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 -----
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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