Flame Math


Scott Draves 3/6/2003









Iterated Function Systems

A kind of fractal.

Earliest example is Cantor's Dust (1882)


All base-3 numbers in [0,1] without a 1.



Sierpinski generalized to 2D set made of several copies of itself:





Used to model natural forms by Michael Barnsley (1980s)








Recursive Set Equations

S = U fi(S)
where

fi(x,y) = (ax+by+c, dx+ey+f)
where

fi are contractive.








How to Solve It?

If p is in S then fi(p) is in S.
And so is fj(fi(p)) and fk(fj(fi(p))).

And every point of S is arbitrarily
close to some sequence ijkl...

Algorithm I

draw point (x,y)
pick random i
(x,y) = fi(x,y)
iterate






Algorithm II

since maps are contractive,
any starting point moves toward
S exponentially.  So just pick a
random start and drop the first
20 iterations





Flame

That was the state of the art in 1991.
I made three critical improvements:
  1. Generalized Functions
  2. Log Density Histogram
  3. Structural Coloring



Generalized Functions

fi(x,y) = V(ax+by+c, dx+ey+f)

V0(x,y) = (x,y)                   linear
V1(x,y) = (sin x, sin y)       sinusoidal
V2(r,h) = (1/r, h)                spherical
V3(r,h) = (r, 2h)                 horseshoe
V4(r,h) = (r, r+h)               swirl
V5(x,y) = (h/pi, r-1)           polar

and V(x,y) = Sum wjVj(x,y)
where Sum wj = 1



The Genetic Code

up to 6 functions (xforms)
6 affine coefficients
7 variation coefficients
1 weight

6(6+7+1) = 84 dimensions



Log Density Histogram

Normal IFS algorithm: draw point, iterate.
mono 149

The problem with that is there is information lost
every time you draw a point that has already been drawn.
The obvious solution is to increment a counter and
make the intensity of the image proportional to the
count.  But this doesn't work!





Power Law

The problem is that the density distribution
follows the power law (frequency inversely
proportional to intensity):  a few pixels are white
and the rest are left very dark, effectively black.
We want to visually distinguish 10 from 30 and
1000 from 3000 and 1000000 from 3000000.
So take a logarithm:





Color by Structure

We want continuous color gradation assigned structurally.
Naturally we want to use a color map, [0,1] -> (r,g,b)


Location in structure identified by our series
of choices for f:  ijklmn, etc.  Treat it as a
fractional binary number with most recent as high-order
bit.  This is equivalent to adding a 3rd coordinate
to the fractal which is the color index.
This work is licensed under a Creative Commons License by spot at draves dot org.