  The stable magnetisation configurations of antiferromagnets on quasiperiodic
tilings are investigated theoretically. The exchange coupling is assumed to
decrease exponentially with the distance between magnetic moments. It is
demonstrated that the combination of geometric frustration and the
quasiperiodic order of atoms leads to complicated noncollinear ground states.
The structure can be divided into subtilings of different energies. The
symmetry of the subtilings depends on the quasiperiodic order of magnetic
moments. The subtilings are spatially ordered. However, the magnetic ordering
of the subtilings in general does not correspond to their spatial arrangements.
While subtilings of low energy are magnetically ordered, those of high energy
can be completely disordered due to local magnetic frustration.
