Osher, Your posted reference to J. C, Sprott’s work (Mandelbrot and Julia Sets Explored by Rare Event Theory, 29 Jan 2003) came along just as I was finding my way into related outcomes based on groups. The approach of using the order 4 cyclic group (C4) to generate the Mset in Fractint leads to a (WinXP friendly) formula such as Mset { x=real(pixel), y=imag(pixel), x1=y1=0: a=x1^2-y1^2+x b=2*x1*y1+y x1=a, y1=b z=sqrt(a^2+b^2) z < 10000 } I subsequently found a generalized formula that apparently works for groups of any order. In it, the Mset looks like this: C4e { x=real(pixel), y=imag(pixel), x1=x2=y1=y2=0: a1=x1^2+x2^2+2*y1*y2+x a2=2*x1*x2+y1^2+y2^2-x b1=2*x1*y1+2*x2*y2+y b2=2*x1*y2+2*x2*y1-y x1=a1, x2=a2, y1=b1, y2=b2 z=sqrt((a1-a2)^2+(b1-b2)^2) z < 10000 } But looking back at the first Mset formula, I saw it as a configuration of the pieces of the expansion of (a+b)^2. What would other arrangements lead to? For example: Mtest { x=real(pixel), y=imag(pixel)*p1, a=b=0: a1=a^2+a*b+x b1=b^2+a*b+y a=a1, b=b1 z=sqrt(a^2+b^2) z < 10000 } For p1=1, the result wasn’t all that interesting, but for p1=(0,1), I got a bit of a surprise. That this too produces an Mset led to curiosity about other polynomial expansions, e.g., (a+b+c)^2 = a^2+b^2+c^2+2*a*b +2*a*c+2*b*c. I discovered that I could mix these up in about any arbitrary fashion and get a fractal object, and the more symmetry in the formula statements, the more to be found in the fractal itself. In the following example, p2 is meant to toggle between 1 and (0,1). Ptest {;use floating point x=real(pixel),y=imag(pixel)*p2,v=p1 a=b=c=0: a1=a^2-p2^2*a*b-p2^2*a*c b1=b^2-p2^2*a*b-p2^2*b*c c1=c^2-p2^2*c*b-p2^2*a*c a=a1+x,b=b1+y,c=c1+v z=sqrt(a^2+b^2+c^2) z < 10000} Where p2=1 produces a symmetrical object oriented toward the vertical, p2=i is an Mset again. Something I find interesting about Ptest is p1’s effect on M. Plus/minus real p1 shift M left or right, imaginary p1 shift it up or down, and complex p1 at a corresponding angle. There seems to be zero distortion, as long as p1 isn’t too close to the bailout value. Here are some examples of frm files created from (a+b+c)^3 Ptest2 { x=real(pixel),y=imag(pixel)*p2,v=p1 a=b=c=0: a1=a^3+3*b^2*a+3*c^2*a+2*a*b*c b1=b^3+3*a^2*b+3*c^2*b+2*a*b*c c1=c^3+3*a^2*c+3*b^2*c+2*a*b*c a=a1+x,b=b1+y,c=c1+v z=sqrt(a^2+b^2+c^2) z < 10000 } Ptest2 forms a square for p2=1,for p2=i an object similar to a third degree Mset (i.e., z1=z^3+c) and for p2=(1,1), a nice combination of the two. Ptest3 has an OK fractal in p2=i but nothing too exciting elsewhere that I discerned. Ptest3 { x=real(pixel),y=imag(pixel)*p2,v=p1 a=b=c=0: a1=a^3+b^3+c^3+6*x*y*z b1=3*b^2*c+3*c^2*a+3*a^2*b c1=3*c^2*b+3*a^2*c+3*b^2*a a=a1+x,b=b1+y,c=c1+v z=sqrt(a^2+b^2+c^2) z < 10000 } That’s a cursory look at it, but the idea of a class or classes of fractals based on polynomial expansions may prove eventually to harbor some interesting things... Ciao, Russ _________________________________________________________________ The new MSN 8: advanced junk mail protection and 2 months FREE* http://join.msn.com/?page=features/junkmail