January 2004 - Fractint - mailman.xmission.com

Re: FOTD 28-01-04 (The Quintessential Midget [8])
by Jim Muth 28 Jan '04

28 Jan '04

At 11:39 AM 1/28/04 -0600, Paul Lee wrote: >Just curious, how come your 'Startup Parameters' >in the SSTOOLS.INI file do not already accomplish >such for your saved PAR files?? As in the >following: > > COMMENT=time=$calctime$--$vidkey$_on_a_P200 Yes, I have thought about it. I just have never gotten around to doing it. The problem is that I often calculate the FOTD images on other machines, and must convert the time to the P200 standard. >And, ever think of expanding your COMMENT >parameter to something like: > COMMENT=(c)_Jim_Muth_--_$date$_-_$time$_/ > time=$calctime$_on_a_P200_at_$xdots$x$ydots$_/ > Video=$vidkey$_using_FractInt_$version$.$patch$ I have thought of a lot of things I might do. I like to keep the parameter file uncluttered, and most of the above information is available elsewhere. But, if a great public outcry should arise, I could do this. Jim M.

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Jim, what is going on here?
by Hal Lane 28 Jan '04

28 Jan '04

######################################################### > FOTD -- January 28, 2004 (Rating 8) > Actually, the midget looks as though it could use a few more > iterations. This could easily be done, but if I raised the > maxiter, it would mess up the color palette, which is what > really makes the image. Jim, I tried raising the maximum iterations to 10,000 as I prefer well iterated, clearly defined midgets. And true to your promise -- some of the colors changed in the center 1/4 of the image. I too like the image coloring better when the midget is underiterated. But as I said, I prefer well iterated midgets. Is there a way for me to have my cake and eat it too? <---<< My experience when adding iterations to an inside=0 fractal is that a few more black pixels get colored in as they escape prior to the new higher maxiter value. I don't remember seeing existing pixels change color. I assume that by "mess up the color palette" you mean that some of the image *data* changed when maxiter was increased and thus got mapped to different colors... (Color palettes remain constant entities unless specifically changed -- which I did not do.) If my assumption is right, why do those pixels change color (iteration count)? <---<< In your image, presumably they were assigned a color because they escaped at an iteration count less than your maxiter=2400. Why would those pixels escape at a different iteration count when maxiter was raised? <---<< - Hal Lane ######################### # hallane(a)earthlink.net # ######################### ######################################################### --- Outgoing mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com) Version: 6.0.572 / Virus Database: 362 - Release Date: 1/27/04

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RE: [Fractint] Golden ratio
by Tony (Anthony) Hanmer 28 Jan '04

28 Jan '04

OK everyone, thanks for demystifying my remarks on the Golden Ratio. I'm glad to have it more fully explained, and would be the first to confess (if it weren't already obvious) that my math ability is somewhat less than my artistic ability. NOW, what do you think about the L-systems themselves?! Tony Hanmer _________________________________________________________________ Tired of spam? Get advanced junk mail protection with MSN 8. http://join.msn.com/?page=features/junkmail

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Some new tiles
by Tony (Anthony) Hanmer 27 Jan '04

