; Date: Thu, 17 Nov 2011 14:47:45 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 17-11-11 (Paradox Country [7.5]) ; Id: <1.5.4.16.20111117144901.37ff65a4@pop.mindspring.com> ; --------- ; ; FOTD -- November 17, 2011 (Rating 7.5) ; ; Fractal visionaries and enthusiasts: ; ; Most everyone is familiar with the so-called 'Let's make a Deal' ; paradox, which is really no paradox at all. You do not have ; a 50-50 chance of making the correct choice when one of the ; other doors is eliminated, you have a 67-percent chance of ; winning by switching your choice to the remaining door. The ; true mystery is the fact that it is so hard to explain to others ; why this must be true. (See below for my attempt to explain it.) ; ; For a real paradox, check today's image, which is a scene in the ; left-over debris in the southwest part of the largest bud of the ; minibrot on the west branch of the filament extending from the ; large period-3 north bud on the main bay of the Mandelbrot set. ; The minibrot is in the process of morphing from a quadratic into ; a near-circular order-15 one. ; ; So far, so good. But things go quite awry when we start ; counting. Normal, well-adjusted order-15 minibrots have 14 ; lobes and are surrounded by rings of 15 elements, but the ; minibrots in today's image are anything but normal. begin with ; the minibrot in the upper left corner. A careful count reveals ; the expected 14 lobes. But now count the surrounding elements, ; and surprisingly, you will find 16, not the proper 15. ; ; This is only the start of the craziness. The large minibrot in ; the center of the scene is apparently a merging of two order-15 ; minibrots, and as such should have 28 lobes. Unfortunately, ; when I counted, I found 29. It's quite a mystery where the ; extra lobe comes from. I have not yet tried to count the ; elements surrounding this merged minibrot, but I doubt that they ; equal the expected 30. ; ; The name "Paradox Country" refers to this paradoxical craziness. ; ; The rating of 7-1/2 is all I could do for the image. The longer ; I see the colors, the harsher and muddier they appear. If it ; were not for the math interest, the image would rate something ; more like a 6. ; ; The calculation time of just under 2 minutes will pass quickly, ; bringing only minor restless eye syndrome. ; ; The finished image is posted for instant gratification at: ; ; ; ; It is posted in high definition at: ; ; ; ; The original classic FOTD web site may be accessed at: ; ; ; ; As I wrote at the start of this discussion, the 'Let's Make a ; Deal' paradox is no paradox at all. When you choose a door, you ; have a 2/3 chance of having chosen a losing door. This means ; there is a 2/3 chance that one of the other 2 doors is the ; winning door. When one of the other doors is shown to be a ; loser, nothing has changed with your original choice. There is ; still a 2/3 chance that it is a loser, but now the 2/3 chance ; that one of the other 2 doors is the winner has been combined ; into one door, which now has a 2/3 likelihood of being the ; winner. So switch your choice to the other door and your chance ; of winning will jump from 33 percent to 67 percent. ; ; The clouds continued today here at Fractal Central, while the ; weather remained dry. The fractal cats (of the escape-time ; variety) thought the temperature of 48F +9C should have been ; much warmer. ; ; The fractal people made it through the day with only minor ; inconveniences. Now that the commercial work is winding down, ; the things that once were larger inconveniences now seem less ; inconvenient. The next FOTD will conveniently be posted in 24 ; hours. Until then, take care, and always choose the other door. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= Paradox_Country { ; time=0:01:58.78 SF5 at 2000MHZ reset=2004 type=formula formulafile=basicer.frm formulaname=FinDivBrot-2 function=recip passes=1 center-mag=-0.1647031078033153/+1.037456720136922/\ 7.305788e+011/1/27.5/0 params=15/1e+023/0/0 float=y maxiter=1750 inside=0 logmap=128 periodicity=6 mathtolerance=0.05/1 colors=000IKzHIzGHzGGzFEzFDzECzEAzDCzDFzEKzEPzEUzF\ ZzFczFhzFmzGrzGvzGzzGzzHzzHzzHzzKzzOzvRyrVxmYwh`vc\ cuZfsUhqPkoKnmFqkCtiCwfDzdDvbDt`DsZDpZEmXDjTCgPCfM\ BfJBeGAcDAaAB_9BY9BW9BU9CS9CQ9CN9CK8CI8BG7AF79F79F\ 59E59D59C5AC5AD6AE7AE7BF8BG8CG9DIAFLBGNBHPBHQ78LBH\ GFQBJZ6Ng1Qo4Pl6Oj8NhAMeCLcEKaHJ_JIXLHVNGTPFQREOVD\ MYCK`BHcAFf9Dh9Be8Aa79Y68V67V79V7BV7DW8FX8GX8IY9KZ\ 9MZ9N_9P_AR`ATaAUaBWbBYcB_cCadCbeCdeCffDhfDigDkhEm\ hEoiEpjFrjFtkFvkFw`HkRI`HKP7LEMeYNfZOg_Ph`QiaRjbSk\ cTldUmeVnfWogXphYqiZrj_sk`tlaumavnfkljaknSjrIix0gv\ 8huGitOisWjrcjqkkpsknplmnmllnkjojgpneqmcqlarkZsjat\ m_umYvmXwmWvrRvrRvrQurQurQurPurPtrRtrStrRtrRsrWsrW\ srXsrYswZsw_sw`swaszYszZsz_sz`szaszbszcszdszeszfsz\ gszhsziszjszkszlszmszlrzkrzkqzjqzjpzipziozhozgnzgn\ zfmzfmzelzelzdkzdkzdjzdjzdizdizdhzdhzdgzcgzcfzcfzc\ ezcezcdzcdzcczcczcczcczcc } frm:FinDivBrot-2 { ; Jim Muth z=(0,0), c=pixel, a=-(real(p1)-2), esc=(real(p2)+16), b=imag(p1): z=(b)*(z*z*fn1(z^(a)+b))+c |z| < esc } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;