In Dr. J's laboratory, as we saw last night, are what he calls experiments. These look to me like apparatus of some kind, all lined in long rows which seem to stretch off into infinity. I asked if we could see them all.
He replied "Sure, one at a time or all at once?"
"Well we don't have time to see them one at a time, so how about all at once? How do we do that?" I asked, figuring to stump him this time.
"You can't!" He replied, "Not really, but I can give you a good sampling. Let me get out my FractoStroboScope."
Have you ever seen the old movies where the wagon wheels appear to rotate backwards? It is the result of the movie frames (24 per second, as I recall) taking a picture (a sample) at a rate slightly faster than the rotation rate of the wheel (actually there is a factor of the number of wheel spokes in there somewhere). So the wheel does not quite get around far enough to look right. It seems to be backing up while the wagon goes forward.
In the laboratory, the apparatus thingys are all lined up and nearly evenly space, about 2/pi apart. In tonight's picture, taken with Dr. J's FractoStroboScope, we see one scan line from each of the apparatus setups. The top line of the image is the most distant. The reason this works is that each apparatus differs by a small amount from the next. And with each line we scan a slightly displaced part of the adjacent apparatus. The final result is an image with properties of the close ones at the bottom and the far ones at the top. Put the par file (see below) into your FractintoScope Mod 19.6 and you can scroll upwards and see additional parts even further out.
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Figure 1. Dr. J's experiment.
Tonight's formula file uses exponential and inverse transformations such that the cardioid and circle are unrolled and stretch out into a line. With image cropping, only a small part of the cardioid and circle components gets into the image. The portions which do appear in the image are quickly disposed of using exact tests to determine if pixels are in the cardioid or the circle.
We shall be taking other close looks at the experiments in Dr. J's laboratory later. So until then...
Stay warm,
Jay
frm:Stretch1 { ; Flattened cardioid by Jay R. Hill, 1998
; Classic Mandelbrot set fractal, transformed
p = 0.5*exp(Pixel), c=p*(1-p), z = 0, done = 1, zc = 0
s=|c|, t1=(256*s - 96)*s + 32*Real(c), t2=16*s + 32*Real(c) + 16
z=z + 250*(t1<=3) + 251*(t2<=1) ; set colors
if(z>0) ; for periods 1 and 2.
done=-1 ; color is set for c in a component, skip iterations
endif
: ; initialization.
zc=sqr(zc) + c ; standard MSet iteration
if (|zc| >= 4) ; Bailout at 4
done=-1 ; Set flag to force an exit.
endif
done >= 0 ; Continue if the flag >=0.
}
Stretch800x600 { ; Jay Hill, 1998
; use 800x600, about 2 hours P200
reset=1960 type=formula formulafile=color101.par
formulaname=Stretch1 corners=0/2.77/0/382.9268 float=y
maxiter=100000 inside=0 outside=summ invert=1/0/-3.14159265359
periodicity=0
colors=wKA000<76>00v00w00w01w<18>0Fw\
0Gw0Hw0Iw0Iw0Jw<52>0vw0ww0ww1ww<77>m\
mw<9>555<5>000
savename=Stretc1
}
Stretch640x480 { ; Jay Hill, 1998
; use 640x480, takes an hour on P200
reset=1960 type=formula formulafile=color101.par
formulaname=Stretch1 corners=0/2.77/0/306.5324 float=y
maxiter=100000 inside=0 outside=summ invert=1/0/-3.14159265359
periodicity=0
colors=wKA000<76>00v00w00w01w<18>0Fw\
0Gw0Hw0Iw0Iw0Jw<52>0vw0ww0ww1ww<77>m\
mw<9>555<5>000
savename=Stretch
}
