September 2003 - Fractint - mailman.xmission.com

[Fractint] Has the California Gubernatorial Campaign Any Lessons for Fractint?
by mdoctorow＠comcast.net 28 Sep '03

28 Sep '03

>From Osher Doctorow Ph.D. A member of Fractint (whose name I can't reveal) has asked me whether the California Gubernatorial Recall Campaign has any lessons for Fractint. Here are a few. 1. In USA politics, it's apparently worse for an excellent actor to ask for questions in advance of a debate than for a Governor and Lieutenant Governor who are poor actors to put the State into catastrophic financial disaster. 2. Ethnic prejudice only works from majority to minority, not in the reverse direction if we are to believe the current Recall speeches and actions. 3. Indian gambling casinos in California have apparently decided to choose Bustamante with enormous donations because Spain originally conquered the Indians. 4. Kennedy relatives are only good if they move to the left; their right-sided profiles apparently don't stand up as well in the public image. 5. If you watch your numbers carefully, you too may invest in an Indian Gambling Casino and thereby become President (or at least Governor). Osher Doctorow

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A (Mathematical) Biology Archive at the Los Alamos arXiv
by mdoctorow＠comcast.net 28 Sep '03

28 Sep '03

>From Osher Doctorow Ph.D. The Los Alamos arXiv:gr-qc, arXiv:astro-ph, etc., are Archives on the internet of preprints in mathematics and physics mostly, but they have just added arXiv:q-bio.QM, a Mathematical Biology Archive that remarkably goes back from September of this year quite a few years backward. Two papers of this September (both September 16, 2003) might be of interest to members of Fractint: "Multi-Stability in Monotone Input/Output Systems" arXiv:q- bio.QM/0309002 and "Monotone control systems," same place/0309001, both by authors David Angeli of U. Florence, Italy, and Eduardo D. Sontag of Rutgers University. I will simply mention for now that monotonicity is a rather interesting property that seems to be related to growth-expansion-contraction rather than constancy or cyclic behavior as I've developed them in my postings and elsewhere in Rare Event Theory/Geometric Physics. The authors don't use any of my explicit ideas, but we seem to be going in somewhat similar directions in some ways. Osher Doctorow Ph.D.

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Still on IFS...
by Andrew Coppin 27 Sep '03

27 Sep '03

http://mathworld.wolfram.com/BarnsleysFern.html Does anyone know of a good interactive IFS editor? I found a copy of F-Design, but eeeek! That thing is SO not fun to use... Any URLs? Thanks. Andrew.

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Re: [Fractint] The Feigenbaum Constant and the Weak Nuclear Force
by JackOTradez 27 Sep '03

27 Sep '03

You say: " I seem to have forgotten what the 666 represents " Let me enlighten you. There is a number. It is the number of a man. He shall be the AntiChrist. Let all who have wisdom understand. His number shall be 666. Find him. Hunt him down. Kill him!

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Classic FOTD 09-04-98 (Pink Midget)
by Jim Muth 27 Sep '03

