September 2003 - Fractint - mailman.xmission.com

The "bug "is not a bug!!! Is not! Is not!
by Tim Wegner 26 Sep '03

26 Sep '03

OK folks, listen up. :-) For you Fractint is a fractal program, for me it is the World's Best Quick And Dirty Reference to Complex Transcendental Functions. Press F1, go to "Fractals and the PC", and navigate down to trig identities. I put these in the help because I went through a LOT of books finding these and wanted to collect them all in one place in order to code them up. Some books are too simple and so don't have the formulas, and other more advanced books are too abstract and so don't have them either. The formula for the complex power is as follows: z^w = e^(w*log(z)) (here"^" is ASCII text shorthand for exponentiation, in this case z^w means "raise the complex number z to the power of the complex number w") where: e^(x+iy) = (e^x) (cos(y) + i sin(y)) and log(x+iy) = (1/2)log(x^2 + y^2) + i(atan(y/x) + 2kPi) for k = 0, -1, 1, -2, 2, ... (The log function refers to log base e, or ln. The expression atan(y/x) is an angle between -pi and pi in the quadrant containing (x,y) implemented in C as the atan2() function.) The point of these formulae from a programmer's perspective is that they define the complex valued power function in terms of the real versions of e^x, log(x), sin(x), and cos(x) that are available to the C programmer. The interesting and important thing here is that the value of the complex log is not unique. There are infinitely many different values, with the imaginary members. Put another way: the complex log is not single valued. Let's look at the log. In the "bug" illustration, the complex argument is: (-5.0, 0.0) The assembler log function gives: (1.6094379124341,-3.141592653589793) Coding by hand in C gives: (1.6094379124341, 3.141592653589793) BOTH ANSWERS ARE RIGHT!!! The imaginary parts differ by 2*PI, which is acceptable. What actually causes this is that the log implementation normalizes the result to a particular 2*PI segment. Jim's fatal mistake was when he assumed that given a = 1, b = -5, g = 1, h = -1, j = 1/6, that (-a*b*g*h) is -5. Yes, it's very close to -5 in the actual computer calculation, but "close" wins only in horseshoes. The problem is that the complex logarithm of (-5,0) has an imaginary part exactly on the logarithm software implementation's boundary of 2*PI normalization (namely PI). So if the calculation (-a*b*g*h) drops a bit here or adds a bit there, even a bit of the lowest significance, it can change the imaginary component from ~ PI to ~ -PI. I leave it to the reader to plug both these logarithm values into the rest of the formula for the exponential power, and see that the result is (1.132467200411351, -0.653830243005915) in the one case and (1.132467200411351,+0.653830243005915) in the other, as Jim discovered. Here's the main point - a recurring theme in trying to create fractals with complex transcendental functions. The complex power function x^y is discontinuous because of the ambiguity in the many- valued complex log. Each decision to normalize the log to a particular 2*PI interval in an effort to define a single-valued function creates a discontinuous tear in the result in a different way. The "bug" Jim found had values right on the boundary of the tear, so that tiny perturbations and rearrangements of the code pushed the calculation over the edge of the discontinuity. Fractint is cleared of the unjust accusation! The verdict: INNOCENT! All is well with the fractal world :-) One single absolute truth of the universe is thus upheld again. It is this: There are NO software bugs, ONLY USER ERRORS!!! (Sorry Jim ... -no, not *really* sorry, I lied :-) I suggest the future Fractal of the Day explore a fractal based on a discontinuous complex transcendental function in such a way that the fractal itself has visually manifest tears. There are several examples in Fractal Creations, and I am sure many of the artists here, including the honorable Mr. Muth, know of or can discover examples. Tim Wegner

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FOTD 24-09-03 (Sir Fractal Galahad [7])
by Jim Muth 25 Sep '03

