Triterions revisited
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\ xu7zz5zz4zz3<4>zj4zg4zc4<3>zS5zP5zL6zI6zE6wE6tD6<6>vAIv9Kv9M<3>x\ 7Sy7Uy6Wz6Yz5_<3>f5Ma5IW4E<2>_4H<3>P3DN2CK2A<3>805G0BO0G<3>g0Ul0\ Xq0`v0d<2>y0fz0gy0h<3>u1m600enG000t1ns1or2q<2>v2u<3>c5dZ6`U6X<2>\ F9LAAG7FO4KW0Qc<3>5Xc6Zc7`c7aa7c_7eY<3>FcgHciJbl<3>S_v<3>civflvi\ nvlqvotvsww<2>zzz<3>rwzovzhsw<3>Gch8_d8Xa<3>8JP8FL8CI<3>SCCXCAaB\ 9<3>uA2zA0xA0<12>SA1QA1NA1<3>D92<3>NN7PQ8SU9<3>`fDhrI<3>UnXRm_Nl\ cJkgFfpAaz<3>PJoSFlWAi_5fc0ce7_jFRoNIzb0<2>zz1zzC<2>zzz<3>jqdfo_\ blU<3>Mb8<6>JF8 } 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 colors=000XkG<3>hoAkp8nq6<2>wt1<6>Zf8Vd9SbA<3>DVF<4>CPCCNBCMA<2>\ BI9BH8BH8<3>AE79D6DE6<3>UI9YJAaLBeMCgODjQF<9>tgGuiGvjG<2>ypGzrFz\ rK<3>yufxukxvqwww<3>rhh`F0<2>ZD0YD0WC0<8>N50M40L30<2>I10I10J20<1\ 3>TA2TA2UB2UB2VC2<3>YE3<3>Q82O61M41<3>Q71R81S91fM5<3>iO5jP5kQ6<7\ >YE4WD3UB3S92Q82<3>I10931A00<3>VK0_P0dU0<3>zm0<3>zb0z_0zX0zU0<3>\ `F0UC0O80H40<2>A4773A73C<3>56K47M48O39Q2AS<8>9QaASbATc<3>E`hFbiG\ di<3>JklKmmJkj<8>BSOAQM9OJ<2>6GB5E86G9<9>EWHFYIGZJ<2>JdMKfNLhM<2\ >UkH } Bifrost { ; 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I get nothing but black screens for all these Triternions images! Is it just me? Lee Skinner
I found my problem. When I copied the formula in Netscape Mail (Windows 98) with <ctrl>C and pasted it into my formula file with <ctrl>V, the lines with exponents became: a = x2+2*y*v b = v2+2*x*y c = y2+2*x*v z=x2+y2+v2 The exponent operator was lost. x**2 became x2, etc. Lee Skinner
Russell, Thankyou for presenting the triternion M-set,J-set, concept here. This idea really spawns a lot of speculation on my part, eg. how far can the approach be generalized and still get recognizable sets, etc. It took me a little while to understand precisely what you were doing, so I'm glad you included the earlier "C6 group" frm-- your idea is nothing short of revolutionary. Kudos to you for sharing it with us. Initially, I saw that you were using the C6 group members as a basis for the 6-space in which you were doing the z->z^2+C iteration, but I didn't grasp why you were doing the particular mapping: on 6/26/05 10:06 PM, Russell Walsmith at russw@lycos.com wrote:
x1=a1+x, x2=a2-x y1=b1+y, y2=b2-y v1=c1+v, v2=c2-v
After contemplating your reduction of the original frm to the "Triternions" frm, though, it became clear to me that you did this particular mapping because those particular linear combinations of the C6 basis form an order 3 subspace which has closure under the "z^2+C" operation. That's just beautiful. Several questions evolve, upon which I hope you will enlighten me: (1) Can the method be extended to other groups (D3, for instance), possibly of order not equal to 6, or to generalized semigroups; (2) Are there other reductions than that particular set which maintain closure, and is there a systematic way to find them; (3) Is there a name for the target object in the 3-space (It isn't a subgroup, since the multiplication table in the reduced frm:
a = x^2+2*y*v b = v^2+2*x*y c = y^2+2*x*v doesn't have an identity element, but looks very similar to that of a cyclic group), and is there some algebra that lets us generate these objects from the parent group?
What I especially like about your concept is the following. In the past when we have attempted to look at different views of order>3 escapetime-generated fractal objects, we have always relied upon the "slice" method in some form-- 2D or 3D linear slices of the object embedded in the n-space. But this is a departure from that method; rather, you are looking at self consistent subobjects of the parent object which have order 2 or 3, something like an eigenvalue view. This is a new way of looking at these objects, and I wonder if it can lead to better understanding them. In particular, what are the rules for domain spaces and iteration operators that allow M-sets to exist at all? On the complex plane a little while ago Ray Filiatreault showed that other iteration functions than polynomials create M-sets topologically conformable to the z^2+C M-set, and it turns out that all that is required for this is a fairly broad set of conditions to be true of the surface limned by the iteration function. So what about this in the order 6 space? The same topological conditions can't be true, can they?
PPS: If anyone has a clue as to how the TMan may be rendered in 3D, lemme know...
Well, when JoTz does this I hope he will illustate the method of doing so and not just render an image with arcane methods. I do know a boilerplate method of rendering 3D objects in fractint, as you probably do since you suggested it earlier concerning the variation of {x,y,v}, but it is very inefficient. Briefly, (1) choose a directed vector (angle,angle) =D in polar coords( this allows all possible directions to be hypothetically viewed), (2) form arbitrary unit vectors E,F perpindicular to it, (3) go out a user chosen distance on D, establish an origin=O (4) iterate over the screen with O at the center and coords P=O+linear comb(E,F), (5) within each pixel iteration do an inner loop proceeding incrementally toward the origin adding small deltas in direction -D each time: do the iteration- if it fails to converge for every pixel in the range the pixel iteration bails out; as soon as it converges for any point the pixel is established as an inside point-- the number of inner loop iterations is the relative distance and can be communicated to the fractint rendering routine as a z-value = color in the right rendering method. Unforturately this may entail ca. one billion points= a lot of calculation time. Perhaps bifurcation of the segment traversed in the inner loop could be used to check many fewer points. More satisfactorily, JoTz will show us a better way to do this, for the general case. ======= Hiram Berry
participants (3)
-
bluemac -
Lee H. Skinner -
Russell Walsmith