Well now, I guess it's going on toward three years since I first
showed up on this list with the "triternion" concept. The basic
idea was that the cyclical group C6 could be configured as a number
system based on ordered triplets and then used to generate 3D
fractals. I was sure it would work, although I had nothing to show
for it at the time, save for a vague formula that generated a
nondescript, amorphous blob.
Luckily though, Morgan Owens had also contemplated this idea, and
he quickly published a formula that generated a fractal even
uglier, but yet more complicated, than mine. Then Jim Muth got to
tweaking the formulas, creating variants that at last produced some
worthwhile images. Lee Skinner took it from there, and came up with
a remarkable series of fractals based on Muth's variations, and Jack O'Tradez
was inspired by that to produce some compelling animations. There
were others, as I recall, who threw in on this; e.g., Gerald K.
Dobiasovsky, for one, had some helpful comments and
clarifications...
The upshot is that there was a wave of creative energy flowing
through the list that I was able to surf ashore. That is, I managed
to develop a template whereby any finite group, once set up into
its table form, could be imported directly into a .frm file. The C6
group appears in the example immediately below: For p1=0 and
floating point on, it gives a view of an object I call the TMan, as
p1 varies from about -0.6 to 0.3.
C6 group{
x=real(pixel), y=imag(pixel), v=p1
x1=x2=y1=y2=v1=v2=0:
a1 = x1^2+ x2^2+2*y1*v1+2*y2*v2
a2 = 2*x1*x2+2*y1*v2+2*y2*v1
b1 = 2*x1*y1+2*x2*y2+v1^2+v2^2
b2 = 2*x1*y2+2*x2*y1+2*v1*v2
c1 = 2*x1*v1+2*x2*v2+y1^2+y2^2
c2 = 2*x1*v2+2*x2*v1+2*y1*y2
x1=a1+x, x2=a2-x
y1=b1+y, y2=b2-y
v1=c1+v, v2=c2-v
z = (x1-x2)^2 + (y1-y2)^2+ (v1-v2)^2
z < 1000 }
(Where y=real(pixel), v=imag(pixel), x=p1 will change the view to
another axis.) This was all well and good, even if it was not at
all the form that I had expected the formula to take. Moreover, no
one could figure out to view this object in 3D, and so after a
while, interest waned. I played around with it intermittently, and
found some neat
variants, but just the other night the light bulb came on again.
I.e., I finally saw that while the multiplication table I had tried
to construct originally presents complications, to square a
triternion is a simple, special case. Thus I was able to
reconfigure the formula to
Triternions{
c1=real(pixel), c2=imag(pixel), c3=p1
x=y=v=0:
a = x^2+2*y*v
b = v^2+2*x*y
c = y^2+2*x*v
x=a+c1, y=b+c2, v=c+c3
z=x^2+y^2+v^2
z < 1000 }
As far as I have determined, save for differences in scale, these
two formulas give identical results. After more investigating, I
noticed that the orbits screen gives only horizontal lines. Rather
odd... anyway, I tried the fmod options, inside and out, and the
results are (IMO) spectacular. So I'm sending along some par files
that maybe list members will enjoy.
PS: Thanks to Tim Wegner for directing me to this list in the first
place, and Jonathon Osuch and others who maintian Fractint.
PPS: If anyone has a clue as to how the TMan may be rendered in 3D,
lemme know...
Ciao, Russell
--------------------------Start par files--------------------------
AsgardsGate { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=-0.250474/-0.242195/3.917342/1/-135.000000000000455/-\
6.23001650268406593e-013 params=0.5/0/10000/0 float=y
inside=fmod proximity=0.25 outside=fmod
colors=000JC8I88H44G00<3>TA5WD6ZF7aI8eLAhMBiODjQF<3>qeBriAtm9vq8\
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}
TheGrotto { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=-0.0578437/-0.374326/11.30119/1/-134.999999999911097/\
-6.50101500410116273e-011 params=0.6/0/10000/0 float=y
maxiter=3000 inside=0 outside=fmod
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}
Bifrost { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=0.201211/-0.361509/6.91239/1/124.999999999998693/5.85\
607951020250539e-013 params=0/0/10000/0 float=y maxiter=1500
inside=fmod outside=fmod
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}
Kawasaki2010 { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=-0.545464/0.141325/1.384055/1/-135.000000000000853/-3\
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}
RainbowBridge { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=-0.155732/-0.670383/32.88762/1/152.499999999999176/5.\
34718103128994926e-013 params=0/0/10000/0 float=y
maxiter=1500 inside=fmod outside=fmod
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}
Souvenir { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=-0.99377796438866060/+0.27291225592671910/124026.5/1/\
-137.499999998789548/9.3960968033934833e-009
params=0.4/0/10000/0 float=y maxiter=10000 inside=0
outside=fmod
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}
BurningMan { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=Triternions.frm
formulaname=9aaa passes=3
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}
WUrldsUnseen { ; Version 2002 Patchlevel 5
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=dcls10.frm
formulaname=Triternions
center-mag=+0.21680803147823950/-0.39441732514254010/101.1767/1/\
124.999999999998792/4.20219414820621751e-013
params=0/0/10000/0 float=y maxiter=1500 inside=fmod
outside=fmod
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}
frm:Triternions{
c1=real(pixel), c2=imag(pixel), c3=p1
x=y=v=0:
a = x^2+2*y*v
b = v^2+2*x*y
c = y^2+2*x*v
x=a+c1, y=b+c2, v=c+c3
z=x^2+y^2+v^2
z < 1000 }
----------------------------End par files--------------------------