25 Sep '03

FOTD -- September 24, 2003 (Rating 7) Fractal visionaries and enthusiasts: It's been six days since the last FOTD. This does not mean that I am losing my enthusiasm for the FOTD. The reason for the delay is that we had a hurricane here at Fractal Central, and the resulting power failure lasted almost 6 days. This was bad enough, but the work, which was quite a bit behind to start, got much farther behind during the time I could not work on it. I am now so far behind that I am in danger of losing customers due to my inability to deliver their jobs on time. As a result, this will be the last FOTD discussion until I catch up my other work, a task that will take at least several weeks, and possibly over a month of 12 hour days. I must work these hours until the pile of 'work to be done', which now holds over one week's work, has been reduced to a manageable size. In the meantime, I will continue to monitor the Fractint and Philofrac- tal lists. When I find the time, I will inject a few brief com- ments. It will be up to the lurkers to keep the lists alive and active for the duration. As a matter of curiosity, my e-mail box held 1181 letters when I checked it for the first time in 6 days. Most of them were junk, which together took almost 2 hours to download. The most curious letter informed me in broken english that I had been identified as a money launderer, and for some strange reason, requested my credit card number. I replied that I always launder my money. I use mild detergent and luke-warm water. Hot water makes the colors fade. ;-) ;-) Today's quickie fractal subtracts 1.11 parts of (1/Z) from Z^(1.11), then adds C. The parent fractal is a distorted M-set rotated -90 degrees, with a curving tail extending from its East Valley, which in this case is on the northeast side of the main bay. The scene is located at the end of a broad irregularly curving area of chaos on the main bay side of the northwest branch of the Seahorse Valley of the parent M-set. I named it "Sir Fractal Galahad". I have no idea what Sir Galahad has to do with fractals -- the name simply sounds noble in a comical way. And the picture looks good, honestly earning its rating of a 7. The render time of 7 minutes is short enough to be reasonable. The download, which is available on the FOTD web sites at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> is even more reasonable. The fractal cats had perhaps the best time of their lives in the 6 days since the last FOTD was posted. As I predicted, the only casualties the big blow brought to Fractal Central were a 6-day power outage and a yard of broken tree branches. The dynamic cats watched from their window shelf with interest last Thursday night as the wind roared, trees fell, and the sky was illumina- ted by the blue flashes of falling power lines and the distant pops of blowing transformers. When the next day arrived, the cats were ready for a full day's romp in the yard, which, with nothing but clean-up to do, I was able to give them. Since then, the weather has been much better, the only exception being a heavy rain that brought more flooding to the area early Tuesday morning. The cats have had 5 full days in the outdoors with me there to watch over them, making them perhaps the two happiest cats in the world. But it's very disconcerting to have so much backed-up design work waiting to be finished. And with an additional pile of storm- associated stuff likely to come in, I'm not as happy as the cats. I guess that, like the linemen of the local electric company, I'll be working 12 and 14-hour days. I'll be doing it until I clear away enough backed-up work to resume the FOTD in a relaxed manner. Until next time and next FOTD, whenever that is, take care, and see you for certain in the relatively near but not immediate future. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ SirFractalGalahad { ; time=0:07:35.70--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=ident passes=1 center-mag=+0.15079637605670420/-1.480224054715044\ 00/5.344319e+009/1/144.999982895458032/9.399329566\ 45073148e-006 params=1/1.11/-1.11/-1/0/0 float=y maxiter=1200 inside=0 logmap=144 periodicity=10 colors=000WNFXPBYR8ZT4_V1YX5WZ9UaDScHRfLPhPNkSLmWK\ p_IrcGugEwkDynfHjdKkcNlaPm`Sn_VoYXpX_qWarVbnVckVdg\ VddVe`VfYUfUUgRUhNUqKUnGUlDUjASd9Q_8OV7NPALKFJFKIA\ PHDUGGZFJcEMhDPmCSsCUzBXoAZf9aY8cPHNcOMZUMV_MRfMMr\ MIwMErNHmOKgONcPQ`PTXQWTRZQRaMSdJSgFTjCTmEWiFYeG_b\ HaZJcVKeSLgOMiKNkHziJrhKmgLhfMcdOZcPUbQPaRK`SFZUEY\ VDXWCWXCVYFUXHTWKTVMSUPSTRRSURRWQQZQP`POcPNeOMgOMh\ HMjGSkFXlEamDgnCloCqe6rW1rX1pX1oX1mX1lX1jY1iY1gY1f\ Y1dY1cY1bW4`U6ZS9XQBWOEUNGSLIQJLPHNNFQLDSJCUIEPFGL\ CIG9KC6KIEKFCKCAK98K66K45L77M98MC9NEBNHCOJDPMFPOGQ\ RHQTJRWKSYLS`NTbOTdPS_ORVOQROPYZOdiNjsNktNltNltNmt\ NntNntNotNotLnuKmuJluIkuHjuFivEhvDgvCfvBev9dw8cw7b\ w6bw5cw8dsBepDfmGgjJhfLicOj`QkYOlRMmLKnEIo8Gp2Jq8M\ rEPsJStPVuVYv_`webxjdziezhgzghzfjzekzdlzdjzchzcfzb\ dzbbzb`za_zaYz`Wz`Uz`Sz_Qz_Pz_Mz5Jz4Gz4Hz9IzDJzHSz\ T`zcVzhQzmSzlSzTTzPUzMVzI } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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Multifractal_9
by Albrecht.Niekamp＠t-online.de 25 Sep '03

