{"id":183149,"date":"2024-03-29T02:20:00","date_gmt":"2024-03-29T01:20:00","guid":{"rendered":"https:\/\/datascientest.com\/en\/?p=183149"},"modified":"2024-03-14T20:36:37","modified_gmt":"2024-03-14T19:36:37","slug":"fibonacci-sequence-recursion-cryptography-and-the-golden-ratio","status":"publish","type":"post","link":"https:\/\/datascientest.com\/en\/fibonacci-sequence-recursion-cryptography-and-the-golden-ratio","title":{"rendered":"Fibonacci sequence: Recursion, cryptography and the golden ratio"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"183149\" class=\"elementor elementor-183149\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62cca9b e-flex e-con-boxed e-con\" data-id=\"62cca9b\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-8335d65 elementor-widget elementor-widget-text-editor\" data-id=\"8335d65\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<h2>In the world of mathematics, the importance of sequences and series in analysis is well established. Sometimes, it&#8217;s hard to find a concrete application for concepts invented (or discovered?) in the field. The Fibonacci sequence, on the other hand, can be found in many areas of nature, and continues to fascinate researchers thanks to its properties closely linked to the golden ratio.<\/h2>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-faf5bef e-flex e-con-boxed e-con\" data-id=\"faf5bef\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d1d1dc4 elementor-widget elementor-widget-heading\" data-id=\"d1d1dc4\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h3 class=\"elementor-heading-title elementor-size-default\">History and background<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2f9f697 elementor-widget elementor-widget-text-editor\" data-id=\"2f9f697\" data-element_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<p>The<strong> Fibonacci sequence<\/strong> takes its name from the 13th-century Italian mathematician Leonardo of Pisa, also known as Fibonacci. Although he didn&#8217;t invent the sequence, he introduced it to Europe with his book &#8220;Liber Abaci&#8221; (Book of the Abacus or Book of Calculation) in 1202.<\/p><p>In this book, <strong>Fibonacci<\/strong> posed and solved a problem involving the growth of an idealized rabbit population, which leads to this sequence of numbers. It&#8217;s worth noting that the Fibonacci sequence was known long before this date in India, <strong>where the relationships between the numbers in the sequence were already being studied.<\/strong><\/p><p>Fibonacci numbers have many applications in nature and art. They can be observed on a variety of scales, from flower petals to galaxy spirals. Flower petals often have <strong>Fibonacci<\/strong> numbers &#8211; daisies can have 34, 55 or even 89 petals. And sunflower seeds or pine cones are often arranged in spirals following Fibonacci numbers.<\/p><p>In architecture, the golden ratio, directly linked to the <strong>Fibonacci sequence<\/strong>, has been used to create works that are pleasing to the eye thanks to their &#8220;perfect&#8221; proportions. For example, the Parthenon in Greece and the Taj Mahal in India both seem to use the golden ratio in their construction.<\/p><p><strong>It&#8217;s this universality that makes the Fibonacci sequence so fascinatin<\/strong>g &#8211; a simple sequence of numbers that finds its way through mathematics, nature and art, uniting fields that might seem remote at first glance. In the next section, we&#8217;ll take a closer look at what these numbers are and how we can generate them ourselves using the <a href=\"https:\/\/datascientest.com\/en\/python-the-most-popular-language\">Python programming language.<\/a><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3f7c6b8 elementor-widget elementor-widget-image\" data-id=\"3f7c6b8\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t<figure class=\"wp-caption\">\n\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" src=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/image1.jpg\" title=\"\" alt=\"\" loading=\"lazy\" \/>\t\t\t\t\t\t\t\t\t\t\t<figcaption class=\"widget-image-caption wp-caption-text\"><\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\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-a34c356 elementor-widget elementor-widget-text-editor\" data-id=\"a34c356\" data-element_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<p>Romanesco cabbage hides a secret &#8211; count the number of spirals, if you can!