The Orbit of Triangles

Let A,B,C be the verticies of an arbitrary non-equilateral triangle 
with edge lengths AB=a, AC=b, and BC=c.  Place the edge AB on the 
segment [0,a] of the x-axis of a cartesian coordinate system and mark 
the location of the C vertex.  Then place AC on the x-axis at [0,b] 
mark the location of B.  Finally, place BC on the x-axis at [0,c] 
and mark the location of A.  The three marks constitute a new 
triangle, with edge lengths a', b', and c'.

This process can be repeated indefinitely.  At each stage it is
convenient to normalize the size of the triangle by magnifying or
shrinking it so the perimeter has length 1.  The sequence of triangles
produced by iteration of this procedure then consists of a sequence
of triples {a,b,c} with a+b+c=1.  Taking a,b,c as coordinates, each
triangular shape can be plotted in 3-D space, and the condition
a+b+c=1 implies that all the points fall on a single plane.

The attractor of these points is a beautiful closed curve that somewhat
resembles a trefoil knot, as shown below.


A few people have asked for some clarification of the construction,
noting that in some implementations the constructed triangles seem
to degenerate into straight lines.  This is due to the choice of
permutations for the edge lengths.

To clarify, each step begins with the three edge lengths {a,b,c} of a
given triangle and produces a new set of edge lengths {a',b',c'} of a
new triangle.  Let [a], [b], [c] denote the verticies of the new 
triangle constructed by placing the edges a, b, and c respectively 
on the x axis.  Then let [a][b] denote the length of the line segment 
from [a] to [b], and so on for the other segments.

The edge lengths of the new triangle (prior to normalization) are then 
given by

                        /                 (b^2 - c^2)^2 - (ac)^2 \
   q =   [a][b] =  SQRT( a^2 - ab + c^2 + ----------------------  )
                        \                            ab          /


                        /                 (a^2 - b^2)^2 - (bc)^2 \
    r =  [a][c] =  SQRT( b^2 - ac + c^2 + ----------------------  )
                        \                            ac          /


                        /                 (a^2 - c^2)^2 - (ab)^2 \
    s =  [b][c] =  SQRT( a^2 - bc + b^2 + ----------------------  )
                        \                            bc          /


Therefore, the normalized egde-lengths of the new triangle are the 
three values given by

          q/(q+r+s)        r/(q+r+s)         s/(q+r+s)

We have six choices for how to assign these values to the new set of
edge lengths {a',b',c'}.  These choices are shown below

                        [a][b]    [a][c]    [b][c]
                        ------    ------    ------
                 1        a'        b'        c'
                 2        b'        a'        c'
                 3        c'        a'        b'
                 4        a'        c'        b'
                 5        b'        c'        a'
                 6        c'        b'        a'

Choices 1, 3, and 5 each produce the "trefoil attractor".  If we 
choose 2 or 4 the resulting iteration converges on a 3-cycle of 
segments of length {0.309016, 0.190983, 0.500000}, which corresponds 
to a degenerate triangle with three co-linear verticies, one of which
cuts the segment connecting the other two in the "golden proportion"
phi = 1.61803....  If we choose permutation 6 the sequence converges
on a 1-cycle of this same "golden" line segment.

Taking choice 1, we have

        a' = q/(q+r+s)       b' = r/(q+r+s)      c' = s/(q+r+s)

Taking these values as the xyz coordinates of a point in space, and
noting that a'+b'+c' = 1, we see that each point falls on a plane.
Therefore each point {a,b,c} can be plotted in two dimensions by means 
of the transformation

             x  =  (a - b)/sqrt(2)

             y  =  (a + b - 2/3)/sqrt(6)  -  (c - 1/3) sqrt(2/3)

This gives the "trefoil attractor" described above.