Fractal visionaries and enthusiasts: 

This phrase is not only the name of today's image, but it is also a harsh fact of the image.  The tiny hole at the center of the scene is a minibrot in the parent fractal that came about when I calculated the expression Z^sqrt(2)+C at a height of 14 levels up the hyperladder with no function applied.

This parent is a shapeless thing, impossible to describe in words, with no resemblance at all to a recognizable Mandelbrot set.  Today's image lies on a filament extending from a mis-shapen bud on the northeast side of the parent.

I chose sqrt(2) as the exponent of Z because the minibrots in this family of fractals are sometimes surrounded by two- and four-way symmetry, making them easier to find, though nowhere near as easy as straight quadratic minibrots.

I sometimes wonder what separates a minibrot from the countless other holes that parent fractals are often filled with.  As I see it, the difference is that true minibrots do not fill in regardless of how high the maxiter is raised.  Also, minibrots almost always lie in a basin, with the number of surrounding elements increasing without limit as the edge of the open area is approached.  The other holes that fill fractals are often simply random open areas that never fill in, or open areas at the center of bottomless spirals that will fill in with a higher maxiter.

Most minibrots are of the quadratic variety, even in fractals created by a combination of exponents other than 2.  This quadratic shape appears to be the generalized shape of minibrots, much as parabolas are the generalized shape of curves in the graphs of many functions.  Today's minibrot is of order 1.414... a variety which has no generalized shape at all, and is therefore quite interesting.

The art rating of an 8 shows that I am quite satisfied with the colors.  The math rating of a 7 was given a boost by the colors.  There is little new math stuff in the image however.

The calculation time of 2-3/4 minutes is slower than I would have preferred, but the FOTD web sites can eliminate the slowness.

The day began with a biting cold temperature of 23F -5C but the clear sky and resulting strong sun bumped it up to 43F +6C by afternoon.  The fractal cats, who are just learning to play together, were too busy chasing each other up and down the fractal hallway to take advantage of the afternoon sunlight flooding their shelf in the southwest window.  The humans, much less playful, spent the day tending to more pressing but less interesting things.

The next FOTD will be posted when the time is right.  Until whenever that might be, take care, and I admit that fractals are awesome, but are they groovy?

Jim Muth
jimmuth@earthlink.net


START PARAMETER FILE=======================================

This_is_a_Minibrot { ; time=0:02:45.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandelbrotBC3 function=ident passes=1
  center-mag=+0.864016578592/+0.61309334532/9.9e+007\
  /0 params=1.414213562375/0/14/0 float=y
  maxiter=3200 inside=0 logmap=295 periodicity=6
  colors=0000BY0Dd0Hf0Lt0Qt0Vs2Zs6cq9hqBlqGqoKvoOznR\
  znVznTziRxdQqaOkYNdTLZQLRLKLIIGEHAAG47E02E00I04L09\
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  B40910600200000000000000000000000200900D00H00L00R0\
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  qtsttvxtxztzztzztzztzztzstz`nzIsz1vz0sz0ot0ln6ihDh\
  aKdVRaOZZIfWDnV7tRDvQIxNOyLVzI`zHdzGkzDqzBxz9zz7zz\
  6zz9zzByzDqyGktHcqKWlNQhOIdRB`T6Yz02o6ETEQ7Na0Wl0a\
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  t0Qt0Vt0Zt0ct0ht0lt0qt0vt0zt0vz0tz0tz0tz0tz0tz0tz0\
  tz0tz0tz0tz0tz0tz0tz4tzBtzILzQOzYRzdVzaWz`ZzZ`zYaz\
  WdzVfzThzRkzOlzNozLqzKszI }

frm:MandelbrotBC3   { ; by several Fractint users
  e=p1, a=imag(p2)+100
  p=real(p2)+PI
  q=2*PI*fn1(p/(2*PI))
  r=real(p2)+PI-q
  Z=C=Pixel:
    Z=log(Z)
    IF(imag(Z)>r)
      Z=Z+flip(2*PI)
    ENDIF
    Z=exp(e*(Z+flip(q)))+C
  |Z|<a }

END PARAMETER FILE=========================================