Received: by mail.san.rr.com for ehill1 (with Cubic Circle's cucipop (v1.21 1997/08/10) Wed Jan 28 20:26:55 1998) X-From_: owner-fractint@lists.xmission.com Wed Jan 28 20:21 PST 1998 Received: from lists.xmission.com (lists.xmission.com [198.60.22.7]) by mail-atm.san.rr.com (8.8.7/8.8.8) with SMTP id UAA25044; Wed, 28 Jan 1998 20:21:48 -0800 (PST) Received: from domo by lists.xmission.com with local (Exim 1.73 #4) id 0xxlT1-000013-00; Wed, 28 Jan 1998 21:20:43 -0700 Date: Wed, 28 Jan 1998 21:20:33 -0700 (MST) From: Kerry Mitchell To: fractint@lists.xmission.com cc: fractint@xmission.com Subject: Re: (fractint) 2 Dumb ?? In-Reply-To: <3.0.3.32.19980128194601.0079d940@mail.earthlink.net> Message-ID: MIME-Version: 1.0 Sender: owner-fractint@lists.xmission.com Precedence: bulk Reply-To: fractint@lists.xmission.com Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Length: 3325 The Farey series provides a convenient way to identify the disks along the cardioids in the Mandelbrot set. For review, the Farey sequence starts: 0/1, 1/1 in the first generation, followed by 0/1, 1/2, 1/1; 0/1, 1/3, 1/2, 2/3, 1/1; etc. For each subsequent level, every second term is a combination of two terms from the previous generation. For example, the 1/3 comes from (0+1)/(1+2) from the previous step. For the Mandelbrot set, let 0/1 represent the cusp (0.25,0), 1/2 is the large disk at -1, and 1/1 is back to the cusp. Looking at the third level above, between 0/1 and 1/2 is 1/3. This corresponds to the large disk at the top of the cardioid (and 2/3 is the other disk at the bottom). Between any two disks with adjacent Farey entries at generation j, the largest disk between those two has the Farey number between those two at generation j+1. In other words, the largest disk between disks 1/3 and 1/2 is disk (1+1)/(2+3) = 2/5. Aside from labelling, the Farey numbers also serve as a coordinate. Let theta = 2 * pi * m/n, where m/n is the Farey fraction for the given disk. Then, r = 0.5 * (1 - cos(theta)), x = r * cos(theta) + 0.25, y = r * sin(theta), c = x + i*y c is the point of tangency between the main cardioid and the m/n disk. Since all fractions eventually show up in the Farey series, all fractions have corresponding disks. Also, if we replace the rational number m/n with an irrational number (such as phi = 0.618034...), then the corresponding c will be a cardioid boundary point with no tangent disk. Therefore, it will have a chaotic orbit. (An easier way to get chaotic points is to use a rational number for theta. Since pi is irrational, that guarantees that the factor (m/n in the Farey case) will also be irrational.) The point abouve above chaotic orbits brings up another Farey connection. Each disk has a corresponding Farey number m/n, which is in lowest terms. The denominator, n, indicates the periodicity of the disk tangent to the cardioid at the point c. For example, with m/n = 1/2, the tangency point is at the disk of periodicity 2 at -1. The 1/3 and 2/3 disks are tangent to the period 3 disks at the top and bottom. These are the connections between the Farey sequence and the Mandelbrot set of which I'm aware. Any others? ------------------------------------------------------------------------------- Kerry Mitchell lkmitch@primenet.com ------------------------------------------------------------------------------- On Wed, 28 Jan 1998, Peter Jakubowicz wrote: > 1)What does the Farey series have to do with the Mandelbrot set? > 2)How long is the coast of the Mandelbrot set? > Thank you. > > > - > ------------------------------------------------------------ > Thanks for using Fractint, The Fractals and Fractint Discussion List > Post Message: fractint@xmission.com > Get Commands: majordomo@xmission.com "help" > Administrator: twegner@phoenix.net > Unsubscribe: majordomo@xmission.com "unsubscribe fractint" > - ------------------------------------------------------------ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@xmission.com Get Commands: majordomo@xmission.com "help" Administrator: twegner@phoenix.net Unsubscribe: majordomo@xmission.com "unsubscribe fractint"