{"id":948,"date":"2019-12-05T15:28:28","date_gmt":"2019-12-05T18:28:28","guid":{"rendered":"https:\/\/www.blogs.unicamp.br\/zero\/?p=948"},"modified":"2023-08-24T17:15:03","modified_gmt":"2023-08-24T20:15:03","slug":"o-misterio-de-%cf%804","status":"publish","type":"post","link":"https:\/\/www.blogs.unicamp.br\/zero\/948\/","title":{"rendered":"O mist\u00e9rio de \u03c0=4"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"948\" class=\"elementor elementor-948\" data-elementor-settings=\"{&quot;ha_cmc_init_switcher&quot;:&quot;no&quot;}\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-852b7a6 elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"852b7a6\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2bf4f7e jltma-glass-effect-no\" data-id=\"2bf4f7e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-08d0e5b jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"08d0e5b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/www.blogs.unicamp.br\/zero\/2019\/12\/05\/the-mystery-of-%cf%80-4\/\">(Translate)<\/a><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d9be277 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"d9be277\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">\u03c0 deve ser o mais famoso dos n\u00fameros\u2026 as pessoas parecem fazer diversas associa\u00e7\u00f5es estranhas quanto ao seu significado. Existe at\u00e9 mesmo um filme de fic\u00e7\u00e3o cient\u00edfica no qual conseguem descobrir o \u00faltimo d\u00edgito decimal de \u03c0 (apesar deste \u201cd\u00edgito\u201d hipot\u00e9tico n\u00e3o existir), e isto possibilita o protagonista prever as varia\u00e7\u00f5es nas bolsas de valores do mundo inteiro. Apesar de toda esta popularidade, o significado de \u03c0 \u00e9 um pouco \u201cmais sem gra\u00e7a\u201d do que parece. \u03c0 significa o per\u00edmetro de um c\u00edrculo de di\u00e2metro 1 unidade.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-81c749d jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"81c749d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"272\" height=\"299\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/c\u00edrculo.png\" class=\"attachment-large size-large wp-image-1764\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7fc137d jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"7fc137d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"center\"><i>Linha vermelha tem comprimento <\/i>\u03c0 unidades.<\/p>\n<p align=\"justify\">Por\u00e9m, o per\u00edmetro deste c\u00edrculo \u00e9 um N\u00famero Irracional, assim somente pode ser tratado nos c\u00e1lculos decimais pela sua aproxima\u00e7\u00e3o. Apesar de algumas pessoas acharem legal lembrar de muitas casas decimais da aproxima\u00e7\u00e3o de \u03c0, sinceramente para qualquer problema que voc\u00ea precise de uma aproxima\u00e7\u00e3o de melhor do que 3,141592 provavelmente estar\u00e1 resolvendo com algum <i>software<\/i>, que te dar\u00e1 umas 100 casas do \u03c0 automaticamente.<\/p>\n<p align=\"justify\">Neste texto discutimos a respeito de um meme que circula envolvendo o \u201cc\u00e1lculo\u201d de \u03c0 de forma que ele d\u00ea 4.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-75dcf6d jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"75dcf6d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"425\" height=\"600\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/provapi4.jpg\" class=\"attachment-large size-large wp-image-950\" alt=\"\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/provapi4.jpg 425w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/provapi4-213x300.jpg 213w\" sizes=\"(max-width: 425px) 100vw, 425px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-967032c jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"967032c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">Em algumas respostas a este meme, quem o critica sugere que a figura formada deveria ser um losango\u2026 por\u00e9m esta conclus\u00e3o est\u00e1 errada. Podemos realmente ter um \u201cc\u00edrculo\u201d de di\u00e2metro 1 e per\u00edmetro 4. Mas para entender como \u00e9 poss\u00edvel construirmos um \u201cc\u00edrculo\u201d nestas condi\u00e7\u00f5es, \u00e9 interessante primeiro discutirmos um pouco sobre fractais.<\/p><p align=\"justify\">Imagine um tri\u00e2ngulo equil\u00e1tero de lado 1. Sem nenhum mist\u00e9rio, sabemos que seu per\u00edmetro \u00e9 igual a 3.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f994267 jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"f994267\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"202\" height=\"182\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/F0.png\" class=\"attachment-large size-large wp-image-955\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a53b600 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"a53b600\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">Ent\u00e3o segmentamos cada uma de suas arestas em tr\u00eas partes. E criamos no centro de cada aresta, um novo tri\u00e2ngulo equil\u00e1tero com lado 1\/3. Agora esta nova figura tem 12 arestas, cada uma com tamanho 1\/3, dessa forma seu per\u00edmetro \u00e9 4.<\/p><p>\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7d32f7c jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"7d32f7c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"214\" height=\"246\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/F1.png\" class=\"attachment-large size-large wp-image-956\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cce1885 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"cce1885\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">Ent\u00e3o segmentamos cada uma de suas arestas em tr\u00eas partes. E criamos no centro de cada aresta, um novo tri\u00e2ngulo equil\u00e1tero com lado 1\/27. Agora esta nova figura tem 192 arestas, cada uma com tamanho 1\/27, dessa forma seu per\u00edmetro \u00e9 aproximadamente 7,11.