fraktal-ve-sanat","tag-fraktaller","tag-geometrik-doga","tag-geometrik-evrim","tag-geometrik-yineleme","tag-kan-damarlari","tag-kar-tanesi","tag-koch-kar-tanesi","tag-koch-kar-tanesi-fraktali","tag-kristaller-ve-fraktallar","tag-kutsal-geometri","tag-mandelbrot","tag-matematik-ve-doga","tag-matematiksel-modelleme","tag-noronlar","tag-sierpinski-ucgen-fraktali","tag-sierpinski-ucgeni","tag-sonsuz-geometri","tag-yaprak-fraktali"],"jetpack_featured_media_url":"https:\/\/leyhatlari.com\/wp-content\/uploads\/2020\/04\/R10-e1745045339204.gif","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/posts\/61","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/comments?post=61"}],"version-history":[{"count":1,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/posts\/61\/revisions"}],"predecessor-version":[{"id":2881,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/posts\/61\/revisions\/2881"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/media\/65"}],"wp:attachment":[{"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/media?parent=61"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/categories?post=61"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/leyhatlari.com\/index.php\/wp-json\/wp\/v2\/tags?post=61"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}