Hi Gilles,
<br>
<br>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
<br>
<br>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). 
<br>
<br>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
<br>
<br>real_nD_Mset {
<br>A1, A2... An = elements of n-dimensional pixel
<br>a1 = b1 = c1... = n1 = 0:
<br>  a2 = a1^2 - b1^2... - n1^2 + A1
<br>  b2 = 2*a1*b1 + A2
<br>  c2 = 2*a1*c1 + A3
<br>  ...
<br>  n2 = 2*a1*n1 + An
<br> a1=a2, b1=b2, c1=c2... n1=n2
<br>z = a1^2 + b1^2 + c1^2...+ n1^2
<br>z < 2^n }
<br>
<br>Based on this generalized structure, we can define subsets such as
<br>
<br>real_2D_Mset {
<br>ca = real(pixel), cb = imag(pixel)
<br>a1 = b1 = 0:
<br>  a2 = a1^2 - b1^2 + ca
<br>  b2 = 2*a1*b1 + cb
<br> a1 = a2, b1 = b2
<br>z = a1^2 + b1^2
<br>z < 4 }
<br>
<br>real_3D_Mset {
<br>ca = real(pixel), cb = imag(pixel), cc = p1
<br>a1 = b1 = c1 = 0:
<br>  a2 = a1^2 - b1^2 + ca
<br>  b2 = 2*a1*b1 + cb
<br>  c2 = 2*c1*c2 + cc
<br> a1 = a2, b1 = b2, c1 = c2
<br>z = a1^2 + b1^2 + c1^2
<br>z < 8 }
<br>
<br>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.
<br>
<br>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...?
<br>
<br>Ciao, Russell
<br>
<br>
<br>---------[ Received Mail Content ]----------
<br>>Subject : Re: [Fractint] Neo T-set
<br>>Date : Fri, 06 Oct 2006 06:17:31 -0400
<br>>From : GillesNadeau <gillesnadeau56@videotron.ca>
<br>>To : Fractint and General Fractals Discussion <fractint@mailman.xmission.com>
<br>>
<br>>Hello Russel, 
<br>>
<br>>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.
<br>>
<br>>Regard,
<br>>
<br>>Gilles
<br>>
<br>>
<br>>Neo-T_set2 { 
<br>>cz=pixel, cv=real(p1)
<br>>z = v = 0 
<br>>: 
<br>>a = z*z - conj(v)*v
<br>>b = z*v + conj(z)*v 
<br>>z = a +cz
<br>>v = b +cv 
<br>>(|z|+|v|) < 8 }
<br>>
<br>>
<br>>  ----- Original Message ----- 
<br>>  From: Russell Walsmith 
<br>>  To: Fractint and General Fractals Discussion 
<br>>  Sent: Tuesday, October 03, 2006 23:11
<br>>  Subject: [Fractint] Neo T-set
<br>>
<br>>
<br>>  Fractal Folk, 
<br>>
<br>>  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: 
<br>>
<br>>  4D M-set { 
<br>>  X=real(pixel), Y=imag(pixel), V=real(p1), W=imag(p1) 
<br>>  x1 = y1 = v1 = w1 = 0: 
<br>>  x2 = x1^2-y1^2-v1^2-w1^2 + X 
<br>>  y2 = 2*x1*y1 + Y 
<br>>  v2 = 2*x1*v1 + V 
<br>>  w2 = 2*x1*w1 + W 
<br>>  x1=x2, y1=y2, v1=v2, w1=w2 
<br>>  z = x1^2+y1^2+v1^2+w1^2 
<br>>  z < 16 } 
<br>>
<br>>  We recall that the M-set generates from real numbers by 
<br>>
<br>>  2D M-set { 
<br>>  X=real(pixel), Y=imag(pixel) 
<br>>  x1 = y1 = 0: 
<br>>  x2 = x1^2-y1^2 + X 
<br>>  y2 = 2*x1*y1 + Y 
<br>>  x1=x2, y1=y2 
<br>>  z = x1^2+y1^2 
<br>>  z < 16 } 
<br>>
<br>>  and interpolate to find 
<br>>
<br>>  Neo T-set { 
<br>>  X=real(pixel), Y=imag(pixel), V=real(p1) 
<br>>  x1 = y1 = v1 = 0: 
<br>>  x2 = x1^2-y1^2-v1^2 + X 
<br>>  y2 = 2*x1*y1 + Y 
<br>>  v2 = 2*x1*v1 + V 
<br>>  x1=x2, y1=y2, v1=v2 
<br>>  z = x1^2+y1^2+v1^2 
<br>>  z < 8 } 
<br>>
<br>>  Inserting this last formulation into Gerald D's marvelous 3D T-set formula gives the image at 
<br>>
<br>>  http://ixitol.com/NeoT-set.GIF 
<br>>
<br>>  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. 
<br>>
<br>>  A bit rushed at the moment, I'll post some pars later. 
<br>>
<br>>  Ciao, Russell 
<br>>
<br>>
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<br>>
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