  In the context of magnetism, frustration arises when a group of spins cannot
find a configuration that minimizes all of their pairwise interactions
simultaneously. We consider the effects of the geometric frustration that
arises in a structure having pentagonal loops. Such five-fold loops can be
expected to occur naturally in quasicrystals, as seen for example in a number
of experimental studies of surfaces of icosahedral alloys. Our model considers
classical vector spins placed on vertices of a subtiling of the two dimensional
Penrose tiling, and interacting with nearest neighbors via antiferromagnetic
bonds. We give a set of recursion relations for this system, which consists of
an infinite set of embedded clusters with sizes that increase as a power of the
golden mean. The magnetic ground states of this fractal system are studied
analytically, and by Monte Carlo simulation.
