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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