A Plane-Tiling Triple Spiral

It's possible to tile the plane with unequal "30-60-90" triangles
by arranging them according to the pattern shown below:

            

The numbers indicate the sequence of triangles according to size,
increasing by a factor of f = 1.085196156832515... on each step.
By examining the pattern of triangles we see that f is the real 
root of
                                     ___
                 2x^5  =  x^3    +  / 3

This arrangement can be generated by braid of three right-angled
spirals, as illustrated below:

            

By examining the pattern of triangles again we can see that each
segment of a spiral contains the vertices of one of the other spirals,
in a three-way arrangement, and the vertex cuts the segment into two
parts proportional to f^3 and sqrt(3) respectively.  Thus we need
only interpolate between the vertices of one spiral, with an 
interpolation factor of sqrt(3)/(2f^5) = 0.57542589..., to find 
the vertices of another spiral.