{"id":1438,"date":"2011-08-30T16:00:16","date_gmt":"2011-08-30T23:00:16","guid":{"rendered":"https:\/\/www.reenigne.org\/blog\/?p=1438"},"modified":"2011-08-20T19:54:12","modified_gmt":"2011-08-21T02:54:12","slug":"fractals-on-the-hyperbolic-plane","status":"publish","type":"post","link":"https:\/\/www.reenigne.org\/blog\/fractals-on-the-hyperbolic-plane\/","title":{"rendered":"Fractals on the hyperbolic plane"},"content":{"rendered":"<p>Some amazing images have been made of fractal sets on the complex plane, but I don&#8217;t think I&#8217;ve ever seen one which uses hyperbolic space in a clever way. I&#8217;m not counting <a href=\"http:\/\/www.hiddendimension.com\/FractalMath\/TessellationFractals.html\">hyperbolic tessellations<\/a> here because the Euclidean analogue is not a fractal at all &#8211; it&#8217;s just a repeated tiling.<\/p>\n<p>The hyperbolic plane is particularly interesting because it is in some sense &#8220;bigger&#8221; than the Euclidean plane &#8211; you can tile the hyperbolic plane with regular heptagons for example. Now, you could just take a fractal defined in the complex plane and map it to the hyperbolic plane somehow, but that doesn&#8217;t take advantage of any of the interesting structure that the hyperbolic plane has. It&#8217;s also locally flat, so doesn&#8217;t add anything new. If you use some orbit function that is more natural in the hyperbolic plane, I think something much more interesting could result. I may have to play about with this a bit.<\/p>\n<p>Similarly, one could also do fractals on the surface of a sphere (a positively curved space &#8211; the hyperbolic plane is negatively curved and the Euclidean plane has zero curvature).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Some amazing images have been made of fractal sets on the complex plane, but I don&#8217;t think I&#8217;ve ever seen one which uses hyperbolic space in a clever way. I&#8217;m not counting hyperbolic tessellations here because the Euclidean analogue is not a fractal at all &#8211; it&#8217;s just a repeated tiling. The hyperbolic plane is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18],"tags":[],"class_list":["post-1438","post","type-post","status-publish","format-standard","hentry","category-fractals"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/1438","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/comments?post=1438"}],"version-history":[{"count":2,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/1438\/revisions"}],"predecessor-version":[{"id":1440,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/1438\/revisions\/1440"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=1438"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=1438"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=1438"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}