25 Sep '03

Hi, the image is the message. Fractal greetings, Al. --------------------------------start-frm-------------------------------- FRM:Multifractal_9 { ; Albrecht Niekamp 170903 ;p1 (spider)julia-seed ;real(p2) 5digits : (1)shape (2)outside (3)inside1 (4)inside2 (5)inside3 ; 0_off 1_secant 2_mand 3_bees 4_jul 5_m_mods 6_phoen 7_newt 8_spider ; input2 : 2digits_many-mods 2digits_phoenix 2digits_spider ; 5digits_reset : 0_no 1_z-reset 2_iter-reset 3_both +5_warp ;imag(p2)(-) 5digits_colour(bailout) number ; input2 : 4digits(-)_mand/jul 2digits_secant 4digits_bees ;real(p3) 2digits_newt 4digits_colour1, 5digits_colour2 5digits_colour3 ;imag(p3) shape : factor (fn1), 5digits_colour4 5digits_colour5 ;real(p4) outside: factor (fn2), 4+1digits_bord-out 4digits+fract_bord-in ;imag(p4) inside1: maxit1, 1digit_use:1_maxit 2(7)_bord-out 3(8)_bord-in ; 5digits_factor1 (fn2) 4digits+fract_border1 ;real(p5) inside2: maxit2, 1digit_use:1_maxit 2(7)_bord-out 3(8)_bord-in ; 5digits_factor2 (fn3) 4digits+fract_border2 ;imag(p5) inside3: maxit3, 1digit_use:1_maxit 2(7)_bord-out 3(8)_bord-in ; 5digits_factor3 (fn4) 4digits+fract_border3 ;fn(1) shared by many-mods+bees ; da=real(p2) dd=trunc(da) da=round((da-dd)*100000000000)+11111 dd=dd+11111 d=trunc(dd/10000) dd=dd-d*10000 d3=(d==4)+(d==5)+(d==8)+(d==9) d4=d3==0 vb=d>5 sc=d==2 mo=d==6 px=d==7 ab=px+(d==3)+(d==5)+(d==9) d=trunc(dd/1000) dd=dd-d*1000 ex1=d>1 sc1=d==2 mo1=d==6 px1=d==7 v1m=mo1+px1 v1j=d>7 dd1=v1j+(d==4)+(d==5) ab1=px1+(d==3)+(d==5)+(d==9) d=trunc(dd/100) dd=dd-d*100 ex2=d>1 sc2=d==2 mo2=d==6 px2=d==7 v2m=mo2+px2 v2j=d>7 dd2=v2j+(d==4)+(d==5) ab2=px2+(d==3)+(d==5)+(d==9) d=trunc(dd/10) ex3=d>1 sc3=d==2 mo3=d==6 px3=d==7 v3m=mo3+px3 v3j=d>7 dd3=v3j+(d==4)+(d==5) ab3=px3+(d==3)+(d==5)+(d==9) d=dd-d*10 ex4=d>1 sc4=d==2 mo4=d==6 px4=d==7 v4m=mo4+px4 v4j=d>7 dd4=v4j+(d==4)+(d==5) ab4=px4+(d==3)+(d==5)+(d==9) ; mm=trunc(da/1000000000) da=da-mm*1000000000 ph=trunc(da/10000000)/100 da=da-ph*1000000000 sp=trunc(da/100000)/100 da=da-sp*10000000 d=trunc(da/10000) ex0=d>4 da=da-d*10000 d=d-5*ex0 ir0=(d==3)+(d==4) d=trunc(da/1000) w1=d>4 da=da-d*1000 d=d-5*w1 rs1=(d==2)+(d==4) ir1=(d==3)+(d==4) d=trunc(da/100) w2=d>4 da=da-d*100 d=d-5*w2 rs2=(d==2)+(d==4) ir2=(d==3)+(d==4) d=trunc(da/10) w3=d>4 da=da-d*10 d=d-5*w3 rs3=(d==2)+(d==4) ir3=(d==3)+(d==4) d=round(da) w4=d>4 d=d-5*w4 rs4=(d==2)+(d==4) tt=ex1+ex0+ex2+ex3+ex4 ; d=real(p3) dd=trunc(d) da=(d-dd)*10000000000 pp=trunc(dd/10000) ba1=dd-10000*pp ba2=trunc(da/100000) ba3=da-100000*ba2 ; d=imag(p3) sfac=trunc(d) da=(d-sfac)*10000000000 ba4=trunc(da/100000) ba5=da-ba4*100000 ; d=real(p4) ofac=trunc(d) da=(d-ofac)*10000000000 bh=trunc(da/100000)/10 bl=(da-bh*1000000)/10 bs=bl/2 ; d=imag(p2) t=d<0 if (t) d=-d endif dd=trunc(d) da=round((d-dd)*10000000000) d=trunc(dd/10000) dd=dd-d*10000 bb0=ba1*(d==1)+ba2*(d==2)+ba3*(d==3)+ba4*(d==4)+ba5*(d==5) d=trunc(dd/1000) dd=dd-d*1000 bb1=ba1*(d==1)+ba2*(d==2)+ba3*(d==3)+ba4*(d==4)+ba5*(d==5) d=trunc(dd/100) dd=dd-d*100 bb2=ba1*(d==1)+ba2*(d==2)+ba3*(d==3)+ba4*(d==4)+ba5*(d==5) d=trunc(dd/10) dd=dd-d*10 bb3=ba1*(d==1)+ba2*(d==2)+ba3*(d==3)+ba4*(d==4)+ba5*(d==5) d=round(dd) bb4=ba1*(d==1)+ba2*(d==2)+ba3*(d==3)+ba4*(d==4)+ba5*(d==5) ; d=da p0=trunc(d/100000000)/10 d=d-p0*1000000000 p6=trunc(d/1000000)/10 d=d-p6*10000000 if (t) p6=-p6 endif p7=trunc(d/10000)/10 d=d-p7*100000 p8=trunc(d/100)/100 d=d-p8*10000 p9=d/100 ; d=imag(p4) mi1=trunc(d) da=(d-mi1)*100000000000 d=trunc(da/10000000000) bt1=d>6 da=da-d*10000000000 d=d-5*bt1 dt1=d>1 iv1=d==3 fac1=trunc(da/100000) da=da-fac1*100000 bo1=(da/100000)/10 ; d=real(p5) mi2=trunc(d) da=(d-mi2)*100000000000 d=trunc(da/10000000000) bt2=d>6 da=da-d*10000000000 d=d-5*bt2 dt2=d>1 iv2=d==3 