27 Jan '04

The family of sets of tiles I describe below is the first of a few announcements I intend make this year in L-systems and IFS fractals, my patience with other ways of releasing my work having run out. I hope to see a renewed interest in these two types, which I see as underdeveloped and worthy of much more attention. S-Tiles There are 3 regular tiles in 2 dimensions: equilateral Triangle, Square and Hexagon. Take one side of any of these, a straight line. Distort it from this configuration in any manner which leaves it with 180-degree rotational symmetry, the same symmetry which is found in the letter S (or Z, or N). It must also be non-crossing (nc). Remake the tile using these altered lines as sides, and it will still function as a tile. I call these S-Tiles. (Does anyone have a better name?) Several examples are known already - you will find them in Benoit Mandelbrot's seminal book, "The Fractal Geometry of Nature", pp. 50-55. I include below a few of my own pieces. Paste them into a file, call it STs.l for example, save it in your Fractint directory and you can then run the fractals as type Lsystem. I have hugely expanded the family of sets as L-systems. The sets are infinite, because a line, the side of a tile, can be made as long and as convoluted as one wishes. (Fractint's capacity to produce these tiles, however, is not infinite, because it has a necessary limit on the number of characters in a line of L-systems code: 160 for versions up to the last developer's release, and 255 for the latter.) My L-systems file, STiles.l, is available zipped from http://spanky.triumf.ca/pub/fractals/params/ for those who are interested. Download and unzip it into your Fractint directory, and you're ready to explore these new tiles in Fractint; and to add more of your own. The easiest way to programme a side of any S-Tile, once you've drawn it on squared paper, is to write the set of L-system transformations ("f=f+f-ff"... etc) for the first half of the line; then reverse both the order of this line's fs and the polarity of its turns, + becomes - and - becomes +. This process could be automated to speed things up. Then, in your L-systems file, paste the 2 halves of the line (call them a and z) together, as az, one string. In your Axiom, write the required number of fs to make the tile, in different colours - e.g. a Square tile would have Axiom c9f+c10f+c12f+c14f if you used Angle 4, or c9f++c10f++c12f++c14f if you used Angle 8, etc. Then run the L-system to check that it is nc (non-crossing). The individual segments (fs) in one's side of an S-Tile need not have uniform length. Altering their lengths can increase or decrease their degree of convolutedness (which, I think, is their fractal dimension, is it not?). In some of the cases I've included in STiles.l, the Golden Ratio (phi: c. 1.6180339887499) is a limit which leads to tiles which touch themselves, and numbers slightly varying from this number lead either to not-quite-touching tiles (the goal), or to self-crossing curves. How could one automate the process of calculating fractal (Hausdorff) dimension for S-Tiles, especially for those whose segments are not uniform in length? These tiles can also be made in a vector drawing programme like CorelDraw or Illustrator, as closed curves; then filled, copied and tiled. But the number of "nodes" (segment endpoints) can quickly grow beyond such programmes' ability to handle, as I have found. More than 50000 nodes and the thing may crash. Further variations, less "orthodox", could have islands and lakes, making them into dusts - discontinuous curves, though they would still need to be tiles. Another type of variation will be the subject of a future announcement - it is a whole family of infinite sets in its own right, and even more fascinating than the whole S-Tiles family of sets, I think. S-Tiles should also work in *3* dimensions, with the 5 regular polyhedra (and with regular polytopes in all higher dimensions!), though I've yet to begin such experiments, not having the programming skills or the programme itself to do them... Anthony Hanmer Tbilisi, Republic of Georgia PS More news to follow - this is only the start. ST30016* { ; Anthony Hanmer 7/2002 Angle 6 ; nc; and 3 together make a tile Axiom c9f++c12f++c14f f=@2f@.5--f++f+@2f@.5-f--f++@2f@.5 } ST30023* { ; Anthony Hanmer 7/2002 Angle 6 ; nc; and 3 together make a tile Axiom c9f++c12f++c14f f=ff+f+f--f-f-f+f+f++f-f-ff } ST30040* { ; Anthony Hanmer 7/2002 Angle 6 ; nc; and 3 together make a tile Axiom c9f++c12f++c14f f=@2f@.5+f++f--@2f-f--f++f+f@.5++f--f-@2f@.5 } ST40001* { ; Anthony Hanmer 2000 (was T105aa, my first ST and first square ST) Angle 4; non-crossing (nc); and 4 together make a tile Axiom c9f+c10f+c12f+c14f f=ffff+fff+ff+f+f-f-ff-fff-ffff } ST40009* {; Anthony Hanmer 6/2002 (was T105aa03b) Angle 4 ; nc; and 4 together make a tile Axiom c9f+c10f+c12f+c14f f=f+@iq2f+@iq2f-@q2f-@q2f } ST40465* { ; Anthony Hanmer 7/2002 Angle 4 ; nc; and 4 together make a tile Axiom c9f+c10f+c12f+c14f f=f+f-f+f-f-f+f-f-@2f@.5+f+f-f+f+f-f+f-f } ST60020* { ; Anthony Hanmer 6/2002 Angle 6 ; nc; and 6 together make a tile Axiom c1f+c2f+c3f+c4f+c5f+c6f f=ff+f+f+ff-f-f-ff } ST60059* { ; Anthony Hanmer 6/2002 Angle 6 ; nc; and 6 together make a tile Axiom c1f+c2f+c3f+c4f+c5f+c6f f=f-f+f+@2f@.5+f+f-f-@2f@.5-f-f+f } ST60062* { ; Anthony Hanmer 6/2002 Angle 6 ; nc; and 6 together make a tile Axiom c1f+c2f+c3f+c4f+c5f+c6f f=f-ff+f+f+f+f-f-f-f-ff+f } _________________________________________________________________ MSN 8 helps eliminate e-mail viruses. 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Golden ratio
by Raymond Filiatreault 27 Jan '04

