{"id":51,"date":"2016-01-25T22:05:39","date_gmt":"2016-01-25T22:05:39","guid":{"rendered":"https:\/\/courses.candelalearning.com\/math4libarts\/?post_type=chapter&p=51"},"modified":"2019-05-30T16:33:24","modified_gmt":"2019-05-30T16:33:24","slug":"module-2-overview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/module-2-overview\/","title":{"raw":"Why It Matters: Fractals","rendered":"Why It Matters: Fractals"},"content":{"raw":"

Why learn about fractals?<\/h2>\r\nFractals are everywhere! \u00a0If you don\u2019t believe me, just take a look outside your window. \u00a0From the shapes of trees and bushes to the jagged profiles of mountains to the irregular coastlines, many features of our natural world seem to be modeled by fractal geometry.\r\n\r\nBut what exactly is a fractal? \u00a0As you will learn in this module, a fractal<\/strong> is an object that displays self-similarity at every level. \u00a0That is, when you zoom in on one section, it resembles the whole image. \u00a0This self-similarity doesn\u2019t have to be exact; in fact many fractals show some variation or randomness. \u00a0Below is a video illustrating how the Mandelbrot set, a well-known fractal, displays self-similarity.\r\n\r\n \r\n

https:\/\/www.youtube.com\/embed\/G_GBwuYuOOs<\/p>\r\n \r\n\r\nWhile some fractals (like the Mandelbrot set) could pass for works of art, the true beauty of fractals is in how such intricate designs and patterns can result from very elementary generating formulas or rules.\r\n\r\nIn this module, you will learn how to create fractal patterns such as the Mandelbrot set using a simple formula such as:\r\n

[latex]z_{n+1} = z_n^2 + c[\/latex]<\/p>\r\n \r\n\r\nOf course there are many details that still need to be explained, such as the relationship between fractals and complex numbers. The values of [latex]c[\/latex], [latex]z_n[\/latex] \u00a0and [latex]z_{n+1}[\/latex] in the above formula are supposed to be complex numbers<\/strong>, that is, numbers that include the imaginary unit<\/strong>, [latex]i = \\sqrt{-1}[\/latex].\r\n\r\n \r\n\r\nThe imaginary number [latex]i[\/latex] is something completely different than any number you have ever seen. \u00a0In fact, [latex]i[\/latex] does not show up on the number line at all! \u00a0Instead, as you will soon discover, the imaginary unit lives on its own separate number line, called the imaginary axis<\/strong>, which is perpendicular to the usual number line (or real axis<\/strong>).\r\n\r\n \r\n\r\nThe Mandelbrot set itself is made up of the complex numbers that satisfy a certain rule related to a simple equation. \u00a0The resulting picture is amazing, and just gets more and more fascinating as you zoom in!\r\n\r\n[caption id=\"attachment_2305\" align=\"aligncenter\" width=\"574\" class=\"center \"]\"A<\/a> A Mandelbrot set in the Complex plane.[\/caption]","rendered":"

Why learn about fractals?<\/h2>\n

Fractals are everywhere! \u00a0If you don\u2019t believe me, just take a look outside your window. \u00a0From the shapes of trees and bushes to the jagged profiles of mountains to the irregular coastlines, many features of our natural world seem to be modeled by fractal geometry.<\/p>\n

But what exactly is a fractal? \u00a0As you will learn in this module, a fractal<\/strong> is an object that displays self-similarity at every level. \u00a0That is, when you zoom in on one section, it resembles the whole image. \u00a0This self-similarity doesn\u2019t have to be exact; in fact many fractals show some variation or randomness. \u00a0Below is a video illustrating how the Mandelbrot set, a well-known fractal, displays self-similarity.<\/p>\n

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