----- Original Message -----

From: Russell Walsmith

To: Fractint and General Fractals Discussion

Sent: Thursday, October 12, 2006 13:33

Subject: [RE]Re: [Fractint] Neo T-set

Hi Gilles,

Re Ron Barnett's 'Neo-T_set2' formula: Yes, ingenious. There's more than one way to skin a fractal, as the old saying goes... At the moment, I prefer the real-number version, as it

1) Fits nicely into Gerald K. Dobiasovsky’s 'Rot3d_T-set' formula (which uses real numbers) and is also quite easy to tweak for certain effects. (See my reply to Jack Of Tradez, to post later).

2) Scales uniformly from 2D to infinite dimensions. In fact, we can extrapolate to state a general formula for an n-dimensional Mandelbrot. I.e., given a space defined by n orthogonal axes labeled A1, A2... An, we have

real_nD_Mset {
A1, A2... An = elements of n-dimensional pixel
a1 = b1 = c1... = n1 = 0:
a2 = a1^2 - b1^2... - n1^2 + A1
b2 = 2*a1*b1 + A2
c2 = 2*a1*c1 + A3
...
n2 = 2*a1*n1 + An
a1=a2, b1=b2, c1=c2... n1=n2
z = a1^2 + b1^2 + c1^2...+ n1^2
z < 2^n }

Based on this generalized structure, we can define subsets such as

real_2D_Mset {
ca = real(pixel), cb = imag(pixel)
a1 = b1 = 0:
a2 = a1^2 - b1^2 + ca
b2 = 2*a1*b1 + cb
a1 = a2, b1 = b2
z = a1^2 + b1^2
z < 4 }

real_3D_Mset {
ca = real(pixel), cb = imag(pixel), cc = p1
a1 = b1 = c1 = 0:
a2 = a1^2 - b1^2 + ca
b2 = 2*a1*b1 + cb
c2 = 2*c1*c2 + cc
a1 = a2, b1 = b2, c1 = c2
z = a1^2 + b1^2 + c1^2
z < 8 }

and so forth. In fact, 'real_3D_Mset' is a more appropriate name for this formula than 'Neo_T-set' because T-sets are based on the triternion number system which derives from the cyclic group C6.

At any rate, thanks for your interest. Have you rendered any fractals based on these 3D formulas? If so, why not post a link so that we may view them...?

Ciao, Russell


---------[ Received Mail Content ]----------
>Subject : Re: [Fractint] Neo T-set
>Date : Fri, 06 Oct 2006 06:17:31 -0400
>From : GillesNadeau
>To : Fractint and General Fractals Discussion
>
>Hello Russel,
>
>Here a version of your formula based on a model of quaternion by Ron Barnett. The conditions of bailouts are different but it is your differently written formula. I hope not to annoy you with that. Said me what think you.
>
>Regard,
>
>Gilles
>
>
>Neo-T_set2 {
>cz=pixel, cv=real(p1)
>z = v = 0
>:
>a = z*z - conj(v)*v
>b = z*v + conj(z)*v
>z = a +cz
>v = b +cv
>(|z|+|v|) < 8 }
>
>
> ----- Original Message -----
> From: Russell Walsmith
> To: Fractint and General Fractals Discussion
> Sent: Tuesday, October 03, 2006 23:11
> Subject: [Fractint] Neo T-set
>
>
> Fractal Folk,
>
> Pondering how quaternions (q = (n,i,j,k) could be represented by real numbers, I realized that when q is squared the noncommutative elements cancel out: e.g., ij + ji = k - k = 0. Therefore, letting X, Y, V, and W represent the axes corresponding to n, i, j,and k respectively, we can generate a 4D M-set by:
>
> 4D M-set {
> X=real(pixel), Y=imag(pixel), V=real(p1), W=imag(p1)
> x1 = y1 = v1 = w1 = 0:
> x2 = x1^2-y1^2-v1^2-w1^2 + X
> y2 = 2*x1*y1 + Y
> v2 = 2*x1*v1 + V
> w2 = 2*x1*w1 + W
> x1=x2, y1=y2, v1=v2, w1=w2
> z = x1^2+y1^2+v1^2+w1^2
> z < 16 }
>
> We recall that the M-set generates from real numbers by
>
> 2D M-set {
> X=real(pixel), Y=imag(pixel)
> x1 = y1 = 0:
> x2 = x1^2-y1^2 + X
> y2 = 2*x1*y1 + Y
> x1=x2, y1=y2
> z = x1^2+y1^2
> z < 16 }
>
> and interpolate to find
>
> Neo T-set {
> X=real(pixel), Y=imag(pixel), V=real(p1)
> x1 = y1 = v1 = 0:
> x2 = x1^2-y1^2-v1^2 + X
> y2 = 2*x1*y1 + Y
> v2 = 2*x1*v1 + V
> x1=x2, y1=y2, v1=v2
> z = x1^2+y1^2+v1^2
> z < 8 }
>
> Inserting this last formulation into Gerald D's marvelous 3D T-set formula gives the image at
>
> http://ixitol.com/NeoT-set.GIF
>
> This is evidently a solid of revolution, more or less the image I was expecting to see when we first got on to this "triternion" thing way back when.
>
> A bit rushed at the moment, I'll post some pars later.
>
> Ciao, Russell
>
>
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