27 Jan '04

Tony Hanmer wrote: 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. First, the ratio is 1.61803398874989... Secondly, your two equations 1/x = x-1 and x^2 = x+1 are just variants of one another. In other words, they are identical and any solution will obviously satisfy *both*. They can also be written as: x^2-x-1=0. Solving for x yields two roots, one being the positive and referred to as the Golden ratio, the other being negative and equal to -0.61803398874989... Raymond Filiatreault _________________________________________________________________ Protect your PC - get McAfee.com VirusScan Online http://clinic.mcafee.com/clinic/ibuy/campaign.asp?cid=3963

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A couple more
by Tony (Anthony) Hanmer 27 Jan '04

27 Jan '04

Another pair of small L-systems files I have included below in their entirety are Trvar.l and TTiles.l. The former is another set of my variations on others' work - Trees, Bushes and Plants. The latter contains a few tiles I have found - some of which may be new, others not. A few of them are also mlr - multiple-layer redundant; in other words, they have several layers of the same thing on top of each other, and are in need of simplification to remove this extra unnecessary repetition. But all of them are interesting. If any of the tiles are, in fact, my rediscoveries of others' tiles, I'd like to be informed about it. Paste the files separately into a text editor and call them Trvar.l and TTiles.l; then save them into your Fractint directory and you're ready to use their contents as type Lsystem. Tony Hanmer Tbilisi, Republic of Georgia Trvar.l _README {; Anthony Hanmer 2001 angle 4; Variations on Trees, Bushes and Plants axiom f f=f } Tree01 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling tree X=@iq2[-fx]+fx } Tree01c1 {; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4 ; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx ; a pseudo-spacefilling tree, colour version 1 X=@iq2[>1-fx]<1+fx } Tree01c2 {; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4 ; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx ; a pseudo-spacefilling tree, colour version 2 X=@iq2[>1-fx]>1+fx } Tree02 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to inverse of sqrt 2 - axiom +++FX; a Levy's Curve variation X=@iq2[--FX]++FX } Tree02a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to Golden Ratio - axiom +++FX; X=@.6180339887449[--FX]++FX } Tree03 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; a Levy's Curve variation X=@iq2[-FX]+FX } Tree03a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to Golden Ratio - axiom ++FX; X=@.6180339887449[-FX]+FX } Tree04 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; X=@iq2[--FX]+FX } Tree04a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to Golden Ratio - axiom ++FX; X=@.6180339887449[--FX]+FX } Tree05 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; Twindragon tree X=@iq2[---FX]+FX } Tree05a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to Golden Ratio - axiom ++FX; X=@.6180339887449[---FX]+FX } Tree06 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; mirror image of Tree04 X=@iq2[-FX]++FX } Tree07 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; Twindragon tree, mirror image of Tree05 X=@iq2[-FX]+++FX } Tree08 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; X=@iq2[--FX]+++FX } Tree08a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to Golden Ratio - axiom ++FX; X=@.6180339887449[--FX]+++FX } Tree09 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 - axiom ++FX; mirror image of Tree08 X=@iq2[---FX]++FX } Tree10 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling... smoke? X=@iq2[-gx]+fx } Tree10c1 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling... smoke? (Colour version 1) X=@iq2[>1-gx]<1+fx } Tree10c2 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling... smoke? (Colour version 2) X=@iq2[>1-gx]>1+fx } Tree10a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to Golden Ratio - axiom +fx; X=@.6180339887449[-gx]+fx } Tree11 { ; Anthony Hanmer 2001, adapted from angle 8; Tree10, angle changed to 8 - Twindragon tree axiom ++FX X=@iq2[---GX]+FX } Tree12 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling ... dust? X=@iq2[-fgx]+fx } Tree13 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling ... dust? X=@iq2[-fgx]+fgx } Tree14 { ; Anthony Hanmer 2001, adapted from angle 8; Tree12, angle changed to 8 - Twindragon tree axiom ++fx X=@iq2[---fgx]+fx } Tree15 { ; Anthony Hanmer 2001, adapted from angle 8; Tree1, angle changed to 8 - Twindragon tree axiom ++fx X=@iq2[---fgx]+fgx } Tree16 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to .615 axiom +++fx X=@.615[--fx--]++fx++ } Tree17 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, angle changed to 6, @ to .5 axiom fx X=@.5[fx+]@2fx[+fx] } Tree18 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, angle changed to 6, @ to .615 axiom fx X=@.615[fx-][fx][+fx] } Tree19 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to .615 axiom ++++++fx X=@.615[fx--][--fx++] } Tree20 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to .615 axiom fx++fx X=@.615[fx-][-fx+] } Tree21 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to .5 axiom fx; X=@.5+fx[-fx]@2+fx } Tree22 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to .615 axiom +++++fx X=@.615[gx--][--fx++] } Tree23 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to .5 axiom fx; X=@.5+fx[-gx]@2+fx } Tree24 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to .5 axiom fx; X=@.5+fx[-fx]@2+gx } Tree25 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to .5 axiom fx; X=@.5+gx[-fx]@2+gx } Tree26 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to .615 axiom ++++++fx X=@.615[fx--][--fgx++] } Tree27 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to .615 axiom fx X=@.615[fx-][fgx][+fx] } Tree28 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, angle changed to 6, @ to .615 axiom +++fx X=@.615[fx-][-fx+] } Tree29 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 3; Tree1, angle changed to 3, @ to .615 axiom [fx]+[fx]+[fx]; X=@.615[-fx]+fx } Tree30 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 axiom ++fx; X=@iq2fx[-fx+]@q2fx } Tree30a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 axiom fx-fx-fx-fx; X=@iq2fx[-fx+]@q2fx } Tree30b { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 axiom fx++fx; X=@iq2fx[-fx+]@q2fx } Tree31 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4 axiom ++fx; X=fx[-fx+]fx } Tree31a { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4 axiom fx-fx-fx-fx; X=fx[-fx+]fx } Tree32 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling... dust? X=@iq2[-gfxg]+gfxg } Tree33 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling tree X=@iq2[-ffx]+fx } Tree33c { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling tree (colour version) X=@iq2[>1-ffx]<1+fx } Tree34 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, angle changed to 4, @ to inverse of sqrt 2 - axiom +fx; a pseudo-spacefilling... dust? X=@iq2[-fgfx]+fx } Tree35 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to inverse of sqrt 2 - axiom +++fx; X=@iq2[--ffx]++fx } Tree36 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 12; Tree1, angle changed to 12, @ to inverse of sqrt 2 - axiom +++fx; X=@iq2[--fgfx]++fx } Tree37 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 axiom +fx; X=@iq2[--ffx]+fx } Tree38 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 axiom +fx; X=@iq2[--fgfx]+fx } Tree39 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 axiom +fx; X=@iq2[-ffx]++fx } Tree40 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 8; Tree1, angle changed to 8, @ to inverse of sqrt 2 axiom +fx; X=@iq2[-fgfx]++fx } Tree50 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@iq2[--fx]+fx } Tree51 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@iq2[--fx]++fx } Tree52 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@iq2[--fx]-fx } Tree53 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@iq2[-fx]++fx[-fx] } Tree54 