;     Date: Tue, 05 Aug 2014 15:16:53 -0400
;     From: Jim Muth <jimmuth@earthlink.net>
;  Subject: [Fractint] FOTD 04-08-14 (Squarely Rooted [A-8,M-5])
;       Id: <1.5.4.16.20140805151840.299712cc@earthlink.net>
; ---------
; 
; FOTD -- August 04, 2014 (Rating A-8,M-5)
; 
; Fractal visionaries and enthusiasts:
; 
; NEW FORMULA ALERT!
; 
; -----------------------------------------------------
; 
; MandelbrotSqrt2BC   { ; by several Fractint users
;   p=p1, q=2*PI*(p/(2*PI)),
;   r=p1-q, Z=C=Pixel:
;     Z=log(Z)
;     IF(imag(Z)>r)
;       Z=Z+flip(2*PI)
;     ENDIF
;     Z=exp(1.414213562373*(Z+flip(q)))+C
;   |Z|<36 }
; 
; -----------------------------------------------------
; 
; Today's formula is a new one, an addition to the 'BC'or 
; 'branch-cut' group of formulas, which are supposed to draw the 
; other fractals resulting from the multi-valued nature of the 
; complex logarithm.  Since Z^(sqrt(2)) is the exponent I use most 
; frequently, I have simplified the formula, and in doing so, I 
; have also speeded up the calculation a little.
; 
; But even from the beginning, the math of the 'BC' formulas has 
; been questionable.  The formula for the complex logarithm as 
; listed in the second edition of "Fractal Creations" is . . .
; 
;     (1/2)In(x^2+y^2)+i(arctan(y/x)+2kPI
; 
; where k can be any real positive or negative integer.
; 
; The value I have entered for 'k' in today's image, -9.42477, 
; shows the math flaw, since it is not an integer, but despite the 
; flawed math, this fractional value still creates a fractal.  
; (I'll not even try to guess what errors might be created by the 
; spurious trig functions that I sometimes use.)
; 
; The parent fractal of today's image is a chaotic indescribable 
; thing with a long zig-zagging filament extending from its 
; northeast side.  Today's scene lies in the 'East Valley' area of 
; a tiny minibrot on this filament.
; 
; I named the image "Squarely Rooted" because the exponent of Z is 
; the square root of 2.  The artistic aspect is a little better 
; than most recent images and rates an honest 8.  Except for the 
; new formula, I could find nothing new in the math, which rates a 
; boringly average 5.  The calculation wait of 3-5/6 minutes may 
; be avoided by checking the web sites.
; 
; Check the finished image on the FOTD web sites at:
; 
;       <http://www.crosscanpuzzles.com/Archives.html>
; 
; <http://www.emarketingiseasy.com/TESTS/FOTD/jim_muths_fotd.html>
; 
;        <http://www.Nahee.com/FOTD/>
; 
;   <http://user.xmission.com/~legalize/fractals/fotd/about.html>
; 
; Today brought a mix of clouds and sun, and a temperature of 81F 
; 27C to Fractal Central, the historic center of fractal activity. 
; After a relaxing week-end, the fractal humans got back to work; 
; after a relaxing week-end, the fractal cats continued relaxing.
; 
; The next FOTD will be posted in a day or two, or maybe three.  
; Until whenever, take care, and when watching the media news, try 
; to separate the propaganda from the facts.
; 
; 
; Jim Muth
; jimmuth@earthlink.net
; 
; 
; START PARAMETER FILE=======================================

Squarely_Rooted { ; time=0:03:50.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandelbrotSqrt2BC passes=1
  center-mag=+1.381338385604301/+0.8059894596556334/\
  6.963546e+011/1/-50/0 params=-9.42477/0 float=y
  maxiter=1800 inside=0 logmap=441
  periodicity=6 mathtolerance=0.05/1
  colors=00000B01D02F03H04J05L06N17P28R39T4AV5AX6AZ7\
  A`8Ab8Ad8Ae7Af7Ag7Ah7AiAAjCBkFDlHGmKInMKoOMpRPqTRr\
  WTsYVs_Yrb_rdbrgerihrkkrnmvpozsqzutzwwzvzzuzztzztz\
  zszzrzzrvvqsrppmpmhomcnlbnlamk_lkZljYkjXjiVjiUihTh\
  hRhgQggPVkfYlg`mibnjakiahhaehabga_faYfaVeaSdaPdaMc\
  aIbaFbYDWVBQRAKO4EK08H02GALFMcEXvUbuhguwluxmnymgwk\
  euictharf_pdYocWmaUk_SjZQhXOfVMeUKcSIaQG`PEbODcNCd\
  MBeLBfKAgK9hJ9iI8jH7kG7lG6mF5nE5oD4pC3qC3YU4eT3mS1\
  tR0uK7uEEv8Lv2SlAVcHYVO`MVcDaf4hi6kb8nX9pRBsLDvFEx\
  9CuJBsS9p`8niCheGbbKX_NRXRLUVFRZ9Oa4Li9AqE0sAGu7Wv\
  4kuGktSksckrokqzkoxdmwZkuTitNhsHloFplEtiCxfBtdCqbD\
  m`EjZEfUFcPG_KHXFH_AMaEQc7UJ8ZK8WF8JB86hbulUqpKltc\
  hw2dmcKtKjsSmq`pphsoqvnyykuqizjgUcdnWboP`pIUqGNrFG\
  sE9hD3uC9vHEwMKxRPyWUz_TzcTzfzzjzzmzzpzztzzwzzzzzr\
  zzkzzczzXzzQzzIzzBzz4zz3zz2zz2zz1zz0zz0zzBzzHzzNzz\
  TzzYzzczzizznzzjzzgzzdzza }

frm:MandelbrotSqrt2BC   { ; by several Fractint users
  p=p1, q=2*PI*(p/(2*PI)),
  r=p1-q, Z=C=Pixel:
    Z=log(Z)
    IF(imag(Z)>r)
      Z=Z+flip(2*PI)
    ENDIF
    Z=exp(1.414213562373*(Z+flip(q)))+C
  |Z|<36 }

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