Read Me.

Both pictures are drawn by iterating a map of the
flat torus  R^2/ Z^2 which is defined by an integer matrix of 
determinant -1.    

The `Fib' pictures, Fib1, Fib2 etc are drawn using the Fibonacci map
(x, y) \mapsto (y, x+y) with x, y taken mod 1. 
The matrix of the lift to R^2 is the matrix

0 1
1 1

Its square is the Arnold cat map

2 1 
1 1

after the involution x<-> y, 
which is to say after conjugating the square by

0 1
1 0

The inverse of the fibonacci matrix is

-1 1
 1  0

which defines the inverse of the Fibonacci map and hence the pictures
FibInverse1, etc.    

One of the points of the drawing is to
see the stable and unstable manifolds of the fibonacci map within the picture.
Iterating F forward, the map squeezes the disc along the UNSTABLE
eigenspace of the matrix for F, the one whose eigenvalue is greater than 1 in absolute
value, while iterating F^{-1} smears the disc along the STABLE manifold
for F, which is the orthogonal line field.  

Why orthogonal?  Well the cat map C = F^2  is symmetric, 
so has orthogonal eigenspacesand
one checks by hand that if F  has an eigenline with eigevalue
\lambda then C = F^2 also has L as eigenline, 
but now w eigenvalue \lambda^2.   distribution
parallel to he stable manifold of F,