"Jim Muth" <jamth@mindspring.com> wrote in yesterday's FOTD:
Today's fractal pictures a quadratic midget. There is nothing unusual in having a quadratic midget at the center of the screen. Most of my images have them.
Okay, Jim. But what does the "quadratic" part of "quadratic midget" mean? Does it just relate to the symmetry splitting you describe later on? If it's only an observational quality, then where do we look for them?
The first indicator of a midget is the presence of two symmetrical elements that split into four similar but smaller elements. This splitting continues in the series 2,4,8,16,32... as the elements approach the midget.
I seem to find that kind of symmetry more prominently the deeper the zoom is, and more in evidence for midgets not on the main spiral arms, but that's completely empirical. What is the definition of the "quadratic" description? Is the midget shown in the following par/frm a quadratic one? It shows symmetry splitting, though not all apparently in the 2^n form, possibly because it's from the z^6 power M-set. Regards, Hiram /***************PAR BEGIN*****************/ pentahex { ; Is this a "quadratic" midget? ; Fractint Version 2003 Patchlevel 1 reset=2003 type=formula formulafile=exprmntl.frm formulaname=3arySlcPwrJulibrot passes=1 center-mag=-0.66798411674660230/+0.31990623903628910/1.456509e+011 params=0/0/6/0/0/0/90/0/0/0 float=y maxiter=2000 inside=0 logmap=yes colors=CCC<15>gCgiCikCk<3>sCs<16>MCs<4>CCs<16>Cis<4>Css<16>CsM<4>CsC<16>\ isi<4>sss<16>ssM<4>ssC<13>sSC<4>sIC<5>sIC<4>sIC<3>dGCaFCYFCUECQDCMCC<2>G\ CC<3>dGC<3>WCTTBYYBecAm<3>www<3>jRqkEWm0A<2>www<3>obvmYukTu<2>eCs<4>WCs<\ 4>MCs<2>SPsUUsUSs<3>TJsTHsTFsSCs<3>KDsIDsGDsEDsCEs<9>CYsC_sCas<3>CisCs_C\ sY } /*************PAR END******************************/ /*************FRM BEGIN****************************/ 3arySlcPwrJulibrot { ; p1=z0(init 0),p2=a(init 2),p3=C(init 0) of anchor point ; p4=latitude(init 90),p5=longitude(init 0) (each in double angle form ; ,converted internally to complex angle form) of the slice's direction ; p4 varies ( [-90,90],(-90,90) ), p5 varies ( [-180,180],(-90,90) ) ; real(lat,long):(0,0|180|-180) Julia dir,(0,90|-90) Power dir(z0,C const) ; ,(90|-90,dontcare) Mandelbrot dir ; C1 linear slice of a C3 parameter phase space ; slices the power Julibrot (the quadratic Julibrot is the subset ; with a=(2,0)=const over its C2 plane) ; The phase space is of {z0,a,C} in the process z->z^a+C ; p1,p2,p3 are complex coords of an anchor point in the space ; p4,p5 are the complex latitude,longitude direction of the line ; emanating from the anchor point. IF ( isinit == 0 ) ; once per screen initialization maxit = maxit ; force floating point lim = 100 ; following cnvt dbl angles to complex angles: lat = real(p4)*pi/180+flip(tan(imag(p4)*pi/360)*pi/2) long = real(p5)*pi/180+flip(tan(imag(p5)*pi/360)*pi/2) Muz= cos(long) * cos(lat) Mua= sin(long) * cos(lat) Muc= sin(lat) ; preceding are slice's polar coord unit direction in C3 isinit =1 ENDIF k = pixel ; pixel position is the parameter of variation z=p1+k*Muz, a=p2+k*Mua, c=p3+k*Muc ; find pos in (z0,a,c) space : z = z ^ a + c |z| <= lim } /***********FRM END**********************/