{"id":411,"date":"2025-07-11T16:29:10","date_gmt":"2025-07-11T15:29:10","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=411"},"modified":"2025-07-11T16:34:10","modified_gmt":"2025-07-11T15:34:10","slug":"introduccio-als-fractals","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/introduccio-als-fractals\/","title":{"rendered":"Introducci\u00f3 als fractals"},"content":{"rendered":"\n<p>Una fractal \u00e9s un objecte matem\u00e0tic caracteritzat per l\u2019<strong>auto-similitud<\/strong>, \u00e9s a dir, la repetici\u00f3 d\u2019una estructura similar a diverses escales d\u2019observaci\u00f3. A difer\u00e8ncia de les formes convencionals de la geometria cl\u00e0ssica, les fractals poden presentar una <strong>dimensi\u00f3 no enter<\/strong>, anomenada <em>dimensi\u00f3 fractal<\/em>, i que expressa la seva complexitat. Les fractals exhibeixen una <strong>estructura infinitament detallada<\/strong> a mesura que s\u2019amplia una part de l\u2019objecte. Aquesta riquesa geom\u00e8trica es troba en objectes naturals com les ramificacions d\u2019arbres, les xarxes vasculars, o encara les l\u00ednies costaneres. <\/p>\n\n\n\n<p>Les fractals s\u00f3n objectes matem\u00e0tics fascinants tant per la seva est\u00e8tica com pel seu contingut te\u00f2ric profund. A m\u00e9s il\u00b7lustren de manera espectacular com les matem\u00e0tiques poden descriure el caos aparent del m\u00f3n real amb una precisi\u00f3 elegant.<\/p>\n\n\n\n<p>Aqu\u00ed teneu algunes l\u00ednies de treball que es podrien explorar en aquesta proposta de TFG sobre les fractals.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>An\u00e0lisi de les propietats geom\u00e8triques<\/strong>: dimensi\u00f3 fractal, auto-similitud, recurr\u00e8ncia, invari\u00e0ncia d\u2019escala.<\/li>\n\n\n\n<li><strong>Construcci\u00f3 de fractals cl\u00e0ssics<\/strong>: conjunt de Mandelbrot, conjunt de Julia, triangle de Sierpi\u0144ski, floc de Koch.<\/li>\n\n\n\n<li><strong>Relaci\u00f3 amb la teoria del caos<\/strong>: com les fractals ajuden a descriure comportaments din\u00e0mics complexos.<\/li>\n\n\n\n<li><strong>Modelitzaci\u00f3 de processos naturals<\/strong>: estructura de xarxes biol\u00f2giques (pulmons, vasos sanguinis), geologia (formacions rocoses, l\u00ednies costaneres).<\/li>\n\n\n\n<li><strong>Simulaci\u00f3 de fen\u00f2mens f\u00edsics<\/strong>: turbul\u00e8ncies, distribuci\u00f3 de gal\u00e0xies, patrons de creixement org\u00e0nic.<\/li>\n<\/ul>\n\n\n\n<p><strong>Bibliografia orientativa<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em>Fractal Geometry: Mathematical Foundations and Applications<\/em> \u2013 Kenneth Falconer<\/li>\n\n\n\n<li><em>The Fractal Geometry of Nature<\/em> \u2013 Beno\u00eet B. Mandelbrot<\/li>\n<\/ul>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Una fractal \u00e9s un objecte matem\u00e0tic caracteritzat per l\u2019auto-similitud, \u00e9s a dir, la repetici\u00f3 d\u2019una estructura similar a diverses escales d\u2019observaci\u00f3. A difer\u00e8ncia de les formes convencionals de la geometria cl\u00e0ssica, les fractals poden presentar una dimensi\u00f3 no enter, anomenada dimensi\u00f3 fractal, i que expressa la seva complexitat. Les fractals exhibeixen una estructura infinitament detallada [&hellip;]<\/p>\n","protected":false},"author":75,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,1],"tags":[44],"class_list":["post-411","post","type-post","status-publish","format-standard","hentry","category-geometria-i-topologia","category-general","tag-florent-balacheff"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/411","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/75"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=411"}],"version-history":[{"count":12,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/411\/revisions"}],"predecessor-version":[{"id":432,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/411\/revisions\/432"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=411"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=411"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}