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July 2005

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Re: [Fractint] Triterions revisited
by Russell Walsmith 02 Jul '05

02 Jul '05

----- Original Message ----- From: bluemac <burningb(a)burningbridges.com> To: "Fractint and General Fractals Discussion" <fractint(a)mailman.xmission.com> Subject: Re: [Fractint] Triterions revisited Date: Thu, 30 Jun 2005 19:47:14 -0400 Hiram, Thank you for your interest in and enthusiasm for the 'triternions' concept. The search for these began some years ago when I perceived a "gap" between the ordered pairs called complex numbers and the systems of ordered 4-tuples called quaternions and hypercomplex numbers. I've also played around with 3D vectors, and perhaps these will prove useful for fractal generation as well, but I presently find the properties of mathematical groups to be most fascinating. My efforts so far are clearly just a bare beginning, and it seems certain that a lot of room remains for discovery and development here. It will be interesting to see where it goes. Other comments are inter-dispersed below... ----- Original Message ----- From: bluemac <burningb(a)burningbridges.com> To: "Fractint and General Fractals Discussion" <fractint(a)mailman.xmission.com> Subject: Re: [Fractint] Triterions revisited Date: Thu, 30 Jun 2005 19:47:14 -0400 > > Russell, > > Thank you 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(a)lycos.com wrote: > > x1=a1+x, x2=a2-x > > y1=b1+y, y2=b2-y > > v1=c1+v, v2=c2-v OK, therein lies a tale, and there are about a hundred ways to tell it. I'll try to present an outline here, and if you are still interested I can send you a paper, written back when, that really gets into the minutiae. First, for convenience, let's set a=1, b=-1, c=i and d=-i. Then the complex numbers are arrayed in the table below. |a b c d ________ a|a b c d b|b a d c c|c d b a d|d c a b In examining this table, we see that there are four ways to generate each element; e.g., a=a^2=b^2=c*d=d*c. So it occurred to me to incorporate this table into a formula like so: C4 { x=real(pixel), y=imag(pixel) a=b=c=d=0: x1=a^2+b^2+2*c*d x2=c^2+d^2+2*a*b y1=2*a*c+2*b*d y1=2*a*d+2*b*c a=x1+x, b=x2-x, c=y1+y, d=y2-y z=(a-b)^2+(c-d)^2 z < 1000 } It worked! This gives the M-set, and among the many other experiments that this inspired, I thought to put Klein's four-group into a formula as well. The group table for "K4" is: |a b c d ________ a|a b c d b|b a d c c|c d a b d|d c b a Hence, K4 { x=real(pixel), y=imag(pixel) a=b=c=d=0: x1=a^2+b^2+2*c*d x2=2*a*b+2*c*d y1=2*a*c+2*b*d y1=2*a*d+2*b*c a=x1+x, b=x2-x, c=y1+y, d=y2-y z=(a-b)^2+(c-d)^2 z < 1000 } The K4 fractal is just a square, but it's proof of concept so far. In an effort to eliminate the obvious redundancy in C4, I tried this revised version: C4r { x=real(pixel), y=imag(pixel) a=b=0: x1=a^2-b^2 y1=2*a*b a=x1+x, b=y1+x z=a^2+b^2 z < 1000 } The M-set again. There are, moreover, other interesting things about C4r; e.g., the variant x1=a^2+b^2 gives the K4 fractal, and b=x1+x, a=y1+x gives an unique object that is a sort of complement to the M-set (the C-set?). The C-set is also given (in reverse orientation) by C-set { z=0, c=pixel: z=conj(z^2)+c z < 1000 } > 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), Yes, the D3 table too can be fashioned into a formula, but since it's non-abelian, it seems unlikely it will reduce. D3 { x=real(pixel), y=imag(pixel), v=p1 x1=x2=y1=y2=v1=v2=0: a1 = x1^2+x2^2+y1^2+y2^2+2*v1*v2 a2 = 2*x1*x2+y1*v2+y2*v1+v1*y1+v2*y2 b1 = 2*x1*y1+x2*v1+y2*v2+v1*y2+v2*x2 b2 = 2*x1*y2+x2*v2+y1*v1+v1*x2+v2*y1 c1 = v2^2+2*x1*v1+x2*y1+y1*y2+y2*x2 c2 = v1^2+2*x1*v2+x2*y2+y1*x2+y2*y1 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 } > possibly of order not equal to 6, or to generalized semigroups; Any cyclic group of order C2j, where j is odd, seems reducible; e.g., C10 { c1=real(pixel), c2=imag(pixel) c3=p1, c4=p2, c5=p3 u=v=w=x=y=0: a=u^2+2*v*y+2*w*x b=x^2+2*u*v+2*w*y c=v^2+2*u*w+2*x*y d=y^2+2*u*x+2*v*w e=w^2+2*u*y+2*v*x u=a+c1, v=b+c2, w=c+c3 x=d+c4, y=e+c5 z=u^2+v^2+w^2+x^2+y^2 z < 1000 } The basic idea is that a cyclic group of odd order j has an identity element (I) and j-1 elements (e) of degree j (i.e., e^j=I). If each element is also given a negative sign, then we have in toto a group C2j. Thus, opposing those elements of degree j across the origin from elements of degree 2j is the same as ruling each axis with positive and negative numbers. But you don't necessarily have to do that; e.g., here is C6 with degree 6 elements on the same axis: C6v {;degree 6 elements are opposed on same axis x=real(pixel), y=imag(pixel), v=p1 x1=x2=y1=y2=v1=v2=0: a1 = x1^2+x2^2+2*y1*y2+2*v1*v2 a2 = 2*x1*x2+2*y1*v2+2*y2*v1 b1 = 2*x1*y1+2*x2*v1+ y2^2+ v2^2 b2 = 2*x1*y2+2*x2*v2+ y1^2+ v1^2 c1 = 2*x1*v1+2*x2*y1+2*y2*v2 c2 = 2*x1*v2+2*x2*y2+2*y1*v1 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 } > (2) Are > there other reductions than that particular set which maintain closure, and > is there a systematic way to find them; See above for C2j; but even-order C-groups and the non-cyclics are another matter... > (3) Is there a name for the target object in the 3-space Not yet, that I know of, but I sometimes call it the T-Man or the T-set. > (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? Actually there is an identity element, but it gets disguised in this context. Think of a,b,c above as a special case of multiplying ordered triplets with the rule that (a,b,c)X(x,y,z) = (ax+bz+cy,ay+bx+cz,az+cx+by), so the identity element would be (1,0,0). > 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. There are several definitions of 3D rendering in the Fractint context, and I think that what Jotz refers to is the technique where a point on a 2D fractal takes a Z-axis value according to a criterion such as escape time. It would be most interesting to see if the method that you've outlined above could provide us another 'slant' on the T-set. Ciao, Russell

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