User talk:Dvgrn
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Hmm, maybe this is where I should have put the stuff about Billiard Ball Earth that ended up in a blog.
Well, possibly this way some non-Wikipedian will find it vaguely interesting...
Just got sucked in to editing out a lot of duplication from the Conway's Game of Life article, came up for a breath of air after an hour and clicked "Submit", and found this paragraph still clutched in one grimy hand, like a mechanic putting a car all back together and having parts left over:
Life has a number of recognised patterns which emerge from particular starting positions. Soon after publication, several interesting patterns were discovered, including the ever-evolving R-pentomino (more commonly known as "F-pentomino" outside the game), the self-propelling "glider", and various "guns" which generate an endless stream of new patterns, all of which led to increased interest in the game.
Gliders and guns are defined elsewhere, I seem to recall the R-pentomino was studied before Gardner's article came out (and it's not "ever-evolving", really) -- but maybe there's a sentence or so remaining that I should work back in...
Well, it can probably wait until I dive in again to fix the funny numbering in Conway's three criteria (and whatever else pops up).
Have to figure out how to format this attempted contribution to the chaos game article:
To draw the above fractal fern using the chaos game, start with any X and Y values -- say X=0 and Y=0, but any values will work.
Choose the next step randomly, as follows:
85% of the time, take
new X' = .85X +.04Y new Y' = -.04X + .85Y + 1.6
(this affine transformation has the effect of moving any point one segment up the fern -- for example, it would move a point on the first stem segment to the same location on the second stem segment, or the tip of a leaflet to the tip of the next higher leaflet.)
7% of the time, take
new X' = .2X - .26Y new Y' = .23X + .22 + 1.6
(this maps the entire fern onto the first branch on the left -- the transformation is a slightly skewed counterclockwise rotation combined with a reduction in length by about a factor of three. For example, given a point on the stem of the fern at the bottom of the second segment, these equations produce a point on the fern's first right-hand leaflet, at the bottom of its second segment.)
7% of the time, take
new X' = -.15X +.28Y new Y' = .26X +.24Y + .44
(this maps the entire fern onto the first branch on the right. This is a clockwise rotation, mirror-reflection, and again a reduction in length to about a third of the original -- but just slightly larger than the left branch, as the numbers indicate.)
1% of the time, take
new X' = 0 new Y' = .16X
(this simply squashes any point on the fern into a straight line -- so (X', Y') will always end up somewhere on the segment at the base of the stem.)
Put a single dot at (X', Y'), then run the same process again with the new numbers as the next X and Y values. The first few dozen points are generally thrown away, to give the iterative process time to converge to a point somewhere on the fern -- or at least close enough to it that any discrepancy is invisible.
After thousands of iterations, the shape of the fern will emerge naturally, as if by magic. The only math required is the simple multiplication and addition described above.
To do: [Brief summary of how to generate the numbers in the above equations.] [Create screen shots for recursive golden-b tiling and a good two-transform spiral, or maybe a dragon.]
