A classification of two-dimensional quasiperiodic tilings obtained with the grid method

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Abstract
Two-dimensional quasiperiodic tilings (QPT) obtained from n grids are classified into local isomorphism (LI) classes by using the invariants of the grids. The number of the invariants is given by n- phi (n) if n is odd or n- phi (2n) if n is even, where phi is Euler's function in number theory. All the QPT obtained from n grids with n=2k(k>or=2) belong to a single LI class, whose point symmetry is D2n. When n is not a power of 2, the QPT are classified into several LI classes. Of the LI classes, two have the highest point symmetry, D2n; one of the two classes is associated with n grids in which all the invariants vanish and the other with n grids in which all of them take 1/2. In the case of an odd n, there is also one continuous series of QPT with point symmetry Dn. The author also presents a general formula of the tile statistics of the quasiperiodic tilings obtained with the grid method.

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