  We derive the rational generating function that enumerates the angels and
devils in M. C. Escher's {\it Circle Limit IV} according to their combinatorial
distance from the six creatures whose feet meet at the center of the disk. This
result shows that the base of the exponential rate of growth is $1.582\ldots$
(the largest root of the polynomial $1 - z^2 - 2z^3 - z^4 + z^6$).
