Chaos 3 explanation, by Jud McCranie, February 10, 1989.

An explanation of the pattern generated randomly, as seen on Nova
- The science of Chaos, Jan. 31, 1989.  

The process is as follows : choose points A, B, and C (not on a
line).  Now choose an initial point (the "current point")
anywhere.  Repeat the following process: randomly choose either
A, B, or C.  Call the midpoint of that point and the current
point the new current point and plot it.  Repeat the process with
the new current point.

When the process is repeated many times, a surprising pattern
emerges.  The program CHAOS3 performs this process and plots the
points so you can see the pattern.  Some people have asked me for
an explanation of how you get such a pattern from randomness. 
This is a brief informal explanation that could be made rigorous.

The current point must eventually fall inside triangle ABC.  If
the current point is outside triangle ABC, successive points are
nearer to the triangle.  Once the current point is sufficiently
near ABC, as soon as each of A, B, and C have been chosen the new
current point must be inside ABC.

Now suppose the current point is anywhere inside ABC.  If point A
is now chosen, the next current point (the midpoint of the
current point and A) must be inside AEF.  Similarly, if B was
chosen, the next current point must be in BDE, or if C is chosen
it must be in CDF.  So once the current point is inside ABC, the
next current point cannot be inside DEF.

                                  A
                                  /\
                                /    \
                              /        \
                          E /------------\ F
                          / \            / \
                        /     \        /     \
                      /         \    /         \
                    /-------------\/-------------\
                  B               D                C

Now, without loss of generality, assume that the current point is
somewhere in AEF.  If point A is chosen, the midpoint (and next
current point) must be in AGH.  If point B is chosen, the next
current point must be in EIJ.  If point C is chosen, the next
current point must be in FKL.

                                  A
                                  /\
                              G /----\ H
                              / \    / \
                          E /-----\/-----\ F
                          / \     M      / \
                      I /-----\ J     K/-----\L
                      / \    /  \    /  \   /  \
                    /-----\/------\/------\/-----\
                  B       N       D       O        C

The cases where the current point was in BDE or CDF are
symmetrical, and cause the next current point to be inside BIN,
EGM, DKO, or CLO, FHM, or DJN, respectively.  So now the current
point can't be in DEF, GHM, IJN, or KLO. 

This process is repeated recursively (an infinite number of
times) on each of the triangles that the current point can be in
and it generates the pattern of allowed points.

Question : How much allowed area is left after the recursive
process is repeated an infinite number of times?

 