27 Sep '03

F.O.T.D., April 09, 1998 (Pink Midget) Fractal visionaries: I had planned on discussing in today's FOTD the ultimate ques- tion of life, and the way it relates to the realm of fractal geometry. But I just couldn't muster the proper degree of profundity that one must feel before tackling such a life- altering topic. So instead, I dredged up a picture of a baby Mandelbrot. The guilty formula is my MandelbrotMix4, which automatically calculates a critical point of the formula and sets that value as the starting point of Z. The actual equation being iterated is (4*Z^3)+(2*Z)+C. These values draw a small Mandelbrot set rotated 90 degrees, with a curious tail dropping down from the cleft of East (Elephant) Valley, which has now become a South Valley. The main Mandelbrot figure is fairly conventional, but the East Valley tail is filled with cut-off bits and pieces that demand further investigation. Only a slight magnification is needed to reveal the tell-tale patterns that lead to hidden midgets. At first glance, today's image appears to have come from a spiral in Elephant Valley itself. A quick examination, how- ever, will show that the pattern around the midget is more delicate, with more fine, hairline detail than that which appears around the midgets in Elephant Valley. Also, the features are broken rather than connected, as they are in the valley. I gave the image a pastel-like palette of colors, which empha- sizes the more delicate nature of this midget. The tendrils curving away from the central midget are themselves filled with midgets, but alas, these midgets are beyond the reach of conven- tional DP math. (Hint, hint . . . Fractint developers, wouldn't it be great if arbitrary precision math could be applied to formula type fractals?) Today's harmless little midget has been posted to the usual sites -- alt.fractals.pictures and alt.binaries.pictures.fractals It is also available at Paul Lee's web site at: http://home.att.net/~Paul.N.Lee/FotD/FotD.html and Scott's site at: http://sdboyd.dyndns.org/~sdboyd/fotd/index.html Tomorrow, I might gather my profundity and discuss the ultimate question of life, or then again, I might blather on about some- thing of little consequence, but whatever I do, I will have another fractal ready and waiting. Until then, take care, and the next time you see a fractal, view it with reverence, for that fractal might encode the answer to the riddle of the universe. Jim Muth jamth(a)mindspring.com START 19.6 PARAMETER-FORMULA FILE=========================== Pink_Midget { ; 7min on a P200, SF5 reset=1960 type=formula formulafile=apr-98.par formulaname=MandelbrotMix4 passes=1 function=ident center-mag=-0.00218445795113323/-0.2346673882601\ 9790/9.192243e+010/1/-160/1.5611329691467279e-005 params=4/3/2/1/0/0 float=y maxiter=6000 bailout=9 inside=0 logmap=123 colors=00016727929A2CB2EC2FD\ 3HF3IG3KH3LI3MJ4NL4OM4PN4QO4SP4SQ4SR5TT5TU5TV9VX\ DXZHZaL`cPbeUehYhlbkognrlqupsxstxutyuqvuottlrukq\ ujpvhnvgmwflxckxajyZhyXgzVfzUewUbuU`rUYoVVmUTjUQ\ gTNeTLbRI`OGYMDVJATH8QG5OF3PE4QD5SE6TF7VG8WH9YIA\ ZJB_KC_LD`PE_TFbVFcXFeYFe_FV`FVaFVbFVcFVcFTdEReD\ PfCOfBMhALjAJl9Hm8Fp7Dr6Cs6Au59w47w35w24v2KncQam\ VQxO9xOCvOEsNHqNJnNMlNPjNRgNUeMWbMZ`M`YMcWMfUMhR\ LkPLmMLpKFeOWV_bLia5obAobFocKpcOpdTpeYpebpfgpflq\ gpqguqhzqcvn_qjVmgRhdMdaI_YDWV9WSEXZI`dMdkQhqRco\ R_lSVjSRgTMeTHbUD`U8YU4WT3UT3TS3SS3RR3QR3PQ3OQ3N\ P2MP2LO2KO2JN2IN2HM2GM2FM2EN4FO6FO7FP9FQGFRIFSKF\ SMFTOFUQFVSFVUFWWFXXFYZFY`FZbF_dF`fF`hFajFblFcnF\ cqFbrJ`sO_tSYuXXv`VwdUxiSymUymVymXymZym`ylaylcyl\ eylgylhzljzllzlnzlozlqzkszkuzkvzkxzkuzlrzmoznlzo\ izofzpczq`zrXzsUztRzuOzvLzvIzwFzxCzyEzyGzyIzyKzy\ MzxOzxQzxSzxEzz0zz0zz1zz1zz } 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 19.6 PARAMETER-FORMULA FILE=============================

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Multifractal_9
by Albrecht.Niekamp＠t-online.de 27 Sep '03

