The Golden Ratio, also known as the Greek letter phi (pronounced "fee") and several other names, is indeed a fascinating number. It is approximately 1.1680339887499; or is it this number minus 1? Anyway, it is the only answer to the question, Which number satisfies *both* 1/x = x-1 and x^2 = x+1?... Wild. It is also calcualted as ((sqrt 5)-1)/2. It appears in numerous surprising places in nature as well, such as angles of packing of sunflower seeds and ratios of flower petals in many species. I have used the GR in a series of L-systems, where in many cases it acts as a limit, allowing growth to approach a certain point but not overlap. At http://spanky.triumf.ca/pub/fractals/params/ There is a file, gr_iq2_pl.zip, containing 2 files - grpl.l and griq2pl.l The first of these uses the GR, the second the average of the GR and the inverse of the square root of 2. Some examples appear below. Paste them into a text editor, call the file griq2.l, put it into your Fractint directory and it'll be ready to use as type Lsystem. Unzip these files into your Fractint directory and they're ready to be used as type Lsystem. Enjoy. And please let me know what you think or if you have any questions. This is also the place to mention some ideas about colouring L-systems. One way to really jazz up an otherwise flat L-system can be the following type of colouring variation: Before the first f, put >1, making it >1f Before the last f, put <1, making it <1f Other combinations are possible, such as <1/>1, or if there are 4 fs,

1f >1f <1f <1f 1f <1f >1f <1f 1f <1f <1f >1f

I use this a few places in the above files, and much alsewhere also. Try it for yourself. I have coloured versions of many old spacefilling curves, the Dragon Curve and TwinDragon, etc. Tony Hanmer Tbilisi, Republic of Georgia pl03-- { ; Anthony Hanmer 2001 Angle 3; Golden Ratio plant series Axiom x x=f[+!@.6180339887499x-x] } pl03a { ; Anthony Hanmer 2001 Angle 3; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!-x]! } pl04+ { ; Anthony Hanmer 2001 Angle 4; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!+x] } pl04+++ { ; Anthony Hanmer 2001 Angle 4; Golden Ratio plant series Axiom x x=f[+!@.6180339887499x!+x] } pl04b { ; Anthony Hanmer 2001 Angle 4; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!-x]! } pl05++ { ; Anthony Hanmer 2001 Angle 5; Golden Ratio plant series Axiom x x=f[+!@.6180339887499x+x] } pl06- { ; Anthony Hanmer 2001 Angle 6; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!-x] } pl06+ { ; Anthony Hanmer 2001 Angle 6; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!+x] } pl08+++ { ; Anthony Hanmer 2001 Angle 8; Golden Ratio plant series Axiom x x=f[+!@.6180339887499x!+x] } pl08-c1 {; Anthony Hanmer 2001 Angle 8; Golden Ratio plant series Axiom x; colour 1 x=f[+@.6180339887499>1x!-x] } pl08e { ; Anthony Hanmer 2001 Angle 8; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!-x!-@.6180339887499x] } pl12a { ; Anthony Hanmer 2001 Angle 12; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!+++++x] } pl16- { ; Anthony Hanmer 2001 Angle 16; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!-x] } pl16+++ { ; Anthony Hanmer 2001 Angle 16; Golden Ratio plant series Axiom x x=f[+!@.6180339887499x!+x] } pl24+ { ; Anthony Hanmer 2001 Angle 24; Golden Ratio plant series Axiom x x=f[+@.6180339887499x!+x] } pl24b { ; Anthony Hanmer 2001 Angle 24; Golden Ratio plant series Axiom x x=f[+@.6180339887499x+x+x!+@.6180339887499x+x+x]! } _________________________________________________________________ The new MSN 8: smart spam protection and 2 months FREE* http://join.msn.com/?page=features/junkmail