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July 2005
- 13 participants
- 41 discussions
----- 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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