{"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't think I've ever seen one which uses hyperbolic space in a clever way. I'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 - it's just a repeated tiling.<\/p>\n<p>The hyperbolic plane is particularly interesting because it is in some sense \"bigger\" than the Euclidean plane - 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't take advantage of any of the interesting structure that the hyperbolic plane has. It's also locally flat, so doesn'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 - 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't think I've ever seen one which uses hyperbolic space in a clever way. I'm not counting hyperbolic tessellations here because the Euclidean analogue is not a fractal at all - it'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}]}}