fac2=trunc(da/100000) da=da-fac2*100000 bo2=(da/100000)/10 ; d=imag(p5) mi3=trunc(d) da=(d-mi3)*100000000000 d=trunc(da/10000000000) bt3=d>6 da=da-d*10000000000 d=d-5*bt3 dt3=d>1 iv3=d==3 fac3=trunc(da/100000) da=da-fac3*100000 bo3=(da/100000)/10 ; if (d4) z=0 if (vb) if (mo) c=0.4*log(sqr(pixel^mm)) else c=pixel z=pixel endif else c=pixel z=pixel*sc endif else c=p1 z=pixel endif t=0 bo=|z| p=pp z0=p7 zold=(0.0,0.0) cb=p9 ba=bb0 : if (tt) t=t+1 if (ex0) ex0=t<mi1 if (bo>bs) u=2*(fn1(t/sfac)) ex0=0 if (ir0) t=0 endif if (d4) z=z*u if (mo) c=0.4*log(sqr(pixel^mm)) else c=pixel endif else z=pixel cb=p9*u c=p1*u p=pp*u endif tt=tt-1+ex0 endif elseif ((ex1)&&bo>bl) if (bo<bh) d3=dd1 ba=bb1 ab=ab1 ex1=0 tt=tt-1 if (w1) u=2*(fn2(t/ofac)) else u=1,0 endif if (ir1) t=0 endif if (d3) vb=v1j if (rs1) z=pixel cb=p9*u c=p1*u p=pp*u else c=p1 z=z*u cb=p9 endif else vb=v1m if (rs1) c=z*u z=pixel*(sc1+px1) z0=p7*u ph=ph*u else c=z z=z*u endif if (mo1) c=0.4*log(sqr(pixel^mm)) endif endif endif elseif (ex2) if (dt1) if (iv1) d=bo>bo1 else d=bo<bo1 endif if (bt1) d=d+(t>mi1) endif else d=t>mi1 endif if (d) ab=ab2 d3=dd2 ba=bb2 ex2=0 tt=tt-1 if (w2) u=2*(fn2(t/fac1)) else u=1,0 endif if (ir2) t=0 endif if (d3) vb=v2j if (rs2) z=pixel cb=p9*u c=p1*u p=pp*u else cb=p9 c=p1 z=z*u endif else vb=v2m if (rs2) c=z*u z=pixel*(sc2+px2) z0=p7*u ph=ph*u else c=z z=z*u endif if (mo2) c=0.4*log(sqr(pixel^mm)) endif endif endif elseif (ex3) if (dt2) if (iv2) d=bo>bo2 else d=bo<bo2 endif if (bt2) d=d+(t>mi2) endif else d=t>mi2 endif if (d) ab=ab3 d3=dd3 ba=bb3 ex3=0 tt=tt-1 if (w3) u=2*(fn3(t/fac2)) else u=1,0 endif if (ir3) t=0 endif if (d3) vb=v3j if (rs3) z=pixel cb=p9*u c=p1*u p=pp*u else cb=p9 c=p1 z=z*u endif else vb=v3m if (rs3) c=z*u z=pixel*(sc3+px3) z0=p7*u ph=ph*u else c=z z=z*u endif vb=v3m if (mo3) c=0.4*log(sqr(pixel^mm)) endif endif endif elseif (ex4) if (dt3) if (iv3) d=bo>bo3 else d=bo<bo3 endif if (bt3) d=d+(t>mi3) endif else d=t>mi3 endif if (d) ab=ab4 d3=dd4 ba=bb4 ex4=0 tt=0 if (w4) u=2*(fn4(t/fac3)) else u=1,0 endif if (d3) vb=v4j if (rs4) z=pixel cb=p9*u c=p1*u p=pp*u else cb=p9 c=p1 z=z*u endif else vb=v4m if (rs4) c=z*u z=pixel*(sc4+px4) z0=p7*u ph=ph*u else c=z z=z*u endif if (mo4) c=0.4*log(sqr(pixel^mm)) endif endif endif endif endif if (vb) if (d3) if (ab) z=z*z+c ;Spiderjul John Horner c=c*sp+z else z1=z^p-1 ;Qusinewton Pusk s Istv n z2=p*z*z z=z-z1/z2 endif elseif (ab) z1=z*z+ph-0.5*zold ;Phoenix Mike Wareman zold=z z=z1 else z2=fn1(z)+c ;Many_mods Linda Allison z1=cos(z2) z=c*(1-z1)/(1+z1) endif elseif (d3) if (ab) z2=z*z ;Julia Pusk s Istv n z=z2*z2+p6*z2+c-p0 else z1=fn1(z)-cb ;Bees Ray Girvan z2=z1^p8-1 z3=p8*(z1^(p8-1)) z=z-(z2/z3) endif elseif (ab) z2=z*z ;Mandel Pusk s Istv n z=z2*z2+p6*z2+c-p0 else z3=z ;Secant Mike Wareman z1=z0*z0*z0*z0-1 z2=z*z*z*z-1 z=z-z2*(z-z0)/(z2-z1) z0=z3 endif bo=|z| bo<ba } ----------------------------------start-par------------------------------- Phoenix { ; Time 2.56.13.76 ; Albrecht Niekamp Sep, 03 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=mfr.frm formulaname=multifractal_9 function=sin/exp/exp/sin passes=t center-mag=0.739239/0.000686676/19.29681/1/-90/-1.23373533611470521e-014 params=-0.05825983458967864/0.3785210730307932/21227.32595788025/15234.0\ 506030602/152048.0404808048/384.1604894048/384.00482000595/256.100768300\ 1/512.1007683001/768.1007683001 float=y maxiter=3072 inside=maxiter outside=tdis periodicity=0 rseed=-2436 colors=2000QY<10>09C08A068<2>022000211<21>icNkeOmfP<3>unTwpVxqWzsX<25>CB\ 7A96875<3>000400<23>o00q00s00<3>z00<25>C00A00800<3>000<11>LLLNNNPPPQQQSS\ SUUU<6>fffgggiiijjjlll<3>sss<25>BBB999777<2>222000012<24>0bo0cq0es<3>0kz\ <13>0R_ } ---------------------------------end----------------------------------------

