Classification Of Tilings Research Articles

Classification of Self-Affine Lattice Tilings

We consider tilings of the plane by a lattice of translates of some compact set A such that the union of k suitably chosen tiles is similar to A. Affine and metric equivalence of such ‘k-reptiles’ are defined. We show that for every k ⩾ 2 there is a finite number of equivalence classes for which A is homeomorphic to a disk. There are three affine types of tiles with two pieces, and seven types with k = 3.

The classification of face-transitive periodic three-dimensional tilings

It has long been known that there exists an infinite number of types of tile-transitive periodic three-dimensional tilings. Here, it is shown that, by contrast, the number of types of face-transitive periodic three-dimensional tilings is finite. The method of Delaney symbols and the properties of the 219 isomorphism classes of crystallographic space groups are used to find exactly 88 equivariant types that fall into seven topological families.

The classification of 2-isohedral tilings of the plane

Two different methods for enumerating k-isohedral tilings are discussed. One is geometric: by splitting and gluing tiles. The other is combinatorial: by enumerating the appropriate Delaney—Dress symbols. Both methods yield 1270 types of proper 2-isohedral tilings of the plane.

Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability

Two 2D quasiperiodic tilings with generalized tenfold symmetry are derived from the lattice A4R, the reciprocal of the root lattice A4. Both tilings are built from four tiles, triangles in one case, rhombi and hexagons in the other. After a brief description of the tilings and their structures, the authors introduce the equivalence concept of mutual local derivability. They present its key properties and its application to several tenfold tilings and discuss some implications on a future classification of tilings in position space.

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

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.