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@iq2[--fx]++fx[--fx] } Tree55 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@iq2[--fx][++fx]--fx } Tree56 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 6; Tree1, axiom fx; X=@.615[-fx]+@i.615fx } Tree57 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, axiom +fx; X=@iq2[-fx]+@q2fx } Tree58 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 3; Tree1, axiom fx; X=@.615[-fx]+fx } Tree59 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 3; Tree1, axiom fx; X=@.615[-gx]+fx } Tree60 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, axiom +x; a spacefilling tree x=f@iq2[-x!]+fx } Tree61 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, axiom x; x=f@iq2[-x]!+fx } Tree62 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, axiom +x; x=f@.68[+x!]+fx } Tree63 { ; Anthony Hanmer 2001, adapted from Adrian Mariano's angle 4; Tree1, axiom +x; a spacefilling tree x=f@iq2[+x]!+fx } Plant01a { ; Adrian Mariano, from the Algorithmic Beauty of Plants ; Plant-like structure, figure 1.24a p.25 ; also p.285a The Science of Fractal Images angle 4 axiom +f f=F[+F]F[-F]F } Plant01b { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +f f=F[+F]F[-F] } Plant01c { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +f f=F[+F]@iq2F@q2[-F]F } Plant01d { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +f f=[+F]F[-F]F } Plant01e { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +f f=F[+F]+F[+F]-F } Plant01f { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom f+f+f+f f=F[+F]+[-F]-F } Plant01g { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +f f=F[+F]-[-F]+F } Plant01h { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom f f=F[+F]@.5F@2[-F]F } Plant01i { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=F[+F]F[-F]F } Plant01ia { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f]+[f]+[f]+[f]+[f]+[f] f=F[+F]F[-F]F } Plant01j { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=F[+F]F[-F] } Plant01ja { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f]++[f]++[f] f=F[+F]F[-F] } Plant01jb { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f]+[f]+[f]+[f]+[f]+[f] f=F[+F]F[-F] } Plant01k { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=F[+F]@iq2F@q2[-F]F } Plant01ka { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f]+[f]+[f]+[f]+[f]+[f] f=F[+F]@iq2F@q2[-F]F } Plant01l { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=[+F]F[-F]F } Plant01m { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=F[+F]+F[+F]-F } Plant01ma { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f]+[f]+[f]+[f]+[f]+[f] f=F[+F]+F[+F]-F } Plant01mb { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f--f]+[f--f]+[f--f]+[f--f]+[f--f]+[f--f] f=F[+F]+F[+F]-F } Plant01n { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f+f+f+f+f+f f=F[+F]+[-F]-F } Plant01o { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=F[+F]-[-F]+F } Plant01p { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom f f=F[+F]@.5F@2[-F]F } Plant01pa { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 6 axiom [f]+[f]+[f]+[f]+[f]+[f] f=F[+F]@.5F@2[-F]F } Plant02 { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +f f=F[+F]F[-F][F] } Plant03 { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom f f=FF-[-F+F]+[+F-F] } Plant05 { ; Adrian Mariano, altered by Anthony Hanmer 2001 angle 4 axiom +x X=f[+X][-X]FX F=FF } Plant07 { ; Ken Philip, altered by Anthony Hanmer 2001 axiom +Z Z=ZGX[+Z][-Z] X=X[-FFF][+FFF]FX angle 4 } Plant07a { ; Ken Philip, altered by Anthony Hanmer 2001 axiom +Z Z=ZGX[+Z][-Z] X=X[-GFG][+GFG]FX angle 4 } Plant10a { ; Adapted from Adrian Mariano's Plant10, Anthony Hanmer 2001: axiom +f ; a tile f=f[+ff][-f]f[+f][-ff]f angle 4 } Plant10aa { ; Adapted from Adrian Mariano's Plant10, Anthony Hanmer 2001: axiom c9f+c10f+c12f+c14f ; a tile f=f[+ff][-f]f[+f][-ff]f angle 4 } Plant10b { ; Adapted from Adrian Mariano's Plant10, Anthony Hanmer 2001 axiom +f f=f@2[+f]@.5[-f]f[+f][-f]f angle 4 } Plant10c { ; Adapted from Adrian Mariano's Plant10, Anthony