If it isn't one calamity happening to Dr. J, it is another, and if neither he will make something happen. Take last night as case in point. He survives a crash which destroys his Car-dioid landing him in the hospital. He gets out of there early the next day so he can watch the FractoBowl games from the front row. And last night I find out he disrupted the game (well actually the game was halted for a long boring iteration decision). Anyway, he made the outrageous claim that he could make borderline decisions about points being in or out of the FractoBall field in just one iteration.
Now if you have ever waited a seeming eternity for Fractint or any other fractoviewer to compute through the boundary of a solid component, well maybe you know how Dr. J and the other fans felt. But I have my doubts, after all, they have to live with this. Their whole world is, as nearly as I can tell, a construct of some mathematical exercise. Anyway, after the confrontation with officials which was broadcast for all to see and hear (or however information gets around in fractal space), Dr. J had to make good on his boast. And so he has, according to the last email. Wait a sec while I put this into my Fractint-o-Viewer Model 19.6 here.... Let's see the formula file should look like... WAIT a second. This formula is the one he was asking about just yesterday in email!
1) |2*sinh[(asinh(sqrt(-6.75)*c))/3]| <= 1
So that is how he does it. I bet he knew all along! Just simply evaluate with complex arithmetic in this function (Eqn 1), involving hyperbolic functions of the field coordinates. If this function evaluates to less than or equal to 1, the position is on the field. Even a real world person can see how this would really help. No long delays. Absolutely accurate judgment calls. Boy, we could use that in some of our sports. Dr. J wants to plant darker grass (grass? oh yes fractal grass) on the field. He will use this formula to quickly lay down the seeds. We should be able to see this in Figure 1.
Let's see now, to make a formula file here...
frm:Colorit-3f { ; (c) Jay Hill, 1998
; p1= light angle (cos a, sin a)
; use these, float=y inside=0 outside=real periodicity=0
iter=0, zc = 0, c = pixel, dz=1
; 23 is the color of the period 1 component.
z=23*(|(2*sinh(asinh(sqrt(-6.75)*c)/3))|<=1.0)
done=-(z>0) ; done if we know z is not 0
if(p1==0)
p1=1
endif
: ; initialization.
iter = iter+1 ; gotta count the iterations
dz=3*sqr(zc)*dz+1 ; derivative, dz/dc, a slope for shading
zc=zc*sqr(zc) + c ; standard MSet cubic iteration z=z^3+c
if(|zc| >= 1024) ; Bailout
z = z -8 + ((sin(2*pi*iter/256)*Real(p1*dz/zc))>0) + iter
done=-1 ; Set flag to force an exit.
endif
done >= 0 ; Continue if the flag >=0.
}
Oh yes, where was I... Right, let's see what the field looks like. Ah, it is one of the FractoBall fields on an island. How nice.
|
Figure 1. Dr. J 's idea of an ideal FractoBall field.
Stay healthy,
Jay
FractoBowl-field1 { ; (c) Jay Hill, 1998
; Dr. J's FractoBowl Field
reset=1960 type=formula formulaname=colorit-3f
center-mag=0/0/0.8522727/1/-90
params=-0.8/-0.6 maxiter=2560
float=y inside=0 outside=real periodicity=0
colors=UmA0Aw0Aw0Ar0Aw0Am09w0Ah09w0A\
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6wS5h`7wS6h`7wT6h`7wT6h`7wT6g_8wU6g_\
8wU7g_8wU7g_8xV7g_9xV7f_9xV8f_9xW8f_\
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RxtpVTSxuqVTSxusVTT
savename=FRAFIELD
}
