The picture below illustrates a function f(x,y) of two variables which is defined
using determinant of matrices.
For rational y=p/q form f(x,y) = log[  det (L(y) x)  ]/q, where L(y) is a
(q x q) matrix with side diagonal entries 1 and diagonal entries
V(k) = 2 cos(2 k p/q ):
 V(1) 1 0 ... 0 1 
 1 V(2) 1 ... ... 0 
L(y)=  0 1 ... ... ... ... 
 ... ... ... ... 1 0 
 0 ... ... 1 V(q1) 1 
 1 0 ... 0 1 V(q) 

Physically, the xcoordinate is the energy. The y coordinate is related
to a magnetic flux.
The Hofstadter butterfly is the set, where f(x,y)=0. The picture to the right is colored
according to the value of f(x,y).
Mathematicians call the function f(x,y) a Lyapunov exponent. It is also defined for irrational
y through a limit.
See the Java applet.