Hanmer 2001 axiom +f f=f@2[+f][-f]f[+f][-f]@.5f angle 4 } Plant10d { ; Adapted from Adrian Mariano's Plant10, Anthony Hanmer 2001 axiom +f f=f@q2[+f][-f]f[+f][-f]@iq2f angle 4 } Bush01 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom +F F=FF-[-F+G+F]+[+F-G-F] } Bush01a { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom F+F+F+F F=FF-[-F+G+F]+[+F-G-F] } Bush01b { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom [F]+[F]+[F]+[F] F=FF-[-F+G+F]+[+F-G-F] } Bush02 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom +F F=FF-[-F+G+]+[+F-G-] } Bush02a { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom [F]+[F]+[F]+[F] F=FF-[-F+G+]+[+F-G-] } Bush03 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom +F F=FF+[-F-FG]-[+F+FG] } Bush03a { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom F++F F=FF+[-F-FG]-[+F+FG] } Bush03b { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 4 Axiom [F]++[F] F=FF+[-F-FG]-[+F+FG] } Bush04 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom F F=FF-[-F+G+F]+[+F-G-F] } Bush05 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom F F=FF--[-F+G+]++[+F-G-] } Bush05a { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom [F]+[F]+[F]+[F]+[F]+[F] F=FF--[-F+G+]++[+F-G-] } Bush06 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom F F=FF-[-F+G+]+[+F-G-] } Bush06a { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom F+++F F=FF-[-F+G+]+[+F-G-] } Bush06b { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom F+F+F+F+F+F F=FF-[-F+G+]+[+F-G-] } Bush06c { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom [F]++[F]++[F] F=FF-[-F+G+]+[+F-G-] } Bush06d { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom [F]+[F]+[F]+[F]+[F]+[F] F=FF-[-F+G+]+[+F-G-] } Bush07 { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom F F=FF--[-F+G+F]++[+F-G-F] } Bush07a { ; adapted from Adrian Mariano's Bush, Anthony Hanmer 2001 Angle 6 Axiom [F]++[F]++[F] F=FF--[-F+G+F]++[+F-G-F] } ____________________________ TTiles.l _README {; Anthony Hanmer 2000-2002 Angle 4; Miscellaneous tiles, some of them perhaps not new. Axiom f f=f } ADH0794 { ; Anthony Hanmer 2000 Angle 6 ; a tile Axiom f f=+@.5ff+++ff++@2g } ADH01782 { ; Anthony Hanmer 2000 Angle 4 ; DC type Axiom f f=ff++f+f+f } adh6280 { ; Anthony Hanmer 2001 Angle 3 ; a tile; rounding needed Axiom f ;Axiom c9f+c12f+c14f; 3 colours ;Axiom c32[f]+c35[f]+c39[f]+c42[f]+c44[f]+c2[f]; 6-coloured star f=f+f-f-f+f+f-f ;f=>1f+f-f-f+f+f-<1f; colour 1 } adh6543 { ; Anthony Hanmer 2001 Angle 4 ; a new tile! Axiom f ; (though massively layer-redundant, or mlr) ;Axiom c9f|c12f f=+!ff|ff-!@2f@.5 ;f=+!>1ff|ff-!@2<1f@.5; colour 1 ;Axiom x ; (6543 using x instead of f) ;x=+!fx|fx-!@2f@.5 ; Rhombic tile of 1:2 bricks } adh6544 { ; Anthony Hanmer 2001 Angle 8 ; The new tile's Angle can be changed Axiom f ; (mlr) f=+!ff|ff---!@2f@.5 } adh6545 { ; Anthony Hanmer 2001 Angle 8 ; The new tile's Angle can be changed Axiom f ; (mlr) f=+++!ff|ff-!@2f@.5 } adh6672 { ; Anthony Hanmer 2001 Angle 4 ; a tile Axiom +x x=f[@.5x]+![@.5x]+!+!f[@.5x]-![@.5x] ;x=>1f[@.5x]+![@.5x]+!+!f[@.5x]-![@.5<1x]; colour 1 } adh6748 { ; Anthony Hanmer 2001 Angle 4 ; tile w/td"vines", sides ratio 1:GR Axiom x x=f+![@.6180339887499x]+[@.6180339887499x]|[@.6180339887499x] ;x=>1f+![@.6180339887499x]+[@.6180339887499x]|[@.6180339887499<1x]; colour 1 } adh8320 { ; Anthony Hanmer 6/2002 Angle 3 ; nice overlapping hex tile Axiom f f=f+@2f@.5-f ;f=>1f+@2f@.5-<1f; colour 1 } adh8735a { ; Anthony Hanmer 7/2002 Angle 6 ; a hexagonal (tile) echo of 8735 Axiom f f=f+f|@2f@.5|f-f ; Hausdorff dimension: log(5)/log() = } adh9622 { ; Anthony Hanmer 19/10/2002 Angle 6 ; I BELIEVE this is a tile... Axiom f ; but it's hard to prove. f=f++f--g++f|f--g++f+g g=gg } T325 { ; Anthony Hanmer 2000 Angle 4 ; DC type Axiom f;+f+f+f ;Axiom c9f-c10f-c12f-c14f; colour 1 ;Axiom c09f-c12f-c09f-c12f--c14f+f+f+f; colour 2 f=f-ff++f-f } _________________________________________________________________ The new MSN 8: advanced junk mail protection and 2 months FREE* http://join.msn.com/?page=features/junkmail