<\/p><p align=\"justify\">\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-af8eb4c jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"af8eb4c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"221\" height=\"250\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/F2.png\" class=\"attachment-large size-large wp-image-957\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b244c11 jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"b244c11\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">Ent\u00e3o segmentamos cada uma de suas arestas em tr\u00eas partes. E criamos no centro de cada aresta, um novo tri\u00e2ngulo equil\u00e1tero com lado 1\/81. Agora esta nova figura tem 768 arestas, cada uma com tamanho 1\/81, dessa forma seu per\u00edmetro \u00e9 aproximadamente 9,48.<\/p><p>\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-16deb99 jltma-glass-effect-no elementor-widget elementor-widget-image\" data-id=\"16deb99\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"227\" height=\"266\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2019\/12\/F3.png\" class=\"attachment-large size-large wp-image-958\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-700e9fc jltma-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"700e9fc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p align=\"justify\">O processo pode ser continuado indefinidamente. Levando a constru\u00e7\u00e3o de uma figura conhecida como Ilha de Koch, que possu\u00ed \u00e1rea finita, por\u00e9m per\u00edmetro infinito. Uma caracter\u00edstica peculiar desta figura diz respeito a todo o seu per\u00edmetro ser formado por pontos de inflex\u00e3o (bicos), o que a faz n\u00e3o deriv\u00e1vel para qualquer ponto.<\/p><p align=\"justify\">Este mesmo princ\u00edpio pode ser aplicado para transformar um quadrado em um \u201cc\u00edrculo\u201d. Transformando cada v\u00e9rtice com \u00e2ngulo interno de 90 graus, em outros tr\u00eas v\u00e9rtices como apresentado no meme, preservamos o per\u00edmetro da figura, por\u00e9m for\u00e7amos que sua \u00e1rea seja igual a \u00e1rea de um c\u00edrculo.<\/p><p align=\"justify\">Assim, n\u00e3o importa o quanto nos aproximarmos, a figura sempre parecer\u00e1 um c\u00edrculo (pois o processo foi repetido infinitas vezes), contudo assim como o caso da Ilha de Koch, este \u201cc\u00edrculo\u201d \u00e9 todo formado por pontos de inflex\u00e3o (bicos), o que faz dele n\u00e3o deriv\u00e1vel para qualquer ponto (detalhe: c\u00edrculos s\u00e3o deriv\u00e1veis em todos os pontos).<\/p><p>\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0eac341 elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"0eac341\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4c1e76d jltma-glass-effect-no\" data-id=\"4c1e76d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-35989b8 jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"35989b8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/www.blogs.unicamp.br\/zero\/\">Voltar para p\u00e1gina principal<\/a><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-84fb701 elementor-section-boxed elementor-section-height-default elementor-section-height-default jltma-glass-effect-no\" data-id=\"84fb701\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;_ha_eqh_enable&quot;:false}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-19bf016 jltma-glass-effect-no\" data-id=\"19bf016\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a80ff07 jltma-glass-effect-no elementor-widget elementor-widget-heading\" data-id=\"a80ff07\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\"><a href=\"https:\/\/www.blogs.unicamp.br\/zero\/quem-escreve-os-posts\/\">Quem escreve os posts?<\/a><\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>H\u00e1 um meme que mostra como transformar um quadrado em um &#8220;c\u00edrculo&#8221; e concluir que \u03c0 = 4. Mas h\u00e1 algo de errado neste &#8220;c\u00edrculo&#8221;.<\/p>\n","protected":false},"author":434,"featured_media":972,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"colormag_page_container_layout":"default_layout","colormag_page_sidebar_layout":"default_layout","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"pgc_sgb_lightbox_settings":"","_vp_format_video_url":"","_vp_image_focal_point":[],"footnotes":""},"categories":[1208],"tags":[],"class_list":["post-948","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-v-2-ed-1"],"_links":{"self":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/948","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/users\/434"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/comments?post=948"}],"version-history":[{"count":17,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/948\/revisions"}],"predecessor-version":[{"id":5212,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/948\/revisions\/5212"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/media\/972"}],"wp:attachment":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/media?parent=948"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/categories?post=948"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/tags?post=948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}