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Re: [Fractint] FOTD 24-09-03 (Sir Fractal Galahad [7])
by Jim Muth 25 Sep '03

25 Sep '03

At 08:26 PM 9/24/03 -1000, David Jones wrote: >My wife got one informing her that our credit card had >been charged $284 for downloading child pornography, >and they needed our credit card info to reverse the >charge. . . . I received one of these child pornography letters about two weeks ago. I think it was from the same sender. It was immediately apparent that the letter was a scam. How did the sender know it was *my* card that had been charged without already having the number? And why was the letter not addressed to me personally? Also, the letter was in 'html' format, formatted to look like an advertisement. Very unconvincing for such a serious thing, yet these schemes must work or else so many people would not be trying them. BTW, I am thinking of re-posting some old FOTD's from the early days of 5 and 6 years ago to keep the lists active while I am catching up my other work. Does anyone on the list object to seeing some images and discussions that were originally posted in 1997 and 1998? Jim M.

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[Fractint] The Feigenbaum Constant and the Weak Nuclear Force etc. via Jerry
by mdoctorow＠comcast.net 25 Sep '03

25 Sep '03

>From Osher Doctorow Ph.D. Jerry Iuliano send me a new posting today by email on Brun's constant (B), the Omega-2 constant (OM) of elliptic curves, and the fine structure constant a (em). In my notation: 1) (B^12)/[100(OM^(OM))] = sqrt(1/a(em)) In his previous posting, he pointed out that the Weak Nuclear Force G(w) or Fermi-coupling charge = .000011664 is obtained from: 2) (emev/pmev)/(10F) = G(w) where F is the Feigenbaum constant and emev/pmev is the electron-proton energy ratio. He also pointed out in his previous posting that the strong force coupling constant a(s) is given by: 3) a(s) = (omega-2 ^ omega-2)/(.37)^2 if I haven't made any typos (one problem is that he omits parentheses fairly often) in multiplication-division. Since the weak nuclear force is related to radioactive decay, definitely a growth-expansion-contraction process (with the emphasis on contraction of material), while the strong nuclear force binds quarks including protons and enutrons together in opposition to the electromagnetic proton-proton repulsion, the appearance of the Feigenbaum constant is encouraging from my post on its appearance in Mandelbrot successive cardioid subcircle area ratios. The strong nuclear force turns out to be related to the electromagnetic force (both in terms of coupling constants a(s), a(em) respectively) by: 4) ((666/(omega-2^a(s)) - 1)/100 = a(em) I seem to have forgotten what the 666 represents. In any case, the fundamental forces of elementary particle physics appear to be closely related to fractal, chaos, and growth parameters. Osher Doctorow Ph.D.