27 Sep '03

Hi, have you ever seen a circular shaped Julia, even the spirals consist of circular structures ? Or an inverted Mandel, which means the "lake" in fact is outside ? 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------------------------------- Fullmoon { ; Time 0.29.11.30 ; Albrecht Niekamp Sep, 03 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=mfr.frm formulaname=multifractal_9 function=sin/cosxx/cos/exp passes=t center-mag=+0.50651979529167650/-0.61792919511027970/1102.367/1/142.2060\ 56268235727/8.22999403236879257e-006 params=-0.02237006744590594/-0.07473982970671712/20684.64575300005/11541\ .0608020602/80148.03072420481/17384.2204812048/384.00240000045/128.30076\ 80000451/256.300768002431/512.2007680082311 float=y maxiter=3072 inside=maxiter outside=tdis logmap=9 periodicity=0 rseed=-2436 colors=200034<23>0bo0cq0es<3>0kz<25>09C08A068<2>022000211<13>UQFWSGYTH<3\ >e`LgbMidN<3>qjRslSunT<2>zsX<25>CB7A96875<3>000400<23>o00q00s00<3>z00<25\ >C00A00800<3>000<23>eeegggiiijjjlllnnn<2>sss<25>BBB999777<2>222000012 } Inversion { ;Time 1.52.59.62 ;Albrecht Niekamp Sep, 03 reset=2002 type=formula formulafile=mfr.frm formulaname=multifractal_9 function=sin/exp/exp/sin passes=t center-mag=+0.79641132292408450/-0.00131969871981132/2017.308/1/-170.000\ 000000001194/-3.61272123328149064e-012 params=-0.05825983458967864/0.3785210730307932/21222.32595700021/25143.0\ 506030602/51024.02048000094/384.01048089094/384.00048000545/256.10076830\ 01/512.1007683001/768.1007683001 float=y maxiter=3072 inside=maxiter outside=tdis logmap=4 periodicity=0 rseed=-2436 colors=200444<19>bbbdddfffgggiiijjj<3>qqqsssqqq<24>BBB999777<2>222000012\ <24>0bo0cq0es<3>0kz<25>09C08A068<2>022000211<21>icNkeOmfP<3>unTwpVxqWzsX\ <25>CB7A96875<3>000400<23>o00q00s00<3>z00<25>C00A00800<3>000222 } Three_rivers { ; Time 0.57.54.48 ; Albrecht Niekamp Sep, 03 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=mfr.frm formulaname=multifractal_9 function=sin/cosxx/cos/exp passes=t center-mag=+0.51651583292708940/-0.60133383907525690/1107.555/1/142.2060\ 56268216969/8.2300112836941608e-006 params=-0.02122867519150365/-0.06936307870723595/20684.64575300005/11321\ .0608020602/80148.03072420481/17384.2204812048/384.00240000045/128.30076\ 80000451/256.300768002431/512.2007680082311 float=y maxiter=3072 inside=maxiter outside=tdis logmap=9 periodicity=0 rseed=-2436 colors=200068<21>0bo0cq0es<3>0kz<25>09C08A068<2>022000211<13>UQFWSGYTH<3\ >e`LgbMidN<3>qjRslSunT<2>zsX<25>CB7A96875<3>000400<23>o00q00s00<3>z00<25\ >C00A00800<3>000<23>eeegggiiijjjlllnnn<2>sss<25>BBB999777<2>222000012034\ 056 } Juliversum { ; Time 0.45.00.97 ; Albrecht Niekamp Sep, 03 ; Version 2002 Patchlevel 5 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=mfr.frm formulaname=multifractal_9 function=sin/cosxx/cos/exp passes=t center-mag=+0.50651979529167650/-0.61792919511027970/55.37774/1/142.2060\ 56268212706/8.23001490952990356e-006 params=-0.02122867519150365/-0.06936307870723595/20684.64575300005/11321\ .0608020602/80148.03072420481/17384.2204812048/384.00240000045/128.30076\ 80000451/256.300768002431/512.2007680082311 float=y maxiter=3072 inside=maxiter outside=tdis logmap=9 periodicity=0 rseed=-2436 colors=200034<23>0bo0cq0es<3>0kz<25>09C08A068<2>022000211<13>UQFWSGYTH<3\ >e`LgbMidN<3>qjRslSunT<2>zsX<25>CB7A96875<3>000400<23>o00q00s00<3>z00<25\ >C00A00800<3>000<25>iiijjjlll<3>sss<25>BBB999777<2>222000012 } Loop_the_loop { ; 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[Fractint] Summary of Cardioid-Cycloid-Epicycloid-Hypocycloid Relationships
by mdoctorow＠comcast.net 27 Sep '03