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FOTD 27-01-04 (Bouquet of Numbers [8])
by Jim Muth 27 Jan '04

27 Jan '04

FOTD -- January 27, 2004 (Rating 8) Fractal visionaries and enthusiasts: After yesterday's exciting adventure in the world of Z^1.003, I decided to subtract Z^(1.003) from Z^(-1.003) and explore the fractal world created by this combination. Before starting, I had to decide on the escape radius, which has a significant effect on fractals in this range. I chose the escape radius of 250 entirely at random. Apparently, I chose well. The image I found rates an 8 and has an overall value of 43. The resulting parent fractal is an almost-symmetrical Mandeloid rotated 90 degrees. Perhaps I should describe the parent as part of a Mandeloid, for the northern area of the Mandeloid is obliterated by the blue disk that usually appears at the origin of fractals with negative exponents of Z, and is likely an artifact. It's no problem however, because today's scene is located in the East Valley area, which lies on the south side of the Mandeloid. Much of the flowery effect resulted from rendering the scene with the outside set to 'summ' rather than the usual 'iter'. The name "Bouquet of Numbers" came to me as I sat in wonder that all this came from such a simple beginning. The render time of 18 minutes is bearable, but a far better way of viewing the image is to download it from Paul's FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> With the yard buried 20cm deep in snow and ice, and the tempera- ture lurking around 19F -7C, the fractal cats passed an unevent- ful day Monday curled by the nearest source of heat. Today is starting with freezing drizzle, so I predict another uneventful day for the cats. For me, the work is pretty much caught up. Combined with the abysmal outside conditions, this should give me plenty time to ensconce myself indoors and find another good fractal. If I do, the fractal will appear as tomorrow's FOTD. Until then, take care, and beware of extrasensory fractals. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Bouquet_of_Numbers { ; time=0:18:46.52--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=MandelbrotMix2 function=recip passes=1 center-mag=+0.00045100966909858/-0.356152672378903\ 30/6.50686e+009/1/90/3.88578058618804789e-016 params=-1/1.003/1/-1.003/0/150/0/0 float=y maxiter=3000 inside=0 outside=summ periodicity=10 colors=000fbZfb_fc_fc`fd`fe`feaffaffbfgbfgcfhcficf\ idfjdfjefkefkedhfcffbdgabg`_h_YhZWhYUiXRiWPjVNjULj\ TIkSGkRElQClR8kPAlOBlNDmMEmLFnJHnIIoHJoGLpFMpDNpCP\ qBQqARr9Tr7Us6Vs5Xt4Yt0_z2Zw3Zt4Zq5Zn6Zk8Zh9ZeAZbB\ Y`UYYUYVXZSX_P_`M`aJcbGccEhbGh`Ic_KcZMZYNZXPXVRXVT\ UTVUUFUUOUSWSIdQBo39l48i47f55c64`52Y63Z74_85`96`A7\ aA8bB9bCAcDAdEBdECeFDfGEfHFgIGhIHiJIiKIjLJkMKlMLnN\ MpONqPOsQPuSQxRPvQPtPPsOOqNOpMOnMNlLNlKNkJMkIMjHMj\ HLiGLhFLhEKgDKgCKfDHdCKfBNgAQh9Ti9Wk8Zl7am6dn3gq6f\ o8fnBemDelGekIdiLdhNdgQcfSceUcdXbbZbaaa`ca_faZh`Xk\ `Wm`Vp_Ur_Tt_SvaRxbRzfR000000000000HDHHJEHSBH_8Hf5\ Hn2Ld6PZ9TWDXWG`TKdQNhORlLUpIYsG`tIatKatLatNbtObtQ\ btRctTctUctWduXduZdu_euaeubeudfuefugfuhgujgukgvhev\ fdvdcwbbw_awY_xWZxUYxRXyPWyNUyLTzISzGRzEQzCPyERyGS\ yHTxJUxLWxMXxOYwQZwR`wTawUbvWcvYdvZfu`gubhuciuektg\ luinukovmqtmqsmqrnrqnrmhe } frm:MandelbrotMix2 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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Golden Ratio and related plants
by Tony (Anthony) Hanmer 27 Jan '04

27 Jan '04

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

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Derivations
by Tony (Anthony) Hanmer 27 Jan '04

27 Jan '04

The file derivations.zip at http://spanky.triumf.ca/pub/fractals/params/ contains the following L-systems files: alfovar.l feather.l leafvar.l m09vars.l schnvar.l t034ifs!.l These are some of my early variations on the L-systems of others. As I began exploring the amazing world of L-systems programming, this was a great way for me to play and to learn. I appreciate the opportunity to build on other people's hard work and still produce something new, recognising that I am only altering an existing L-system. (Some examples appear below. Paste them into a text editor, call the file deriv.l, put it into your Fractint directory and it'll be ready to use as type Lsystem.) I also wish to thank especially William McWorter and Morgan Owens for some very useful correspondence over the last few years which has helped me with some key questions usually relating to L-systems. I STILL have lots of blank spots in my understanding, but that's not your fault! THANK YOU VERY MUCH. Tony Hanmer Tbilisi, Republic of Georgia Alfonseca029 { ; Manuel Alfonseca rendered for fractint Angle 4 ; altered by Anthony Hanmer 2001 Axiom +A A=f-[A+A]+f[+fA]A f=ff } Alfonseca097 { ; Manuel Alfonseca rendered for fractint Angle 12 ; altered by Anthony Hanmer 2001 Axiom A A=f-[A+A]+!f[+!fA]-A f=ff } Alfonseca120 { ; Manuel Alfonseca rendered for fractint Angle 16 ; altered by Anthony Hanmer 2001 Axiom A A=f-[A+A]+f[+fA]-!A f=ff } _________________________________________________________________ Tired of spam? Get advanced junk mail protection with MSN 8. http://join.msn.com/?page=features/junkmail

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FOTD 26-01-04 (Spiral Mesa [8])
by Jim Muth 26 Jan '04

