Astroid_Mset Part 3 - Astroid Of The Far North

In this installment we will see what happens when the Astroid_Mset is applied to The_Far_North, Jim Muth's Fractal Of the Day for January 14, 2011. Setting the center of the image, zooming in to the required depth, and tweaking a few parameters generated the above image.

You can see the purple asteroid ring has period doubled twice since it has 16 lobes. The astroids are much further from the minibrot in this image than the one in Part 1. I think the shallowest minibrots in the image are located inside the biggest astroids. It doesn't appear that there is a minibrot at the center of any of the spirals although the spiral arms have plenty. The noise in the image is from tiny astroids surrounding an infinite number of unresolved minibrots. The 8 lobe astroid can be found by zooming out to a magnification of 1.2E10 and the 4 lobe is at 3.8E8. You can compare this image to Jim's original at:

http://www.emarketingiseasy.com/TESTS/FOTD/F110114f.gif

In Part 4 we will experiment with a new shape.

The parameter file used to generate the image is below:

Astroid_OTFN_16 { ; Exported from Fracton.
 reset=2004 type=formula formulafile=fracton.frm
 formulaname=Astroid_Mset_v2 passes=1 float=y
 center-mag=-0.128183294678311/0.649523014799861/19\
 3002911147.3112/1/120/0
 params=0.003/0.12/8/30/0/0/0/0/0/0 maxiter=250000
 inside=255 outside=summ
 colors=000fOz<28>I0Kz0f<28>O08z88<28>O00zW0<28>c40\
 zz0<28>aG00zR<28>0C40zz<28>0CCGGz<28>00O000<12>000\
 z88 }

frm:Astroid_Mset_v2 {
 ; Copyright (c) Paul W. Carlson, 1997
 ; Modified for compatibility with Fractint 20.04 and
 ; Fracton by Mike Frazier, 2011
 ;****************************************************
 ; Always use floating point math and outside=summ.
 ;
 ; Parameters:
 ;   real(p1) = a factor controlling the width of the curves
 ;   imag(p1) = radius of the astroid
 ;   real(p2) = number of color ranges
 ;   imag(p2) = number of colors in each color range
 ;
 ; Note that the equation variable is w, not z. 
 ; Initialize cindex to the index of the background color
 ; Formula modified to avoid color index 0 which can not
 ; be used with outside=summ in FractInt v20.04
 ;****************************************************
 w=0,
 c=pixel,
 z=0,
 cindex=254,; Background color 254
 bailout=0,
 iter=0,
 range_num=0,
 i=(0,1),
 r=imag(p1),
 ;****************************************************
 ; In the accompanying par file,
 ; we have 8 color ranges with 30 colors in each range
 ; for a total of 240 colors. The first range starts at
 ; color 1.  Pixels will use color 254 when |w| > 1000.
 ; Other values can be used here as long as the product
 ; of num_ranges times colors_in_range is less than 255.
 ; Color 254 is reserved for the background color and 
 ; color 255 can be used for the inside color.
 ;****************************************************
 num_ranges=real(p2),
 colors_in_range=imag(p2),
 ;****************************************************
 ; Real(p1) controls the width of the curves.
 ; These values will usually be in the range 0.001 to 0.1
 ;****************************************************
 width=real(p1),
 index_factor=(colors_in_range-1)/width:
 ;****************************************************
 ; The equation being iterated.  Almost any equation
 ; that can be expressed in terms of a complex variable
 ; and a complex constant will work with this method.
 ; This example uses the standard Mandelbrot set equation.
 ;****************************************************
 w=w*w+c,
 ;****************************************************
 ; The orbit trap curve.  This example uses an "astroid"
 ; curve (which has absolutely nothing to do with huge
 ; rocks in outer space).  Any two-dimensional curve can
 ; be used which can be expressed in parametric form in
 ; terms of the angle from the origin.
 ;****************************************************
 ang=atan(imag(w)/real(w)),
 astroid=r*(cos(ang)^3+i*sin(ang)^3),
 ;****************************************************
 ; If the orbit point is within some distance of the curve,
 ; set z to the index into the colormap and set the bailout
 ; flag.  Note: the way we use the "distance" here has
 ; the effect of turning the curves inside-out in the image.
 ;****************************************************
 distance=abs(|w|-|astroid|),
 if(distance<width&&iter>1),
 cindex=index_factor*distance+range_num*colors_in_range+1,
 bailout=1,
 endif,
 ;****************************************************
 ; Cycle through the range numbers (0 thru num_ranges - 1)
 ; With two color ranges, even iterations use color
 ; range 0, odd iterations use color range 1.
 ;****************************************************
 range_num=range_num+1,
 if(range_num==num_ranges),
 range_num=0,
 endif,
 ;****************************************************
 ; Since we are using outside=summ, we have to subtract
 ; the number of iterations from z.
 ;****************************************************
 iter=iter+1,
 z=cindex-iter,
 ;****************************************************
 ; Finally, we test for bailout
 ;****************************************************
 bailout==0&&|w|<1000
 }