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Call for FOTD fill-ins!
by Tony (Anthony) Hanmer 25 Sep '03

25 Sep '03

Greetings all, Jim Muth - sounds like you've really got your hands full for the next long while! I really apreciate your daily explorations into fractals - they've become something of a fix for me. But of course life throws the occasional fly into the ointment, beyond our control. I suggest that this may be a critical time for the Fractint forum, the expected month-long absence of FOTDs. Rather than it drying up almost entirely, how about all of the rest of us digging out OUR favourite fractals and posting them? Preferably new work, but not only. There are so many other fractal types which rarely get any action anymore. An impromptu competition? No limit to the number of entries. Here's one of mine, a variation on the famous Dragon Curve rendered as an IFS. The accompanying IFS entry must be cut and pasted into any existing .ifs file, and then the par will run. Dragon_Curve_Var01 { ; SOME good work takes a little time! ; Version 2002 Patchlevel 5 reset=2002 type=ifs ifsfile=dcs01.ifs ifs=dc022b6c passes=1 center-mag=0.370231/4.5196/0.203252 params=0 maxiter=50000 sound=off colors=KAw006<3>00800900A<9>K04M04O03<2>U01W00W00<\ 10>d00e00f00<14>u70v70w80x80z90<79>zm0zm0zn0<43>zx\ 0zy0zy0zy0zy0<68>zzz } DC022b6c {; Tony Hanmer 3/2002, Designed with Brazil: DC variation 0.3535534 0.3535534 0.3535534 0.3535534 -3.5251263 4.9362330 -0.1428571 0.4999245 -0.4999245 0.4999245 0.4999245 0.1356225 2.4994715 0.4901932 -0.4999245 -0.4999245 0.4999245 -0.4999245 4.2405285 7.4996225 0.4901932 0.001 0 0 0.5 5.586 2.284 0.0098068 0.5 0 0 0.001 -3 1.057 0.0098068 } _________________________________________________________________ Add photos to your e-mail with MSN 8. Get 2 months FREE*. http://join.msn.com/?page=features/featuredemail

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[Fractint] Deeper Signifance of Cardioids, Light, Springs, Electromagnetics
by mdoctorow＠comcast.net 24 Sep '03

24 Sep '03

>From Osher Doctorow Ph.D. Following up further on Jerry Iuliano's email (see my previous posting), we find that cardioids and their more general relatives epicycloids relate strongly to optics and curve minimization (and thereby to calculus of variations which in turn relates to general relativity and quantum gravity and many other branches of physics) via such things as caustics, diacaustics, the brachistochrone problem (most rapid travel of a point acted upon only by gravity between two different fixed points), light sources at infinity (Huygens), nephroids, etc. I will let readers who are interested look these up as keywords on the internet and simply mention that when a circle of radius b rolls around a fixed circle of radius a (without slipping), a point on the first circle traces out an epicycloid, and when a = b it traces out a cycloid. How this relates to growth and expansion-contraction could well be regarded as a fascinating open problem. I want to spend the rest of this posting dealing with a rather curious "counterexample" that seems to go against the general idea of growth equations for Rare Events, namely the electromechanical analogy. Basically, a simple system of masses or weights with springs and friction acted upon by a force (simple problems in mechanical vibrations) is completely analogous to a simple electrical system (simple circuit) with a resistor and a capacitor and inductor and source of voltage, with mass ("weight" near the Earth) corresponding to inductance L, damping or friction corresponding to resistance R, spring constant ("springiness" vs "stiffness") corresponding to 1/capacitance = 1/C, impressed force F corresponding to source of voltage E(t) that depends on time t. The equation for the circuit, with Q being electrical charge, is: 1) LDtt(Q) + RDt(Q) + (1/C)Q = E(t) were Dtt(Q) is the acceleration of charge with respect to time, Dt(Q) is the velocity or speed of charge (called the current) with respect to time. Solutions of differential equations like (1), called linear differential equations of the second order, are sums of real and complex exponentials, or in real terms simple sums of exponentials and trigonometric sine and cosine functions with possibly some simple multiplication of exponential and sine or cosine factors. Those are familiar from growth expansion-contraction problems tat I have discussed except for the trigonometric functions. If we write Dt(Q) as I, the current, then Dtt(Q) is the change of current, which we can write as Dt(I), so that if the (1/C)Q term is absent we have an equation of growth in I type. Even if the LDtt(Q) term is absent, we have an equation of growth in Q. What makes equation (1) NOT a growth equation in general is the case where all the terms are not 0 except possibly for E(t). Yet the special cases where one term or two are zero other than E(t) indicate that growth is not entirely unrelated to mechanical vibrations and electrical circuits. What relates growth to mechanical vibrations and electrical circuits at a deep level is WAVES and their simulation. In mechanics, these are Vibrations. In electricity, these are Electromagnetic Waves. Waves tend to go outward from a source, although electrical or electromagnetic circuits tend to "capture" this behavior so that they travel instead circularly or rectangularly or perpendicular to these directions. When growth is "captured", it stops being similar to expansion-contraction as my previous postings have described the latter two and instead either becomes fixed (unchanging) or keeps repeating or oscillating (up and down, etc.) although sometimes the oscillations die down if the voltage source E(t) dies down for example or alternatively is 0 to begin with. This is somewhat similar to the case when planets orbit around stars or the Sun as opposed to comets or asteroids which pass by and never return (although some do return in larger orbits). I pointed out in an earlier posting that growth-expansion-contraction is typical or radiation from a source, biology, the expansion of the universe, development of galaxies (spherical to elliptical, etc.), condensation as from water to ice or gases (water vapor) to liquids (water), consciousness- perception-human memory. But we see that although the early Universe was expanding in growth, later on parts of it became attracted in cyclical and orbiting or oscillating ways to other parts, and we call this the Matter- Dominated Era as opposed to the earlier Radiation-Dominated Era. Radiation still exists to considerable extent in the Matter-Dominated Era, but some of it is "captured" by Matter and near Matter. We see that in Black Holes including photospheres also. We see that in NASA's recent finding of heat-light-to-sound- wave conversion in the Perseus cluster of galaxies which retains instead of diffuses heat outside the cluster. Rare Events are least "oscillating" or repeating - intuitively, if they repeat exactly at fixed intervals, their probabilities could no longer be less than .05 because they are perfectly certain to occur at those intervals or points in time. One could extend probability to include rarity of occurrence even if it is deterministically predictable, but I think that would be a different type of situation and I don't deal with that in my present work. Fairly Frequent Events do, however, begin to have a closer relationship to determinstic predictability because of processes similar to such repeats or oscillations that return to their original states or positions in some way. This is an explanation of why Bayesian (Conditional) Probability-Statistics or BCP, which involves division of probabilities to indicate dependencies, has been roughly fairly successful in Mainstream Fairly Frequent Event Probability- Statistics but not in applications to Rare Events usually. Very Frequent Events, such as random bombardment by molecules on dust particles, curiously enough have an intermediate status between Rare and Fairly Frequent Events/Processes, because there is so much contact between matter and radiation that there is "over-kill" - you can never tell when something will bombard you, so to speak, and these events are called Independent. With Fairly Frequent Events, there is typically only dependence on the immediately preceding Event so to speak - Markov Chains perfectly illustrate that, and have resemblance to Independent Events in that Markov Chains depend on only one preceding Event whereas Independent Events depend on NO preceding events. Very Frequent Events depend on either ALL or INFINITELY MANY preceding events, so that while more "random" in this sense than Independent Events, they also have some more predictability than Markov Chains since more past things are being simultaneously taken into consideration. It is sometimes claimed that Markov Chains do take more than the immediately past Event into consideration, but they do that by one-step-at-a-time paths like checkers or chess or event ants touching one other ant at a time in sequence rather than being simultaneously influenced by more than the immediately past Event. So when Markov Chains are used in Very Rare Event situations, they tend to go astray. Osher Doctorow Ph.D.