26 Jan '04

FOTD -- January 26, 2004 (Rating 8) Fractal visionaries and enthusiasts: In the discussion of the "Whirling Devilish" fractal of two days ago I remarked about the straight edge cutting through the upper right portion of the image. I commented that some very unusual chaos lies in this area but that the exploration there is too slow to be practical. Well, for today's image I have turned impractical and wandered into the chaos along the straight-edge. Today's FOTD is what I found there. I have rarely seen an image with such contrasting parts, the dull reddish rocky lower part contrasting as starkly as it does with the ethereal blue celestial spiral filling the upper part. The lower part of the picture reminds me of one of those mesas found in the southwest desert area of the U.S. The upper part reminds me of nothing other than a fractal spiral. Combining the two impressions, I named the image "Spiral Mesa". I feel that the rating of an 8 is justified, though the extreme slow- ness holds the overall value to a lowly 4. Perhaps the most incredible thing about the image is that it is produced by Z^1.003. The graph of this expression is basically a straight line, and linear expressions do not make fractals. It is only at very large values of Z that the slight curve in the graph becomes noticeable, which is why a maxiter of 20000 is necessary to reveal the chaos. An unfortunate effect of the high maxiter is the 3-hour render time. The wait can be eliminated however by downloading the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> I sent the image file to Paul several hours ago, so the image should be posted shortly after he receives this discussion. As a matter of interest, I made a 0.65-degree change in the rotation of the image since I sent it. This was done so that the border between sky and mesa would be level. To see a perfectly straight horizon, run the parameter file; to see the image without a wait of several hours, download it. The only difference is in the orientation, which is barely noticeable. Thickening clouds and cold temperatures prevailed all day here at Fractal Central on Sunday. The high temperature of 18F -8C and lack of sun kept all thoughts of outdoors from the minds of the fractal cats. They were quite content to lie by the heat source and occasionally ask for food. This morning we have 6in 15cm of fresh snow on the ground and a temperature of 17F -8C, which makes the outdoor conditions worse. The forecast is for freezing rain, sleet and more snow. I feel sorry for the duo. There is no reason to feel sorry for myself however. The work is caught up, leaving me little to do but sweep the powdery snow and explore the world of fractals. Until next time, take care, and don't forget to remember. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Spiral_Mesa { ; time=3:08:09.19--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=MandelbrotBC1 function=floor passes=1 center-mag=+0.84105158773950410/-0.082216007167982\ 33/10307.65/1/-176.85/6.14521628028441569e-012 params=1.003/0/1040/1500 float=y maxiter=20000 inside=255 logmap=4824 periodicity=10 colors=000_00ODNQCMPBLT9JU8IV7HX5GY4FX2FZ3EZ4D_4D_\ 5C`6BW6BY7AZ8A_89a98bA8dA7eB7bC6hC5eD5kE4hF3lE4gR5\ lE6gT7kE8aD9kDAfDBkDCjQDjDEeOFjCG_CHiCIiRJiCKdULhU\ McBNhBOcBPhBQbCPgDPgPPgPOgQOgUObUOgUNbUNgUNgYNgUMg\ NMgNMYOMgPLgQLgRLYSLgTKgTKgUKgVKgWJhXJiYJg0EkYJa1N\ qYSdJWvY`wUexYddIcxdbyAayk`yo_yrZrnUkkQdhMYeIRbEK_\ ALZCMYEMYFNXHNWIOWKOVLPVNPUPQTQQTSRSTRSVSRWTQYTQ_U\ P`UPbVOcVNeWNfWMhXMjXLkYKmYKnZJpZJq`KpaLpbMpdNoeOo\ fPohQoiRnjRnlSnmTmnUmpVmqWmrXltYluZlvZltYmrYmqXmoX\ mmXmlWmjWmhWmgVmeVmdUmbUm`Um_TmYTmWTnVSnTSnRSnQRnO\ RnNQnLQnJQnIPnGPnEPnDOnBOnAOnEQpIRqJQtLSuMTvNVwOWx\ PXyRZzS_zTazUbzVczXezYgzZiz_kz`mzbozcpzdqzerzfszht\ ziuzjvzkwzlwzkwzjvziuzhtzgszfrzeqzdpzcozbnzamz`lz`\ kz_jzZizYgzXezWczVazU_zTYzSWzRUzQUzPUzQUzRUzSUzTUz\ UUzVUzWUzTUzQUzNVzLWzI_zFdzDhzAlz7pz5tzCuzIvzOwzVw\ z`xzfyznzzlyzjxzhwzfvzemz } frm:MandelbrotBC1 { ; by several Fractint users e=p1, a=imag(p2)+100 p=real(p2)+PI q=2*PI*fn1(p/(2*PI)) r=real(p2)-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 20.0 PAR-FORMULA FILE==================================

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