To test this, I made 100 zooms (made possible by Fractint's deep-zooming capabilities) with each magnification 2^2.5 times deeper than the previous zoom, rotating the zoom box by 45 degrees, and shifting the color palette by 2 colors each zoom. I then stitched the 100 gifs into an animated .fli file. The result was a animation sequence that was visually boring, but mathematically exciting. Once it got past the first couple of frames, the animation didn't do anything, with the exception of a few flickers in the thread part of the image. Each frame was almost a perfect match!
Since the color palette had to be rotated on each zoom, instead of using the command line or .par file interface, I used Fractint's autokey feature. The C file, frames45.c, generates an auto.key script to produce the whole sequence of frames via fractint's "g" command. After compiling frames45.c, at the command line prompt, just enter "frames45 > auto.key". Then I started fractint with
video=af3 float=y viewwindows=///128/128 passes=b maxiter=5000 autokey=play
and away it went. When you animate the sequence, you may have to slow it down a tad. Also, there's one frame that seems to be shifted a few pixels. I think it has something to do with how many digits are being used. The resulting animation is zoom45.flc.
Since the image rotates 45 degrees for every magnification factor of 2^2.5, I thought it might be interesting to do this 8 times to make a complete 360 degree rotation. The final image should look just like the first one.
So I made a second 100 frame animation flick starting with a magnification of 1e5 and an ending magnification of 1e5*2^20 (20 = 2.5 * 8, 8 = 360/45). I altered a couple of lines in Fractint so the colors would repeat every 16 colors as Fractint normally repeats every 15 colors in 16 color mode and repeats every 255 colors in 256 color mode. The result is a zoom that ends with an image that looks just like the one with which it started - AN ENDLESS ZOOM! (a 900 kbyte animation)
Since the palette didn't need to be altered, I used the command line interface to generate each frame. The C program, frames.c, was used to generate the batch file, using "frames > tmp.bat" at the command line prompt. I put the following in my sstools.ini and then ran tmp.bat.
batch=y video=f3 sound=off float=y passes=b viewwindows=///128/128 maxiter=1000 colors=000_HcaDhc9n<2>YGkD_HEdH76H<2>50GTWJ<2>YLZA7F<73>YQo000<75>IdU111\ <36>igu102<14>SAi000<13>5815917105A1<14>AJ3
When I stitched them together in an animation, the last frame looks just like the first frame. I think I didn't include the last frame in the animation so that it would look smoother as restarted the sequence. The animation is called endless.fli.
ago, I generated an image with center-mag=0/1/1e1500. I figured that I should be able to use my predictions mentioned above to produce an image that looked "just like it" but at a much shallower zoom. If you divide 1e1500 by (2^20)^248 you get 7784334.86397. The value 248 put the result in a good range for Fractint, 249 would have made it zoomed out too far. Sure enough, using center-mag=0/1/7784334.86397, the image was very very similar, but with a palette shifted 3968 (2*8*248) colors. Since fractint repeats every 255 colors, the colors in the new image were actually shifted 143 (3968 mod 255) colors. The two images are shown here side by side.
and are available for downloading individually as,
top1500s
and
topalike
.
My original reason for started studying this was to be able to accurately predict the location of a baby mset. The jury is still out on this. The ability to predict the next baby mset was only partially successful. Since the babies are only about 1/32 the size of the previous one, you have to zoom 2^5, not 2^2.5 to actually get to see the baby. The images from one zoom to the next are similar, but are not exact. The slight variations in the zooms are magnified to the point of rendering the prediction almost useless. Still work to be done here.