Mandelbrot Set<\/h3>\r\n
For any complex number [latex]c[\/latex], define the sequence:<\/p>\r\n
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If this sequence always stays close to the origin (within [latex]2[\/latex] units), then the number [latex]c[\/latex] is part of the Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number [latex]c[\/latex] is not part of the set.<\/p>\r\n<\/div>\r\n<\/section>\r\n To generate the Mandelbrot Set you'll need to multiply complex numbers.<\/p>\r\n Recall that multiply [latex](a + bi)^2[\/latex] uses the same technique as squaring a binomial such as [latex](a + b)^{2}[\/latex], with the added condition that [latex]i^{2}=-1[\/latex]. When squaring binomials, you can follow the pattern [latex](a+b)^{2} = a^2 + 2ab + b^2[\/latex].<\/p>\r\n When squaring complex numbers, follow a similar pattern that takes into account the fact that\u00a0[latex]i^{2}=-1[\/latex].<\/p>\r\n Determine if [latex]c=1+i[\/latex] is part of the Mandelbrot Set. <\/p>\r\n We can already see that these values are getting quite large. It does not appear that [latex]c=1+i[\/latex] is part of the Mandelbrot Set.<\/p>\r\n The omitted details may be added using this pattern.<\/p>\r\n Determine if [latex]c=0.5i[\/latex] is part of the Mandelbrot Set. <\/p>\r\n While not definitive with this few iterations, it does appear that this value is remaining small, suggesting that [latex]0.5i[\/latex] is part of the Mandelbrot Set.[\/hidden-answer]<\/p>\r\n<\/section>\r\n\r\n[caption id=\"attachment_1737\" align=\"alignright\" width=\"187\"] If all complex numbers are tested, and we plot each number that is in the Mandelbrot Set on the complex plane, we obtain the shape to the right.<\/p>\r\n The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole.<\/p>\r\n In addition to coloring the Mandelbrot Set itself black, it is common to color the points in the complex plane surrounding the set. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than [latex]2[\/latex] units away from the origin. For example, using [latex]c=1+i [\/latex] above, the sequence was distance [latex]2[\/latex] from the origin after only two recursions.<\/p>\r\n For some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, in the image below light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. Greens, reds, and purples can be seen when we zoom in\u2014those are used for numbers that grow very slowly.<\/p>\r\n <\/p>\r\n
\r\n[reveal-answer q=\"703380\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"703380\"]
\r\nWe start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations:<\/p>\r\n
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\r\n[\/hidden-answer]<\/section>\r\n
\r\n[reveal-answer q=\"703381\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"703381\"]
\r\nWe start with [latex]{{z}_{0}}=0[\/latex]. We continue, omitting some detail of the calculations:<\/p>\r\n![\"A](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2017\/02\/23225415\/inkblot.png\")
Figure 1. A Mandelbrot Set graphed on a plane[\/caption]\r\n\r\n![\"The](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4685\/2017\/03\/05015451\/1440px-Mandel_zoom_00_mandelbrot_set.jpg\")
<\/a> Figure 2. A Mandelbrot Set colored in[\/caption]\r\n<\/center>\r\n
![This zoom GIF file<\/a> on the Wikipedia Mandelbrot Set entry illustrates what it looks like to zoom in.<\/p>\r\n<\/section>","rendered":"Mandelbrot Set<\/h2>\nThe Mandelbrot Set<\/strong> is a set of numbers defined based on recursive sequences.<\/p>\n\n\nMandelbrot Set<\/h3>\nFor any complex number [latex]c[\/latex], define the sequence:<\/p>\n <\/p>\n[latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\\quad{{z}_{0}}=0[\/latex]<\/div>\n <\/p>\nIf this sequence always stays close to the origin (within [latex]2[\/latex] units), then the number [latex]c[\/latex] is part of the Mandelbrot Set<\/strong>. If the sequence gets far from the origin, then the number [latex]c[\/latex] is not part of the set.<\/p>\n<\/div>\n<\/section>\nTo generate the Mandelbrot Set you’ll need to multiply complex numbers.<\/p>\n\nRecall that multiply [latex](a + bi)^2[\/latex] uses the same technique as squaring a binomial such as [latex](a + b)^{2}[\/latex], with the added condition that [latex]i^{2}=-1[\/latex]. When squaring binomials, you can follow the pattern [latex](a+b)^{2} = a^2 + 2ab + b^2[\/latex].<\/p>\nWhen squaring complex numbers, follow a similar pattern that takes into account the fact that\u00a0[latex]i^{2}=-1[\/latex].<\/p>\n[latex]\\begin{array}{rcl} (a+bi)^{2} & = & a^2 + 2abi + (bi)^2 \\\\ & = & a^2 + 2abi + b^2(i^2) \\\\ & = & a^2 + 2abi + b^2(-1) \\\\ & = & a^2 + 2abi - b^2 \\\\ \\end{array}[\/latex].<\/div>\n<\/section>\n\nDetermine if [latex]c=1+i[\/latex] is part of the Mandelbrot Set.<\/p>\n](\"https:\/\/en.wikipedia.org\/wiki\/Mandelbrot_set#\/media\/File:Mandelbrot_sequence_new.gif\"){kind=link}