27 Sep '03

>From Osher Doctorow Ph.D. To summarize the relationships between/among cardioids, cycloids, epicycloids, and hypocycloids, after correcting my confusion of cardioid and cycloid, here is the story. All of these are curves generated by rolling circles. While cycloids are generated only by one rolling circle (being the slowly oscillating path traced out by a point on the circumference of a circle that rolls without slipping along a straight line in one direction, cardioids are special cases of epicycloids. Epicycloids are similarly generated by one circle rolling over a second fixed circle on the outside of the latter - the point of tangency (touching) traces out an epicycloid, and for epicycloids the rolling circle is smaller than or equal to the other circle in radius. When the radii are equal, it's called a cardioid. The hypocycloid is like the epicycloid but with the smaller circle rolling inside the fixed larger one. The cardioid has one cusp, the nephroid is like the cardioid except that it has 3 equally spaced cusps, and the ranunculoid is like the others but has 5 equally spaced cusps. The cycloid, epicycloid, and hypocycloid have the following respective equations: 1) x = a(t - sin(t)) y = a(1 - cos(t)) 2) x = (a + b)cos(t) - bcos[(a + b)t/b] y = (a + b)sin(t) - bsin[(a + b)t/b] 3) x = (a - b)cos(t) + bcos[(a - b)t/b] y = (a - b)sin(t) - bsin[(a - b)t/b] The cardioid is the second case when a = b, which results in: 4) x = 2acos(t) - acos(2t) y = 2asin(t) - asin(2t) Osher Doctorow Ph.D.

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Oops!
by mdoctorow＠comcast.net 27 Sep '03

27 Sep '03

>From Osher Doctorow Ph.D. I've confused the cardioid and the cycloid in my last posting and I may have done it previously too. The cardioid is: 1) r = a(1 - cos(u)) nsofar as it is represented by the Mandelbrot cardioids, while the cycloid is: 2) x = a(t - sin(t)) y = a(1 - cos(t)) Although y and r look similar, they're very different curves because of the x equation in (2). The same restrictions and considerations apply as I stated in my last posting since the forms of the equations are so similar. In general, one should be very careful with a closed curve like a cardioid or circle or ellipse or a closed surface like a sphere or ellipsoid in not regarding it as a growth-expansion-contraction object except in very special conditions or for very special reasons, because being closed (an object moving along it returns to its original position and repeats its path in finite time), its equation is cyclic unless it taken as purely a spatial equation - and in the latter case, an additional equation has to be hypothesized to give its time variation. Osher Doctorow Ph.D.

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Coordinate-Dependent Growth Difficulties
by mdoctorow＠comcast.net 27 Sep '03

27 Sep '03

>From Osher Doctorow Ph.D. The equations dy/dt = ky, dy/dt = ky(1 - y) or ky(k2 - y), dy/dz = f(z) + g(z)y + h(z)y^2 and the latter similarly for y^3 in an added term are the "standard" growth equations provided that z can be regarded as a non-oscillating function of time. If z were to oscillate so much that it cancelled out the growth efect of y or time, then it would defeat the intuitive idea of growth-expansion- contraction which is not rapidly oscillating (back-and-forth) increase and decrease although that could belong to some different theory. It is possible for things to expand during "youth" and contract during "old age", but oscillating quantities of usual trigonometric type (sin(t), cos(t), etc.) are generally found in science and mathematics to characterize rapid rather than long-term change at least as we intuitively understand that as human beings. Another type of problem, though sometimes interacting with the previous one, occurs when equations change very much during transformations to polar or spherical or cylindrical coordinates (other coordinates won't be considered here). In that case, as long as the above types of equations work in one coordinate system (Cartesian or rectangular being the most common one, which is the ordinary one used in Euclidean Space), that is fine. With "extended" growth equations that involve cardioids or equations of type dy/dt = k/y or dy/dt = sqrt(1 - y) or sqrt(y), this may fail because the main coordinate system may be polar or spherical or cylindrical in which trigonometric oscillating functions like sin(t), cos(t), etc., are the main type in these coordinates. For example, one of the simplest equations of a circle is in polar coordinates: 1) r = asin(u) where u is the polar angle. Unless we restrict u to be between 0 and 2pi, this oscillates too much. Even with such a restriction, there would have to a very good physical or biological and mathematical reason for making that restriction, and this is generally unlikely since there is so much difference between the restricted and the unrestricted cases. The cardioid which is so important to the Mandelbrot set suffers from this difficulty. In polar coordinates it is of form: 2) r = a(1 - cos(u)) In parametric coordinates, it has the form (with sine and cosine possibly changed to cosine and sine respectively, etc.): 3) x = a(t - sin(t)) y = a(1 - cos(t)) which repeats its graph every 2api intervals from 0 onward on the x axis. In my opinion, we are justified in restricting the interval of t or u to be only one of those intervals, which eliminates the oscillation, but only because the Mandelbrot set is critical to fractals and chaos. Even here, there is a question of whether u or t is taken to be time or not. If it is time, then we are asserting that the physical object - for example, the Universe as a whole - is drawing out a cardioid in spacetime, not just in space. This may or may not be what we want to theorize. If u or t is NOT time but some spatial angle only, then we are not even talking about growth as expansion-contraction in time. We would then have to separately hypothesize how the Mandelbrot set or its cardioids change in time, and this is an entirely different question. Osher Doctorow Ph.D.

