Fractal visionaries and enthusiasts:

Today we enter "Uncharted Territory", which is what I named the image.  Never before have I checked a fractal with an exponent as close to unity as 1.009, making today's image uncharted for sure.

The graph of Z^1.009 is virtually a straight line, so how is it possible that a fractal as rich and varied as today's could come about?  The answer is the multi-valued nature of the complex logarithm.

The complex log is multi-valued.  This means that it has many, in fact an infinity of values, and all of them are 'correct'.  When applied to fractals with fractional exponents, these different values create different fractals, all of which are correct.  But curiously, when applied to exponents very close to unity, as is today's, these different values create some unexpectedly rich images, in effect turning a 'sow's ear' into a 'silk purse'.

Today's image lies in one of these 'silk-purse' fractals.  The parent fractal is an apparently featureless ellipse, but a close examination reveals a tiny bit of chaos on the southeast shore line.  Today's image lies in this chaos.

I used the built-in black-and-white color palette to create the 3-D effect.  Other palettes might have worked better, but an attack of terminal laziness prevented me from trying to find one.  This laziness held the rating to an everyday 7.

Typical midsummer weather made Sunday a typical midsummer day here at Fractal Central.  The fractal cats spent the day doing what cats do when the temperature reaches 88F 31C -- they stretched out and went to sleep.

The humans took it just as easy, though we did stay awake at least most of the time.  The next FOTD is due to be posted in 24 hours.  Until then, take care, and don't ask me what I think is wrong with the new world being brought about by the recent advances in technology.  I just might start to answer.

Jim Muth
jimmuth@earthlink.net


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

UnchartedTerritory { ; time=0:12:09.74-SF5 on P4-2000
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandelbrotBC3 function=recip passes=1
  center-mag=-0.132209132/-1.699446787/4304/1/97.5/0
  params=1.009/0/4.75/1500 float=y maxiter=4200
  colors=000fffgggggghhhhhhiiiiiijjjjjjkkkkkkllllllm\
  mmmmmnnnnnnooooooppppppqqqqqqrrrrrrssssssttttttuuu\
  uuuvvvvvvwwwwwwxxxxxxyyyyyyzzzyyyxxxxxxwwwwwwvvvvv\
  vuuuuuuttttttssssssrrrrrrqqqqqqppppppoooooonnnnnnm\
  mmmmmllllllkkkkkkjjjjjjiiiiiihhhhhhggggggffffffeee\
  eeeddddddccccccbbbbbbaaaaaa``````______ZZZZZZYYYYY\
  YXXXXXXWWWWWWVVVVVVVVVUUUUUUTTTTTTSSSSSSRRRRRRQQQQ\
  QQPPPPPPOOOOOONNNNNNMMMMMMLLLLLLKKKKKKJJJJJJIIIIII\
  HHHHHHGGGGGGFFFFFFEEEEEEDDDDDDCCCCCCBBBBBBAAAAAA99\
  99998888887777776666665555554444443333332222221111\
  11000000000000000111111222222333333444444555555666\
  666777777888888999999AAAAAABBBBBBCCCCCCDDDDDDEEEEE\
  EFFFFFFGGGGGGHHHHHHIIIIIIJJJJJJKKKKKKLLLLLLMMMMMMN\
  NNNNNOOOOOOPPPPPPQQQQQQRRRRRRSSSSSSTTTTTTUUUUUUVVV\
  VVVVVVWWWWWWXXXXXXYYYYYYZZZZZZ______``````aaaaaabb\
  bbbbccccccddddddeeeeeefff }

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=========================================