<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-22763d7 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"22763d7\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-scientist-course\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">A better understanding of the Fibonacci sequence<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-45579b2 e-flex e-con-boxed e-con\" data-id=\"45579b2\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-87d5693 elementor-widget elementor-widget-heading\" data-id=\"87d5693\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Definition and implementation<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-75c998a elementor-widget elementor-widget-text-editor\" data-id=\"75c998a\" data-element_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<p>Formally, the <strong>Fibonacci sequence<\/strong> is a sequence in which each term is equal to the sum of the two preceding elements. Just knowing the first two terms and the recurrence formula is enough to construct the sequence in its entirety.<\/p><p>$F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1}+ F_{n-2}$ for $n \\geq 2$.<\/p><p>Building a program to provide the nth F<strong>ibonacci number<\/strong> is a very good elementary exercise. Before reading on, you can try coding the function yourself!<\/p><p>Here&#8217;s a Python function to solve the problem:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4436fd6 elementor-widget elementor-widget-code-highlight\" data-id=\"4436fd6\" data-element_type=\"widget\" data-widget_type=\"code-highlight.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"prismjs-tomorrow copy-to-clipboard \">\n\t\t\t<pre data-line=\"\" class=\"highlight-height language-python \">\n\t\t\t\t<code readonly=\"true\" class=\"language-python\">\n\t\t\t\t\t<xmp>def fibonacci(n):\r\n    if n in {0, 1}:\r\n        return n\r\n    return fibonacci(n-1) + fibonacci(n-2)<\/xmp>\n\t\t\t\t<\/code>\n\t\t\t<\/pre>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0d2a7e7 elementor-widget elementor-widget-text-editor\" data-id=\"0d2a7e7\" data-element_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<p>This<strong> initial recursive implementation may be satisfactory<\/strong>, but there is scope for optimizing this code by opting for an iterative approach. Recursion is very restrictive in terms of computation time; in fact, the time complexity of this function is O(n\u00b2). In practice, this means that the function struggles to deliver a result within a satisfactory timeframe when n is large (for example, n = 40).<\/p><p>Here&#8217;s a new approach, this time iterative, which produces the same result using a for loop:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2608b8d elementor-widget elementor-widget-code-highlight\" data-id=\"2608b8d\" data-element_type=\"widget\" data-widget_type=\"code-highlight.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"prismjs-tomorrow copy-to-clipboard \">\n\t\t\t<pre data-line=\"\" class=\"highlight-height language-python \">\n\t\t\t\t<code readonly=\"true\" class=\"language-python\">\n\t\t\t\t\t<xmp>def fibonacci(n):\r\n    a, b, c= 0, 1, 0\r\n    if n == 0:\r\n        return 0\r\n    for i in range(2, n+1):\r\n        c = a + b\r\n        a = b\r\n        b = c\r\n    return b<\/xmp>\n\t\t\t\t<\/code>\n\t\t\t<\/pre>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d930d7e elementor-widget elementor-widget-text-editor\" data-id=\"d930d7e\" data-element_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<p>One way of optimizing this code would be to cache the elements of the sequence as they are introduced, so as to avoid redundant calculations each time the function is called.<\/p><p>In addition, there&#8217;s a mathematical formula for calculating the exact term of the sequence without recursion. Called Binet&#8217;s formula, it uses the golden ratio :<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0e9635e elementor-widget elementor-widget-code-highlight\" data-id=\"0e9635e\" data-element_type=\"widget\" data-widget_type=\"code-highlight.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"prismjs-tomorrow copy-to-clipboard \">\n\t\t\t<pre data-line=\"\" class=\"highlight-height language-python \">\n\t\t\t\t<code readonly=\"true\" class=\"language-python\">\n\t\t\t\t\t<xmp>def fibonacci(n):\r\n\r\n\r\n    sqrt5 = math.sqrt(5)\r\n\r\n\r\n    F_n = int((( (1 + sqrt5) ** n - (1 - sqrt5) ** n ) \/ ( 2 ** n * sqrt5 )))\r\n\r\n\r\n    return F_n<\/xmp>\n\t\t\t\t<\/code>\n\t\t\t<\/pre>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-c7fde3e e-flex e-con-boxed e-con\" data-id=\"c7fde3e\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5019008 elementor-widget elementor-widget-heading\" data-id=\"5019008\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Conclusion<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9ac0a33 elementor-widget elementor-widget-text-editor\" data-id=\"9ac0a33\" data-element_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<p>The<strong> Fibonacci sequence<\/strong> is used in a wide variety of fields, such as cryptography and trading. Among other things, it can be used to generate a list of pseudo-random numbers or to create an elementary encryption system.