Fractal Folk,
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<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:
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<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 }
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<br>We recall that the M-set generates from real numbers by
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<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 }
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<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 }
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<br>Inserting this last formulation into Gerald D's marvelous 3D T-set formula gives the image at
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<br>http://ixitol.com/NeoT-set.GIF
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<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.
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<br>A bit rushed at the moment, I'll post some pars later.
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<br>Ciao, Russell