From: MX%"bengtmn@algonet.se" 18-NOV-1995 11:59:34.60 To: MX%"noel@erich.triumf.ca" CC: Subj: Fractal series 2 Return-Path: Received: from plato.algonet.se by Erich.Triumf.CA (MX V4.0-1 VAX) with SMTP; Sat, 18 Nov 1995 11:59:00 PST Received: from mail.algonet.se (bengtmn@sophocles.algonet.se [193.12.207.10]) by plato.algonet.se (8.6.12/hdw.1.0) with SMTP id UAA15902 for ; Sat, 18 Nov 1995 20:53:50 +0100 Date: Sat, 18 Nov 1995 20:53:50 +0100 Message-ID: <199511181953.UAA15902@plato.algonet.se> X-Sender: bengtmn@mail.algonet.se X-Mailer: Windows Eudora Version 1.4.4 MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=====================_816753391==_" To: NOEL_GIFFIN From: bengtmn@algonet.se (Bengt Mansson) Subject: Fractal series 2 X-Attachments: C:\WINCODE\ENCODE\TIP1-300.UUE;C:\WINCODE\ENCODE\TIP1-451.UUE;C:\WINCODE\ENCODE\TIP1-453.UUE;C:\WINCODE\ENCODE\TIP1-535.UUE;C:\WINCODE\ENCODE\TIP1-584.UUE;C:\WINCODE\ENCODE\TIP1-600.UUE;C:\WINCODE\ENCODE\TIP1-602.UUE; --=====================_816753391==_ Content-Type: text/plain; charset="us-ascii" Hi Noel, Here comes my second series of deepzoom images of the (standard) Mandelbrot set. The intention this time has been to obtain a picture of a mini-MS very near the tip of the antenna at c = -2. I attach the gif-pictures, uu-encoded, to this e-mail, and I will also send the final one to a.b.p.f., and a message to sci.fractals. The pictures were calculated by FRACTINT 19.2, using video mode SF6 for the preview format pictures, and SF5 for the full screen picture. The reason for this is that I wanted to have it before Christmas! All 7 pictures are centered around the same mini-MS near c = -2, and show increasing magnification. The first one is just included to indicate the area where the mini-MS under investigation is located. The digits after hyphen in the file names indicate the magnification, TIP1-602.GIF, being the final, and main, one. NOTE: "TIP1" in the file names ends in _digit 1_ (_not_ an "i"). ******************************************************** Below follows a description which you may edit and combine with the pictures the way you like, and include in "my" deep-zoom page, http://spanky.triumf.ca/www/fractint/bmansson.html The final image, TIP1-602.GIF should be made downloadable, I think. DESCRIPTION ########################################### The following seven deep-zoom images are centered around a very small mini-MS in the usual Mandelbrot set, located at a distance of approximately 8.6 * 10^(-306) from the tip of the main antenna, at -2. To find the mini-MS I made a zoom of 10^300 times around the tip at -2 by means of a .par-file, containing { center-mag=-2/0/1e300 }: TIP1-300.GIF Any (high) magnification around -2 gives essentially the same picture, but at the nodes between the colored bubbles small mini-MS:s can be expected. I centered a zoom-box around the leftmost node visible. After roughly one hundred zooms the following picture emerged at magnification 2.96*10^451, showing the first signs of a mini-MS. TIP1-451.GIF Then, for the next hundred zooms or so, an increasing number of radial, colored fields appeared: TIP1-453.GIF , TIP1-535.GIF , TIP1-584.GIF Magnifications in these pictures were 1.72*10^453, 2.28*10^535, and 6.94*10^584, respectively. I have chosen these three as typical examples of new features emerging, essentially the bifurcation of radial fields, and the appearance of circular fields mixed with the radial ones. The colors inside the sectors are, however, changed by color cycling. This mixed circular/radial structure is more clearly seen in this picture: TIP1-600.GIF (magnification 1.24*10^600) where one might also vaguely discern a mini-MS. Magnifying another hundred times I finally arrived at this, maybe the smallest mini-MS, ever seen: TIP1-602.GIF The magnification is 2.09*10^602 , maximum number of iterations 60000, and the calculation took 800 hours on my 90 MHz Pentium computer. Try downloading it, load it into Fractint, look at the parameters and cycle the colors! The (prolonged) circular structures are predominant, and the radial patterns much less distinct than in the earlier published 10^240 times magnifications. Also note the tendency to horizontal patterns just to the right of the center of the image. The picture has a lot of intricate details although it is not artistically very spectacular. I would say that it gives more mathematical ideas than most pictures at lower magnification, maybe because it looks in some sense simpler. One might wonder how much of the details are artifacts from the finite precision of computer calculations. A final note on the location of the mini-MS: The .gif-file shows a truncated value x-value for the center of the image, but saving the parameters into a parameter file, one obtains the corners with full precision. The .par-file is like this: tip1-602 { reset=1920 type=mandel passes=1 corners=-1.9999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999136474120322879759808924890901144822800533270087\ 018470523001410436373876931251560048947248635871465728551062680726512832\ 825326702863861049490116584050325725448630853614756604371799666130177696\ 899410739693986886100632128120562967427942954220226606132501167320298421\ 02443898286156029956025206658194075945853/-1.999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999913647412032287\ 975980892489090114482280053327008701847052300141043637387693125156004894\ 724863587146572855106268072651283282532670286386104949011658405032572544\ 863085361475660437179966613017769689941073969398688610063212812056296742\ 794295422022660613250116732029842102443898286156029956025206658194075933\ 094/-4.7918365e-603/4.7778769e-603 params=0/0 float=y maxiter=60000 inside=0 colors=000SEHSEE<3>SSE<3>ESE<3>ESS<2>EHSKKS<2>QKSSKSSKQSKOSKMSKK<2> SQKSS\ KQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS00G<3>G0G<3>G00<3>GG0<3>0G0<3>0GG <2>04G88\ G<2>E8GG8GG8EG8CG8AG88<2>GE8GG8EG8CG8AG88G8<2>8GE8GG8EG8CG8AGBB G<2>FBGGB\ GGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>BGFBGGBFGBDGBCG000<6>0000 0e0e00eee0\ 0e0eeL0eeeLLLLLzLzLLzzzLLzLzzzLzzz000555<3>HHHKKKOOOSSSWWW___ccchhhmmm ss\ szzz00z<3>z0z<3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zzV<3>VzV<3\ >Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz00S<3>S0S<3>S00<3>SS0<3\ >0S0<3>0SS<2>07SEES<3>SESSEOSEL } From this I calculated the x-coordinate of the midpoint of the final image. Approximately, x-center = -1.99999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999991364741203228797598089\ 248909011448228005332700870184705230014104363738769312\ 515600489472486358714657285510626807265128328253267028\ 638610494901165840503257254486308536147566043717996661\ 301776968994107396939868861006321281205629674279429542\ 202266061325011673202984210244389828615602995602520665\ 819407593 I truncated to 602 decimals, starting with 301 9:s. One unit in the last decimal corresponds very nearly to 500 pixels in the picture, so these all of these decimals are really required if you want to find this particular mini-MS. Changing the last decimal to 1 or 5 will move you completely out of the picture. This may give some feeling for how deeply we have zoomed into the Mandelbrot set! END OF DESCRIPTION ##################################### My next project will be a zoom into the main valley in the biggest mini-MS (at -1.75), to a distance of only 10^(-6) from the real axis. Cheers, Bengt. --=====================_816753391==_ Content-Type: text/plain; charset="us-ascii" Content-Disposition: attachment; filename="TIP1-300.UUE"