[note: I sent this msg to the list last night, but it appears not to have been registered in the archive, thus the resend.] Lee H. Skinner <skinner@thuntek.net> wrote:

The horizontal lines are periodicity effects. Set periodicity=no.

Thankyou, Lee. That's exactly what the lines were-- your remedy totally eliminated them. On closer examination I noticed that there was something else wrong with the second fractal. Areas of the inside to outside border appeared to have the visual quality of perturbed sets, implying that I had not initialized those areas on all the critical points of the generator expression. I found that I had made a mistake in the formula where calculating the principal value of the root that sets the modulus of the nonzero critical points, and had used an exponent of diff instead of 1/diff. Fixing that as well caused the second fractal to render properly. I wish that setting periodicity=0 had changed the result of the first .par as well, but it did not. It rendered substantially the same, with the straight horizontal border between outside and inside sets along the negative real axis, so that conundrum remains. Changing the exponents in the terms so that they differed by an integer, but were themselves rational nonintegers, changed the position of the puzzling behavior but not the fact of its existence. In the following case (with the revised .frm) the exponents are 5.5 and 3.5 instead of 5 and 3.5, and a section of the main body's border in the third quadrant is shown. It doesn't look like the perimeter of an M-set at all; it is smooth and the outside points which usually show the direction of a gradient are divided up into many facets without an apparent gradient direction. I have to wonder if this is a mathematical effect related to taking a fractional power-- I had thought that the sets would vary continuously as the exponents were varied, but this effect suggests discontinuity instead, if it is not spurious. Regards, Hiram ================BEGIN PAR & FRM===================== OddThirdQuadrant { ; z->z^5.5+(0-1i)*z^3.5+C ;The fault line has been replaced by this faceting of the ; outside=iter points and smooth surfaced "midgets" and ; main body coastline in the third quadrant. ; Fractint Version 2003 Patchlevel 1 reset=2003 type=formula formulafile=exprmntl.frm formulaname=bimandeloid2 passes=1 center-mag=-0.716191/-0.354303/8.788233/1/-115/-2.50563458870090017e-014 params=5.5/3.5/0/-1/400/3 float=y inside=0 periodicity=0 colors=000zzzc00<6>n00o00p00<3>w00<8>mGAlIBjJC<3>eRHdTIfTG<3>nTApU8rU7<2\

xU2zV0zX0<5>yi0yk0xn0<2>xt0ww0vw1<9>exKcyLayN`yPZyRXyT<17>WyoWypWyq<2>W\ yuVzwWww<8>lOwmKwoGw<3>w0w<9>e0kc0ja0h<2>W0eU0cU0c<118>TQTTQTTRT<3>SSS00\ 0000 }

Bimandeloid2{ ; vary C over the screen in z->z^A+k*z^B+C ; p1=(A,B)[reals],p2=k[complex],p3=(bailout-4,#critical points) IF(isinit==0) ; image-wide initialization section isinit=1 A = real(p1), B = imag(p1), k = p2 bailout = real(p3) + 4, cps = trunc(imag(p3)) IF(cps<=0) ; give a default value for entry of 0 cps = 1 + ceil(abs(A - B) ) ; works for integers only ENDIF is_agtb = (A > B), diff = abs(A - B), angle = 2 * pi / diff rotfactor = cos(angle) + flip(sin(angle)) IF(is_agtb) ; principal roots of the eqn for cps principal = (-B * K / A) ^ (1/diff) ELSE principal = (-A / B / k) ^ (1/diff) ENDIF ENDIF C = pixel ; Mandelbrot-like type z = currcp = 0, cpctr = 1 ; first critical point is z = 0 isdiv = isalldiv = 0 ; flags for current cp and all cps divergence : IF(isdiv) ; last critical point diverged cpctr = cpctr + 1 IF(cpctr > cps) ; finished, with all cps diverging isalldiv = 1 ELSEIF(cpctr == 2) ; second cp is the principal value z = currcp = principal, isdiv = 0 ELSE ; rotate the last cp by angle on the complex plane currcp = currcp * rotfactor z = currcp, isdiv = 0 ENDIF ENDIF IF(isalldiv != 1) z = z^A + k*z^B + C ; main iteration isdiv = ( |z| > bailout) ENDIF isalldiv == 0 } ===================END=======================