Computing Julia set of exp(z) to really high resolution
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From: mike3 on 4 Jan 2010 23:31 Hi. I was wondering: Is there any way at all to draw the Julia set for the complex exponential exp(z) to high resolution? Note that if one affixes a small finite "bailout", the resulting plot is very "bubby", and doesn't show much structure. The actual Julia set is a collection of infinitely thin hairs or something, I think. Of course one can't draw all that on a computer due to the monitor's resolution, but is there some method to get up to that limit at least? The trouble seems to be that direct iteration leads to very fast growth. Very fast growth = overflow. One idea I thought of was using a "power tower" to represent humongous numbers, but how the heck would you find sin (2^2^2^2^2^2) or something?! So that doesn't seem to be useful.
From: Philippe 92 on 5 Jan 2010 07:51 mike3 a �crit : > Hi. > > I was wondering: Is there any way at all to draw the Julia set for the > complex exponential exp(z) to high resolution? Hi, May be the "Fractint" program gives you hints about how to do ? <http://www.fractint.org/> It allows for tremendous zooming in ... (and has built in Julia sets BTW) Regards. -- Philippe C., mail : chephip, with domain free.fr site : http://mathafou.free.fr/ (mathematical recreations)
From: I.N. Galidakis on 5 Jan 2010 15:50 mike3 wrote: > Hi. > > I was wondering: Is there any way at all to draw the Julia set for the > complex exponential exp(z) to high resolution? Note that if one > affixes a small finite "bailout", the resulting plot is very "bubby", > and doesn't show much structure. The actual Julia set is a collection > of infinitely thin hairs or something, I think. Of course one can't > draw all that on a computer due to the monitor's resolution, but is > there some method to get up to that limit at least? The trouble seems > to be that direct iteration leads to very fast growth. Very fast > growth = overflow. One idea I thought of was using a "power tower" to > represent humongous numbers, but how the heck would you find sin > (2^2^2^2^2^2) or something?! So that doesn't seem to be useful. Didn't we have this discussion before? Do you remember what I told you? The "hairs" are a computer artifact of all "Cantor Bouquet" type fractals. They are infinitely thin, so they accumulate at certain points of the fractal and form "bubbles" or "fingers". You can only "resolve" them further by increasing the bailout value. Increasing the bailout value doesn't solve the resolution problem completely, however. The hairs will accumulate somewhere else, lower on the fractal, because in certain areas they are infinitely close to each other and eventually this will beat your resolution. Apropos of all the above, you really don't need very high bailout or iteration values to see the exp fractal. Here are some results with Maple for the following code: > EF:=proc(x,y) > local m,z; > z:=evalf(x+y*I); > m:=0; > to 100 while abs(z)<1e10 do #100 is the iterations, 1e10 is the bailout > z:=exp(z); > m:=m+1; > od; > m; > end: > plot3d(EF, -4..4, -4..4, grid=[300,300], shading=ZHUE,scaling=CONSTRAINED, style=PATCHNOGRID, orientation=[0,0], axes=BOX); Maple V rel 4: (iterations=100, bailout=1e10) http://misc.virtualcomposer2000.com/expMand100.gif (iterations=500, bailout=1e10) http://misc.virtualcomposer2000.com/expMand500.gif and the same code on Maple 9 with the largest bailout for conventional arithmetic: (iterations=100, bailout=1e18) http://misc.virtualcomposer2000.com/expMandLargeBail.gif Note how the hairs have receded somewhat more on the last fractal. That's about the best that can be done using 18 digits of precision. If your program allows for a larger number of digits, then the hairs will split/recede even further, but at some point they will still form unresolvable "fingers". -- Ioannis
From: Danny73 on 5 Jan 2010 17:20 On Jan 5, 3:50 pm, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > mike3 wrote: > > Hi. > > > I was wondering: Is there any way at all to draw the Julia set for the > > complex exponential exp(z) to high resolution? Note that if one > > affixes a small finite "bailout", the resulting plot is very "bubby", > > and doesn't show much structure. The actual Julia set is a collection > > of infinitely thin hairs or something, I think. Of course one can't > > draw all that on a computer due to the monitor's resolution, but is > > there some method to get up to that limit at least? The trouble seems > > to be that direct iteration leads to very fast growth. Very fast > > growth = overflow. One idea I thought of was using a "power tower" to > > represent humongous numbers, but how the heck would you find sin > > (2^2^2^2^2^2) or something?! So that doesn't seem to be useful. > > Didn't we have this discussion before? Do you remember what I told you? > > The "hairs" are a computer artifact of all "Cantor Bouquet" type fractals.. They > are infinitely thin, so they accumulate at certain points of the fractal and > form "bubbles" or "fingers". You can only "resolve" them further by increasing > the bailout value. > > Increasing the bailout value doesn't solve the resolution problem completely, > however. The hairs will accumulate somewhere else, lower on the fractal, because > in certain areas they are infinitely close to each other and eventually this > will beat your resolution. > > Apropos of all the above, you really don't need very high bailout or iteration > values to see the exp fractal. Here are some results with Maple for the > following code: > > > EF:=proc(x,y) > > local m,z; > > z:=evalf(x+y*I); > > m:=0; > > to 100 while abs(z)<1e10 do #100 is the iterations, 1e10 is the bailout > > z:=exp(z); > > m:=m+1; > > od; > > m; > > end: > > plot3d(EF, -4..4, -4..4, grid=[300,300], shading=ZHUE,scaling=CONSTRAINED, > > style=PATCHNOGRID, orientation=[0,0], axes=BOX); > > Maple V rel 4: > > (iterations=100, bailout=1e10)http://misc.virtualcomposer2000.com/expMand100.gif > > (iterations=500, bailout=1e10)http://misc.virtualcomposer2000.com/expMand500.gif > > and the same code on Maple 9 with the largest bailout for conventional > arithmetic: > > (iterations=100, bailout=1e18)http://misc.virtualcomposer2000.com/expMandLargeBail.gif > > Note how the hairs have receded somewhat more on the last fractal. That's about > the best that can be done using 18 digits of precision. If your program allows > for a larger number of digits, then the hairs will split/recede even further, > but at some point they will still form unresolvable "fingers". > -- > Ioannis Great graphics Ioannis. What is the final resolution for each side of the 8 length square of your last graphic? Is the unresolvable "fingers" because of pixel resolution or because of digit precision problems of both? Dan
From: bvcvideo on 5 Jan 2010 17:39
unfortunatley, due to the wide range of implimentations of the floatingpointware specification (IEEE-754/-854; the first is an article in an IEEE magazine), there is really no such thing as a canonical M-set, or Julia set; can you say, Psychedelic "Magnification?" thus: the usual mathematical term is Universe; contrary to teh extreme exegesis of the Copenhagenskool, there is only one (the rest is Solopsism, per David Deutsch's say-not .-) > I appreciate your help in this matter. thus: strictly untrue; Kepler et al were using the program of [Cardinal] Nicholas of Cusa; of course, since you are apparently British, you've probably been indoctrinated with the secular church of Newton, and/or the Harry Potter PS curriculum of the "Venetian Party" of England. also, I keep on referring to the 2.5-page article in *Math.Mag.* (MAA.org), that proves of the isometry of inductive & deductive proofs, also giving a formula to convert from one to the other. the Royal Society attack on Leibniz was political; he was actively being considered to be the PM, by Queen Anne. (deny that, if you care to .-) > With Gawd getting in the way it's small wonder the Xtian Romans > produced no mathematicians of the earlier Greek calibre. :-)- Hide quoted text - thus: doctor Einstein's essay seems quite confuzed about the electromegnetic properties of matter, but that was a while before our standard textbookoid concepts were put out from the Texas Schoolbook Suppository. thus: he is giving a lot of credit to Lorentz, who may be more responsible, after all, for the time-space crack-up than doctor Minkowski; can you say, Most useless formalism of Century 20.1? however, the real problem is your persistent use -- with whomever else from the past & future -- of the the concept of vacuum, as Pascal first thought of it, which is really, strictly relative or active (as in, That giant sucking sound, you hear, when you're trying to read this ****). > http://www-groups.dcs.st-and.ac.uk/~history/Extras/Einstein_ether.html --Brit's hate Shakespeare, Why? http://wlym.com/campaigner/8011.pdf --Madame Rice is a Riceist, How? http://larouchepub.com/other/2009/3650rice_racist.html --The Riemannian Space of the Nucleus, What? http://www.21stcenturysciencetech.com/Articles_2009/Relativistic_Moon... --In perpetuity clause in healthcare bill, Where? |