Quasiperiodic tilings with low-order rotational symmetry

Using a projection method of de Bruijn [Proc. K. Ned. Akad. Wet. Ser. A (1981), 43, 39-66], Whittaker & Whittaker [Acta Cryst. (1988), A44, 105-112] obtained nonperiodic tilings of the plane with n-fold rotational symmetry, n = 5, 7, 8, 9, 10 and 12. However, when their method was applied to the cases of 3-, 4- and 6-fold rotational symmetry it produced periodic tilings. This might be taken as circumstantial evidence that 3-, 4- and 6-fold rotational symmetry is incompatible with nonperiodicity. It is demonstrated that they are compatible by constructing quasiperiodic tilings with only 3-, 4- and 6-fold rotational symmetry. This approach uses basic eigenvalue methods of matrix theory.