Robert P. Munafo, 2023 Mar 20.
The escape radius is a parameter used in iterating a point under the Mandelbrot Set and Julia Set functions. It is the radius of a circle on the Complex Plane used as a boundary to determine when iteration can stop. The circle is centered at the origin, and has a radius of at least 2.0. As soon as the iteration yields a value that falls outside the circle, iteration can stop and it has been determined that the point that was iterated is not a member of the Mandelbrot Set.
A common optimization is to compare the square of an iterate's magnitude to the square of the escape radius; this is to avoid the square root calculation that would be required to compute the actual magnitude. The square of the magnitude is simply the sum of the squares of the two components, and these are computed as imtermediate results of the iteration calculation.
Minimum Escape Radius of 2
The reason for having an escape radius is to prevent iteration from going on forever. The Mandelbrot iteration is defined to terminate when the iteration becomes unbounded, i.e. goes off to infinity. It is fairly straightforward to show that if a point ends up outside a radius of 2.0 then the iteration will go off to infinity. Therefore any escape radius greater than 2.0 will also suffice. Using larger escape radius also makes the lemniscates more closely resemble the equipotentials.
Here is the proof:
%% need to translate into Mu Ency terminology, try to simplify a bit, eliminate terms "induction" and "triangle inequality".
Original USENET article: -------R-P-M----N-e-w-s-b-l-i-n-k------------------------------ Xref: world sci.math:34837 Path: world!eff!sol.ctr.columbia.edu!zaphod.mps.ohio-state.edu!uwm.edu!biosci!agate!usenet.ins.cwru.edu!math26647.math.cwru.edu!user From: email@example.com (mike hurley) Newsgroups: sci.math Subject: Re: Mandelbrot question Message-ID: <firstname.lastname@example.org> Date: 26 Apr 93 18:58:27 GMT References: <1993Apr26.email@example.com> Followup-To: sci.math Organization: case western reserve u Lines: 43 NNTP-Posting-Host: math26647.math.cwru.edu In article <1993Apr26.firstname.lastname@example.org>, email@example.com (Hein Roehrig) wrote: > > > I currently am working on a (high school) report concerning the mandelbrot > set. In one book (The Emperor's New Mind by Penrose) I learned that the > series is guaranteed to be unbounded if the absolute value of one of its > elements surpasses 1+sqrt(2). Can anybody explain me why? > > Thanks in advance for any advice, including book tips! > > Hein > ---- > eurom: Free Multiline Unix BBS Home of the FSAG > Frankfurt/Main,Germany Data: ++49-69-6312934 The sequence generated by f(z) = z^2 + c beginning with 0 begins 0,c,c^2 + c, ... Call these terms z0, z1, z2, ... . In fact this sequence will be unbounded if any of its terms has magnitude larger than 2. The main idea is that if the magnitude |z| is bigger than both 2 and the magnitude |c|, then |f(z)|/|z| > |z| - 1 > 1, so that by induction the sequence of magnitudes will grow geometrically. To establish the inequality, note that |f(z)|/|z| = |z^2 + c|/|z| >= (|z|^2-|c|)/|z| [ triangle inequality ] = |z| - (|c|/|z|) > |z| - 1 [ |z| > |c| ] > 1 [ |z| > 2 ] If |c| <= 2 and |zn| > 2 then certainly z = zn satisfies these hypotheses, so the sequence is unbounded. If |c| > 2 then |c^2+c| >= |c|^2 - |c| = |c|(|c|-1) > |c| > 2 so that z=z2 satisfies the hypotheses of the argument above. mike hurley firstname.lastname@example.org
Proof that 2.0 is sufficient: Mike Hurley (email@example.com)
revisions: 19971119 oldest on record; 20230320 describe optimisation
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023. Mu-ency index
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Mar 20. s.27