Quasiperiodic tilings generated by matrices

Abstract

Using the inflation method, Watanabe, Ito and Soma [3], Clark and Suryanarayan [4] and Balagurusamy, Ramesh and Gopal [5] have obtained nonperiodic tilings of the plane with n-fold rotational symmetry, n = 2, 3, 4, 5, 8, using two unit prototiles. Fortunately, there is an easier way to generate a more general class of nonperiodic tilings which contains the above-mentioned tilings as special cases. We do this by specifying two matrices of order two which define the two classes of tilings; thus, our approach uses the basic techniques from linear algebra in the study of quasiperiodic tilings and the method can be generalized to obtain tilings that have more than two prototiles. The tilings generated are fractals and their dimensions and the rate of growth are determined.