In a recent perusal of Bradley Beacham's "AN INTRODUCTION TO THE FRACTINT FORMULA PARSER" (the FRMTUTOR.TXT file in Fractint) I came across the 'fibo-A' formula. This .frm introduces Fibonacci-type recursions to the Mset formula, and it inspired me to investigate this idea further. I thus came up with the 'Fibrot' formula, presented (in a generalized version) below: ------------------------------parfile----------------------------- Nautiloid { ; July 23, '03 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=next.frm formulaname=Fibrot1 passes=3 center-mag=-6.42033/2.44459/0.04984424 params=-5/-1/1/0 float=y maxiter=300 colors=000z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0W } ------------------------------frmfile----------------------------- frm:Fibrot1 { z = 0, c = pixel, j=0, k=1: a = p2*j + p1*k j = k k = a z = z^k + c z < 1000 } -------------------------------endpar----------------------------- Soon thereafter, Jack O'TradeZ used this .frm to render the 'Mosaic' viddie now playing at: http://www.sonic.net/~jot/ufvp.html Both the 'stills' and the video demonstrate that complex parameter values in 'Fibrot' induce a shear factor along the real axis that warps circular shapes into nautiloids. It occurred to me that a variant formula might give more insight into this effect, and this led to a series of 'Nautibrot' formulas, some of which are presented below: ------------------------------parfile----------------------------- Nautilii { ; July 23, '03 ; Version 2002 Patchlevel 5 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=next.frm formulaname=Nautibrot2 passes=3 center-mag=-6.84262/0/0.04783259 params=5/1 float=y maxiter=300 colors=000z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0W } ------------------------------frmfile----------------------------- frm:Nautibrot2 { z = 0, c = pixel, k=1, j=-1: j=j*(-1) k=p1*k*j z = z^k + c z < 1000 } -------------------------------endpar----------------------------- Another variant introduces i-axis shear: ------------------------------parfile----------------------------- ShearMadness { ; July 23, '03 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=next.frm formulaname=Nautibrot3 passes=3 center-mag=0/0/0.6666667 params=-1/1 float=y maxiter=300 colors=00000j0j0jj0j00j0j0jjkkkkrkdmwW000W0WW000WW0W0WWFNF<4>rNF\ zNFFVF<4>rVFzVFFbF<4>rbFzbFFjF<4>rjFzjFFrF<4>rrFzrFFzF<4>rzFzzFF\ FN<4>rFNzFNFNN<4>rNNzNNFVN<4>rVNzVNFbN<4>rbNzbNFjN<4>rjNzjNFrN<4\

rrNzrNFzN<4>rzNzzNFFV<4>rFVzFVFNV<4>rNVzNVFVV<4>rVVzVVFbV<4>rbV\ zbVFjV<4>rjVzjVFrV<4>rrVzrVFzV<4>rzVzzVFFb<4>rFbzFbFNb<4>rNbzNbF\ Vb<4>rVbzVbFbb<4>rbbzbbFjb<4>rjbzjbFrb<4>rrbzrbFzb<4>rzbzzbFFj<4\ rFjzFjFNj<4>rNjzNjFVj<4>rVjzVjFbj<4>rbjzbjFjj<4>rjjzjjFrj<4>rrj\ zywccdWWWz000z0zz000zz0z0zzzzz }

------------------------------frmfile----------------------------- frm:Nautibrot3 { z = 0, c = pixel, k=1: k=p1*conj(k) z = z^k + c z < 1000 } -------------------------------endpar----------------------------- Finally, in these "Not-a-Brot" calculations, 2 + 3 = 4: ------------------------------parfile----------------------------- L'escagot { ; July 23, '03 ; Version 2002 Patchlevel 5 reset=2002 type=formula formulafile=next.frm formulaname=Nautibrot4 passes=3 center-mag=-0.58651/0.281617/0.6666667 params=4/1 float=y maxiter=300 colors=000z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0\ zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc\ 00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000zk0kW0Wc00z00zW0zj00W000\ zk0kW0W } ------------------------------frmfile----------------------------- frm:Nautibrot4 { z = 0, c = pixel, k=1, j=-1: j=j*(-1) k=p1*conj(k)*j z = z^k + c z < 1000 } -------------------------------endpar----------------------------- A generalized nautibrot is: ------------------------------frmfile----------------------------- Nautibrot5 { z = 0, c = pixel, k=p2: k=p1*fn1(k) z = z^k + c z < 1000 } -------------------------------endpar----------------------------- That's about the gist of it... I suspect that these efforts haven't been all for 'naut', for there are perhaps many other patterns that will emerge in this context. I hope that list members will find something of interest here, but hey, it got me all in a state of sheer amazement and that's the bottom line... Ciao, Russell (PS: I think I have this .par file thing down, but if they fail to render correctly,lemme know.)