<\/p><p>In finance, a tool called the Fibonacci retracement level is used to estimate how far an asset could fall before resuming its trend movement. In time series analysis, <strong>Fibonacci<\/strong> numbers are used to determine the optimum number of time periods to use in calculating moving averages.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-504deb3 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"504deb3\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-scientist-course\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Mastering the Fibonacci sequence in Data Science<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6587bb5 elementor-widget elementor-widget-html\" data-id=\"6587bb5\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"FAQPage\",\n  \"mainEntity\": [{\n    \"@type\": \"Question\",\n    \"name\": \"History and background\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The Fibonacci sequence takes its name from the 13th-century Italian mathematician Leonardo of Pisa, also known as Fibonacci. Although he did not invent the sequence, he introduced it to Europe with his book \"Liber Abaci\" (Book of the Abacus or Book of Calculation) in 1202.Fibonacci numbers have many applications in nature and art. They can be observed on a variety of scales, from flower petals to galaxy spirals. Flower petals often have Fibonacci numbers - daisies can have 34, 55 or even 89 petals. And sunflower seeds or pine cones are often arranged in spirals following Fibonacci numbers. In architecture, the golden ratio, directly linked to the Fibonacci sequence, has been called upon to create works that are pleasing to the eye thanks to their perfect proportion.\"\n    }\n  },{\n    \"@type\": \"Question\",\n    \"name\": \"Definition and implementation\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Formally, the Fibonacci sequence is a sequence in which each term is equal to the sum of the two preceding elements. Just knowing the first two terms and the recurrence formula is enough to construct the sequence in its entirety. The construction of a program providing the nth Fibonacci number is a very good elementary exercise. Before reading on, you can try coding the function yourself!\"\n    }\n  }]\n}\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>In the world of mathematics, the importance of sequences and series in analysis is well established. Sometimes, it&#8217;s hard to find a concrete application for concepts invented (or discovered?) in the field. The Fibonacci sequence, on the other hand, can be found in many areas of nature, and continues to fascinate researchers thanks to its [&hellip;]<\/p>\n","protected":false},"author":67,"featured_media":183150,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2425],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fibonacci sequence: Recursion, cryptography and the golden ratio<\/title>\n<meta name=\"description\" content=\"The Fibonacci sequence has diverse applications ranging from mathematical analysis to cryptography and design, and its connection to the\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/datascientest.com\/en\/fibonacci-sequence-recursion-cryptography-and-the-golden-ratio\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fibonacci sequence: Recursion, cryptography and the golden ratio\" \/>\n<meta property=\"og:description\" content=\"In the world of mathematics, the importance of sequences and series in analysis is well established. Sometimes, it&#039;s hard to find a concrete application\" \/>\n<meta property=\"og:url\" content=\"https:\/\/datascientest.com\/en\/fibonacci-sequence-recursion-cryptography-and-the-golden-ratio\" \/>\n<meta property=\"og:site_name\" content=\"Data Science Courses | DataScientest\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/DataScientestEN\" \/>\n<meta property=\"article:published_time\" content=\"2024-03-29T01:20:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-03-14T19:36:37+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/datascientest.com\/en\/files\/2024\/03\/Fibonacci-768x422-1.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"768\" \/>\n\t<meta property=\"og:image:height\" content=\"422\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Melanie\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" 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