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Dr. J is feeling good tonight and is showing off pictures of his big Easter party. We can see the eggs lined up the hill, basking in the sun. As you can see, it is a childs dream. There is an unlimited number of eggs, one under each bush. If you look closely, you can see Dr. J in his white lab coat, high up on the antenna above the first Easter egg.
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Figure 1. Easter at Dr. J's Laboratory.
The formula uses the exp function to flatten out the cardioid boundary of the Mandelbrot set. The buds are placed in a line, the spacing is according to the interior angle. Then an inverse (1/x) function centered on the base of the period one component turns the whole thing inside out. This makes buds whose spacing was like 1/2, 1/3, 1/4, ... become 2, 3, 4, ... Exact tests are used to allow skipping expensive iterations in period 1, 2, and 3 components.
Stay healthy,
Jay
PS. When I was about 6 years old, my mom placed Easter eggs under bushes,
on rocks around the fish pond and in planter pots. The real prize was a large
chocolate egg. Then she allowed us kids out of the house to search for them.
We only found about two eggs! Then she saw the family dog, licking her chops.
She had followed my mom around, watching all. When mom went in to get us,
that was her cue!
frm:Stretch3j { ; Flattened cardioid by Jay R. Hill, 1998
; Classic Mandelbrot set fractal, transformed
; p1 = bailout
; use: maxiter=256 inside=249 outside=summ symmetry=xaxis periodicity=0
p = 0.5*exp(Pixel), c=p*(1-p), z = 0, done = 1, zc = 0. bailout=p1
s=|c|, t1=(256*s - 96)*s + 32*Real(c); period 1 cardioid test
t2=16*s + 32*Real(c) + 16 ; period 2 circle test
B=sqrt(-4*c-7), t3=|8+4*c*(1-B)|, t4=|8+4*c*(1+B)| ; period 3 components
test
z=z + 249*(t1<=3) + 250*(t2<=1) + 251*(t3<=1) + 252*(t4<=1) ;
set colors
if(z>0) ; periods 1, 2 and 3 have z>0.
done=-1 ; color is set for c in a known component, skip iterations
endif
: ; initialization.
zc=sqr(zc) + c ; standard MSet iteration
if (|zc| >= bailout)
done=-1 ; Set flag to force an exit.
endif
done >= 0 ; Continue if the flag >=0.
}
Stretch3-j00 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/0.25/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<78>0Cw0Cw0Cw0Cw0Cw<150>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj00
}
Stretch3-j01 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/0.91771/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<78>0Cw0Cw0Cw0Cw0Cw<150>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj01
}
Stretch3-j02 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/1.58542/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<78>0Cw0Cw0Cw0Cw0Cw<150>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj02
}
Stretch3-j03 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/2.25261/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<77>0Cw0Cw0Cw0Cw0Cw<151>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj03
}
Stretch3-j04 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/2.9198/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<77>0Cw0Cw0Cw0Cw0Cw<151>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj04
}
Stretch3-j05 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/3.58751/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<77>0Cw0Cw0Cw0Cw0Cw<151>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj05
}
Stretch3-j06 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/4.25522/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<76>0Cw0Cw0Cw0Cw0Cw<152>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj06
}
Stretch3-j07 { ; (C) Jay Hill, 1998
reset=1960 type=formula formulafile=stretch.par
formulaname=stretch3j center-mag=0.75/4.92241/4/1/-90 params=80/0
float=y maxiter=8000 inside=249 outside=summ invert=1/0/0
periodicity=0
colors=mmmww0mmmhmocmq<6>QmvNlwLlwJkw<76>0Cw0Cw0Cw0Cw0Cw<152>555mKAAqUwm\
cwm0www111000 cyclerange=1/248 savename=stretj07
}
