{"id":469,"date":"2008-05-22T16:02:39","date_gmt":"2008-05-22T23:02:39","guid":{"rendered":"https:\/\/www.reenigne.org\/blog\/?p=469"},"modified":"2021-08-04T08:12:36","modified_gmt":"2021-08-04T07:12:36","slug":"rotating-fractals","status":"publish","type":"post","link":"https:\/\/www.reenigne.org\/blog\/rotating-fractals\/","title":{"rendered":"Rotating fractals"},"content":{"rendered":"

Fractals like z=z^2+c and z=z^3+c are easy to draw because they just involve complex addition and multiplication. What happens if we try to generalize these sorts of fractals to non-integer powers?<\/p>\n

We can do this using the identities
\n\\displaystyle z^w = (e^{\\frac{\\log z}{\\log e}})^w = e^{w\\frac{\\log z}{\\log e}}
\n\\displaystyle e^{x+iy} = e^x(\\cos y + i \\sin y)
\n\\displaystyle \\frac{\\log z}{\\log e} = \\frac{\\log |z|}{\\log e + i(\\arg z + 2n\\pi)}<\/p>\n

The trouble is, we have to pick a value for n. Given a parameter \\theta we can pick n such that \\theta - \\pi \\leq \\arg(x+iy) + 2n\\pi < \\theta + \\pi (i.e. choose a branch cut of constant argument and a sheet of the Riemann surface): \\displaystyle n = \\lfloor\\frac{1}{2}(\\frac{\\theta - \\arg(x+iy)}{\\pi} + 1)\\rfloor. Then as we gradually vary \\theta the fractal we get will change. For w = u+iv, v \\ne 0, increasing \\theta will cause z^w to \"spiral in\" from infinity towards the origin (v<0) or out (v>0). When v = 0, increasing \\theta will have a periodic effect, which will cause the fractal to appear to \"rotate\" in fractally sort of ways with a period of \\displaystyle \\frac{2\\pi}{u}.<\/p>\n

It would be interesting to plot these as 3D images (like these<\/a> with \\theta on the z axis. These would look sort of helical (or conical, for v \\ne 0.)<\/p>\n

Using u < 0 poses some additional problems - the attractor isn't bound by any circle of finite radius about the origin so it's more difficult to tell when a point escapes and when it is convergent but just currently very far out. Clearly some further thought is necessary to plot such fractals accurately - perhaps there are other escape conditions which would work for these.<\/p>\n","protected":false},"excerpt":{"rendered":"

Fractals like and are easy to draw because they just involve complex addition and multiplication. What happens if we try to generalize these sorts of fractals to non-integer powers? We can do this using the identities The trouble is, we have to pick a value for . Given a parameter we can pick such that […]<\/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-469","post","type-post","status-publish","format-standard","hentry","category-fractals"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/469","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=469"}],"version-history":[{"count":4,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/469\/revisions"}],"predecessor-version":[{"id":2122,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/469\/revisions\/2122"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=469"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=469"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=469"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}