{"id":1033,"date":"2019-02-26T17:03:53","date_gmt":"2019-02-27T01:03:53","guid":{"rendered":"http:\/\/blog.ronrecord.com\/?page_id=1033"},"modified":"2019-02-27T14:21:20","modified_gmt":"2019-02-27T22:21:20","slug":"mandelbrot-julia-sets","status":"publish","type":"page","link":"https:\/\/blog.ronrecord.com\/index.php\/mandelbrot-julia-sets\/","title":{"rendered":"Mandelbrot & Julia Sets"},"content":{"rendered":"\n

The <\/strong>Mandelbrot Set<\/strong><\/a> is the set of values of <\/strong>c<\/strong><\/em> in the <\/strong>complex plane<\/strong><\/a> for which the <\/strong>orbit<\/strong><\/a> of 0 under <\/strong>iteration<\/strong><\/a> of the <\/strong>quadratic map<\/strong><\/a><\/p>\n\n\n\n

\"{\\displaystyle<\/figure><\/div>\n\n\n\n

remains <\/strong>bounded<\/strong><\/a>. That is, a complex number <\/strong>c<\/strong><\/em> is part of the Mandelbrot Set if, when starting with<\/strong> z<\/strong>0<\/strong><\/sub> = 0 and applying the iteration repeatedly, the <\/strong>absolute value<\/strong><\/a> of <\/strong>z<\/strong><\/em>n<\/strong><\/em><\/sub> remains bounded however large <\/strong>n<\/strong><\/em> gets.<\/strong><\/p>\n\n\n\n

Images of the Mandelbrot Set exhibit an elaborate and infinitely complicated <\/strong>boundary<\/strong><\/a> that reveals progressively ever-finer <\/strong>recursive<\/strong><\/a> detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the <\/strong>fractal<\/strong><\/a> property of <\/strong>self-similarity<\/strong><\/a> applies to the entire set, and not just to its parts.<\/strong><\/p>\n\n\n\n

The Mandelbrot Set has become popular outside <\/strong>mathematics<\/strong><\/a> both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of <\/strong>mathematical visualization<\/strong><\/a> and <\/strong>mathematical beauty<\/strong><\/a>.<\/strong><\/p>\n\n\n\n

Each point in the Mandelbrot Set defines a corresponding <\/strong>Julia Set<\/strong><\/a>.<\/strong><\/p>\n\n\n\n

The images in this gallery were created by Dr. Ronald Joe Record who holds a Ph.D. in Mathematics from the University of California. Dr. Record’s research focused on applications of Dynamical Systems Theory, an area of Mathematics popularly known as Chaos Theory.<\/strong><\/p>\n\n\n

[foogallery id=”1032″]<\/p>\n\n\n\n

<\/div>\n\n\n\n

All photography and digital photo manipulation by Ronald Joe Record, Copyright 2019, all rights reserved.<\/b><\/p>

Mandelbrot<\/b><\/a><\/td>Lyapunov<\/b><\/a><\/td>Iterated<\/b><\/a><\/td>Fractals<\/b><\/a><\/td>Photos<\/b><\/a><\/td><\/tr><\/tbody><\/table>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

The Mandelbrot Set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map remains bounded. That is, a complex number c is part of the Mandelbrot Set… <\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/pages\/1033"}],"collection":[{"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/comments?post=1033"}],"version-history":[{"count":3,"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/pages\/1033\/revisions"}],"predecessor-version":[{"id":1097,"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/pages\/1033\/revisions\/1097"}],"wp:attachment":[{"href":"https:\/\/blog.ronrecord.com\/index.php\/wp-json\/wp\/v2\/media?parent=1033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}