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An Email From Jerry Iuliano
by mdoctorow＠comcast.net 24 Sep '03

24 Sep '03

>From Osher Doctorow Ph.D. Jerry Iuliano, whose email is JerryIuliano(a)aol.com, posted some very interesting comments to me today on the subject Omega-2 constant and TOE. Jerry is far more familiar with number theory and numerical constants than I am, and he is interested in (and has been pursuing for a long time) both a unified theory of fundamental forces in physics and Fermat's Last Theorem. At a number of points in my theorizing when I have touched on matters related to numerical constants of considerable importance, Jerry has emailed me with extremely valuable ideas, some of which I have pursued and some of which I wished that I had. As a Non-Mainstreamer myself for a long time, I can say with little doubt that Jerry is even more Non-Mainstream than I am in his particular fields - he isn't embarrassed at pointing out Ancient Egyptian and Biblical references to or occurrences of various number constants, while I tend to look over my shoulder in embarrassment at those points at least when discussing them on scientific-mathematical forums. Jerry's posting concerned the Omega-2 and Omega-One Constants and Transformations, but he mentioned their tie-in with Weierstrass Elliptic Functions which are involved in Fermat's Last Theorem's proof and which in turn tie in with Fibonacci numbers which are of considerable interest in plant growth and other contexts. I looked up Jerry's references to Wolfram and Eric Weisstein's World of Mathematics, and reviewed some characteristics of the Weierstrass Elliptic Function that now take on a new significance for me at least. Letting r(z) be the Weierstrass Elliptic Function in my terminology, one of the most remarkable equations is: 1) r'^2(z) = 4r^3(z) - g2r(z) - g3 where g2 and g3 are elliptic invariants which, in the equianharmonic case, are respectively 0 and 1. Readers will recognize the growth equations in (1) except that with r' as dr/dz, r'^2(z) is used instead of r'(z) as in typical growth equations and the cube of r is one degree higher than in most growth equations for the relevant function taking the place of r. The Weierstrass elliptic curve modular form is equivalent to the "golden number" of ellptic modular forms, this number being the Omega-One constant which is (1/2)w2(1 + isqrt(3)) where w2 is called the Omega-Two constant defined by the value 1.529954037... according to Jerry. If we follow up the Golden Mean or Golden Ratio Phi, for example in the Eric Weisstein source mentioned, it turns out to be the limit lim Fn/Fn-1 of successive Fibonacci numbers as n --> infinity, but the latter ratio Fn/Fn-1 also measures fractions of turns between succesive leaves on stalks of various plants in botany. Fn itself turns out to be of considerable interest in biological growth - for example, it is the number of pairs of rabbits n months after a single pair begins breeding if newly born bunnies are assumed to start breeding when 2 months old. Phi turns out to be the most irrational number because of its continued fraction representation entirely by ones. Osher Doctorow Ph.D.

