Local rules for quasiperiodic tilings of quadratic 2-planes inR 4

We prove that quasiperiodic tilings of the plane, appearing in the strip projection method always admit local rules, when the linear embedding ofR ² inR ⁴ has quadratic coefficients. These local rules are constructed and studied. The connection between Novikov quasicrystallographic groups and the quasiperiodic tilings of Euclidean space is explained. All the point groups in Novikov's sense, compatible with these local rules, are enlisted. The two-dimensional quasicrystals with infinite-fold rotational symmetry are constructed and studied.Address after September 1, 1992: Dept. of Physics, Harvard University, Cambridge, MA 02138, USA