On Sat, 25 Jan 2003 00:22:36 +1300 Morgan L. Owens wrote:
"Does that make me a bridesmaid?"
Yo Morgan, since all of us are god and were making the world out of thoughts, you are that which you choose to be. I perceive this Earth, and in fact the entire cosmos, is a venue where our ideas, beliefs, desires, attitudes and expectations precipitate into physical reality as objects and events. So we are, for practical purposes, figments of our own imagination here, and thus we are free to define ourselves in any way we choose. Of course the dogmas of limitation that are so strongly embraced in the conventional wisdom of common social consciousness, will vehemently contradict this suggestion. But as you have apparently looked long and deep into many mind-blowing fractals, you surely have evolved beyond the herd mentality to some extent. You see, I conceive of fractals as Cosmic Mandalas, objects of such hypnotic intricacy and beauty that they shall ultimately prove to be powerful tools in the service of expanding our consciousness and awakening us to our grander identity So lets get on with it and if were going to move so quickly into the new freedom of generating fractals of any dimensionality whatsoever, it seems important to have at least a small inventory of fundamental forms that can be utilized as references. An attempt to create a few of these follows. In the interest of uniformity, certain simplifications are employed that should help make the transition upwards from 2D to 3D to 4D and so forth an intuitive process. The first thing to do is to forget that imaginary numbers ever existed, or remember that they never existed; i.e., unimagine them if you will Thus the complex plane reverts to the Cartesian plane, and we have two axes, X and Y, both measured by real numbers. Let a point in the Cartesian plane be identified by x,y: Then the Mandelbrot set (M) can be generated by the following formula, where brackets indicate a subscript. a[1] = b[1] = 0 s[n] = a[n]^2 + b[n]^2, t[n] = 2*a[n] *b[n] a[n+1] = s[n] + x, b[n+1] = t[n] + y if sqrt(s[n]) < 2, then the point x,y is in M. Back to the familiar .frm form in a moment, but in order to see more clearly whats going on here, well take a quick refresher course in CP. CP in this context stands for cyclical permutation, and as an example take the sequential set of numbers 1 2 3 4. Call this configuration of these four numbers, c1. Now permute c1 cyclically to c2 = 2 3 4 1, then to c3 = 3 4 1 2 and again to c4 = 4 1 2 3. So it may be said of c2 that it is the operation permute the four numbers to the leftward one slot, and so c2^1 = c2, c2^2 = c3, c2^3 = c4 and c2^4 = c1. Hence, c2 is a 4th root of identity. As c3^2 = c1, then c3 is a 2nd root of identity. These four permutations form the order 4 cyclic group (C4). It is isomorphic (i.e., abstractly identical) to the complex number system: Thus c1 = 1, c2 = (0,1), c3 = -1 and c4 = (0,-1). There is much more exposition on these ideas at http://fibonacci-arrays.com/Triternions.pdf At any rate, the point being made here is that the complex number concept is not at all requisite to the process of generating fractals. In the following frm files, the default imaginary values for Y are changed by imag(pixel)*(0,-1) to real. CP4 (XAXIS){ x=real(pixel), y=imag(pixel)*(0,-1), a=b=0: a1=a^2+b^2, b1=2*a*b a=a1+x, b=b1+y z=sqrt(a^2 + b^2) z < 4 } There are other versions of C4, so this formula is called CP4 to identify it as being based on the group generated by the CP process. This file makes an interesting version of M. E.g., note that the inside orbits no longer exhibit any curvature. Moreover, certain color mappings make visible field lines that suggest extensive connections between M and its satellites and even (by single threads) to several points at infinity. Now take c1 = 1 2 3 4 5 6, c2 = 2 3 4 5 6 1 and etc. These form the order 6 cyclic group (C6), which comprises the roots c2^6 = c6^6 = c3^3 = c5^3 = c4^2 = c1. The formula that follows is based on this group. CP6 (XAXIS){ x=real(pixel),y=imag(pixel)*(0,-1),w=p1, a=b=c=0: a1=a^2-2*b*c b1=-c^2+2*a*b c1=b^2+2*c*a a=a1+x,b=b1+y,c=c1+w z=sqrt(a^2+b^2+c^2) z < 4 } Somehow, Fractint seems to have reserved z for comparison with the bailout value; any other symbol Ive tried distorts the set ? The CP6 formula is a close cousin of TMan and TGirl, which are based on a variant of C6, and the familial resemblance is there. There are 58 other variations of C6 that should also generate images, but suffice it to say that, in constructing formulas for many of these, complications mat ensue... On to C8, where c2^8=c4^8=c6^8=c8^8=c3^4=c7^4=c5^2=c1 CP8 (XAXIS){ x=real(pixel),y=imag(pixel)*(0,1),v=p1,w=p2 a=b=c=d=0: a1=a^2-c^2-2*b*d b1=2*a*b-2*c*d c1=2*a*c+b^2-d^2 d1=2*a*d+2*b*c a=a1+x,b=b1+y,c=c1+v,d=d1+w z=sqrt(a^2+b^2+c^2+d^2) z < 4 } There are many, many possible basic variations here, perhaps even thousands. Some interesting results can come of just changing a few signs In closing, well look at C10: c2^10=c4^10=c8^10=c10^10=c3^5=c5^5= c7^5=c9^5=c6^2=c1. Cp10 (XAXIS){ x=real(pixel), y=imag(pixel)*(0,-1), u=p1, v=p2, w=p3 a=b=c=d=e=0: a1=a^2-2*b*e-2*c*d b1=-d^2+2*a*b-2*c*e c1=b^2+2*a*c-2*d*e d1=-e^2+2*a*d+2*b*c e1=c^2+2*a*e+2*b*d a=a1+x,b=b1+y,c=c1+u,d=d1+v,e=e1+w z=sqrt(a^2+b^2+c^2+d^2+e^2) z < 4 } It should be clear from these patterns that imaginary/complex numbers are not inherent to fractals per se. To reintroduce them into any of these formulas, particularly into systems such as CP6 and CP10, where 4th order roots are not already presently, creates a bit of havoc but hey, havoc and chaos may complement one another nicely, e.g., as in Lee Skinners renditions of Jim Muths TMan1. BTW, some of these, such as Fig Tree, are to my eye reminiscent of Japanese watercolors, an impression that fractals dont often convey. Peace, Russell _________________________________________________________________ STOP MORE SPAM with the new MSN 8 and get 2 months FREE* http://join.msn.com/?page=features/junkmail