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I'm back online -- FOTD coming soon
by Jim Muth 23 Sep '03

23 Sep '03

There is good news and bad news from F.C. The good news is that the power is back on and Fractal central survived the hurricane with nothing more than the expected power failure. The FOTD for September 24 will appear in a few hours. The bad news is that the work has fallen so far behind that I will need to suspend further FOTD's until I get caught up. More details in the upcoming FOTD. Jim M.

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[Fractint] Is the Cardioid-like Heart Surface related to Big Bangs and Black Holes?
by mdoctorow＠comcast.net 23 Sep '03

23 Sep '03

>From Osher Doctorow The 3-dimensional generalization of a cardioid is something like G. Taubin's (IBM Thomas J. Watson Center, Yorktown Heights, N.Y.) Heart Surface in his "An accurate algorithm for rasterizing algebraic curves and surfaces," IEEE Graphics Appl. 14, 14-23, 1994. Rasterizing is representing or converting graphical information into or with pixels. The equation of the Heart Surface, which looks like a 3-dimensional cardioid with a cusplike part of the surface on top (in the positive z direction), is the sextic equation: 1) [x^2 + ((3/2)y)^2 + z^2 - 1]^3 - x^2 z^3 - [(3/2)y]^2 z^3/20 = 0 If we use spherical coordinates after changing (3/2)y to a new y coordinate which could be labelled y' (not the derivative of y!), we get an expression for Dz(rho) where rho is the spherical radius coordinate whose square is x^2 + y'^2 + z^2 which resembles the last derivative for the cardioid in the previous posting. In other words, Dz(rho) is an expression with (rho - 1)^2 in the denominator. Mathematical physicists are immediately reminded of a black hole singularity in this context, especially with a chaotic Heart Surface. Although there were various theoretical reasons for various quantum theorists to consider that the Big Bang was not exactly a black hole, quantum theorists have not exactly kept up with analysis of Existence and Change in their own field, and if we add the fact that a similarity to a black hole doesn't imply that something is exactly a black hole, there is plenty of room for theorizing. A Big Bang and a Singularity are rather odd events even for Growth Events/Processes. Most Growth is not Big Bang type or Singularity type, and the fact that division is involved for cardioids and Heart Surfaces suggests that a major split has occurred between additive-subtractive Growth as in exponential and logistic and Riccati differential equations, on the one hand and chaotic-related Mandelbrot and Heart Surface growth with division, on the other hand. This division corresponds to a division between Rare Events and Fairly Frequent Events - Rare Events involve additive-subtractive Growth, Fairly Frequent Events involve division-multiplication in or out of Growth although division is primary. Very Frequent Events appear to be on the borderline between Rare and Fairly Frequent Events with regard to Growth. If you follow most of my postings to various cited places over the past few years, including some publications, you'll probably understand this most rapidly. Also look at researchmathematics(a)yahogroups.com. (Some of my postings have a remarkable tendency to disappear after several years - for example, I posted to anzap-l (an applied probability forum/mailing list in New Zealand-Australia) for several years, only to have the list owner close down the list with the rather cryptic remark which in essence amounted to the list not going in the directed that he had wanted (whatever that was - I'm still not clear about it). Superstringtheory.com is currently closed in the main forum on Strings/M Theory- Duality to which I had posted for quite a long time, apparently due to a deliberate virus by somebody.) Speaking of viruses, there is an indication that Rare Event Growth, which is the most important type of Growth except possibly for Singularities, is more Lifelike than Deathlike or Virus-like. A good model for Rare Event Change and Existence, in other words, is certain lifelike processes and events such as Knowledge (or Knowledge-Information), Perception, Memory, Consciousness. The other types of Change and Existence appear to bring in the Deathlike or Virus- like or Singularity-like regimes, and to be associated with events whose probabilities are greater than .05. The intuitive concept of an "over- bombarded" Existence with higher frequency Events/Processes in central to this. It isn't Knowledge that is the culprit at least in its simple expressions, but more like Complication and perhaps even Imitation and Mediocrity. Are Fractals and Chaos then signs of the breakdown of "beneficial" Change and Existence? We always assumed that Chaos at least involves the breakdown of predictability, but I emphasized the difference between Knowledge as Semantics or Meaning and "Information" as Syntactics or Form of messages. In some sense, becoming less predictable increases Information and Entropy but decreases Knowledge as Semantics or Meaning. To understand the Meaning of something is closely related to our ability to predict something. Rare Events seem to increase in Knowledge, whereas more common Events seem to increase in Entropy and Information. So in a deep sense, Existence and Change near the Big Bang involved a clash between Meaning and Form. The Mandelbrot and Julia sets demarcate the boundaries between the two pairs of clashing tendencies, Meaning vs Form, Existence vs Change. Existence and Change seem to be more compatible in Growth than Meaning and Form, Semantics and Syntactics. Osher Doctorow Ph.D.

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