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Classic FOTD 27-08-97 (S_Curve)
by Jim Muth 27 Sep '03

27 Sep '03

F.O.T.D., August 27, 1997 (s_curve) Fractal visionaries: (Originally posted on the Fractal Art list.) While the debate about whether to organize or not to organize grows in intensity, I stand on the sidelines, contentedly continuing to search for fractal images. I am pretty much neutral on the idea of a fractal artists' guild. Belonging to such a group would give a certain legitimacy to our fractal art field, but I doubt if such a group would suddenly open the general public's eyes to the wonderful things that numbers can do. All too many people disengage their brains when they encounter art. They judge a painting's worth by how photographically realistic it is, a novel by how sensational it is, and a piece of music by the immediate catchiness of the major scale melody or else by the volume of the percussion. This natural human aversion to intellectual effort is in my opinion the chief impediment to the wide acceptance of fractal art, and indeed, to the acceptance of all abstract art. I have been a member of a similar group, the Graphic Artists' Guild, for several years. And to be honest, the greatest effect the guild has had in my life to this point has been when I wrote the checks for the membership fee. If a fractal guild were formed, I would consider the membership fee and what the group might accomplish, before making my decision about joining. My fractal today, as promised yesterday, is a figure I discovered in the Julia plane of the (-Z)^1.007 mandeloid. It's incredible that an exponent so close to unity could produce an image with as much detail as in today's fractal. But here it is, in all its spiral glory. The image will soon be available at: http://home.att.net/~Paul.N.Lee/FotD/FotD.html and at: http://sdboyd.dyndns.org/~sdboyd/fotd/index.html Tomorrow will be my last FOTD until I return from a well- deserved vacation on September 8. I will try to make it a good one, but my good intentions sometimes are not fulfilled. So check then to see what fractal turns up. Jim Muth jamth(a)mindspring.com START 19.6 FILE============================================= s_curve { ; 8 minutes on a P200 SF5 reset=1960 type=formula formulafile=jim.frm formulaname=JulibrotInvZ passes=1 center-mag=0.986183/0.601218/0.7013364/0.7171/90 params=1.007/0/2/1.3/0/0 float=y maxiter=10000 bailout=100 inside=0 logmap=yes symmetry=xaxis periodicity=10 colors=000JTqBFC<12>B32<30>akealf\ `kZ_jRZjJ<42>g`ug`ufat<34>TfnTfnUgm<41>qnm<7>19t\ <28>tP9<2>2vc<27>zna<7>IXw<4>JUr } frm:JulibrotInvZ {; Jim Muth ; Draws oblique slices of order n inverse Julibrot Figure ; P1 defines power of Z, P2 determines center of slice ; P3 determines angle of slice z=pixel, c=p2+(pixel*p3): z=(-z)^p1+c, |z| <= 100 } END 19.6 FILE===============================================

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