A rep-tiling ~is a self replicating, lattice tiling of R". Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, and self-replicating means that there is a non-singular linear map ¢: R"-o R" such that, for each T e J, the image 4~(T) is, in turn, tiled by Y-. This topic has recently come under investigation, not only because of its recreational appeal,...

In this work we study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we indentify restrictions to the problem where either it remains NP-complete or it has a polynomial time algorithm. First we show that an aztec diamond of order n always has an L-tromino tiling...

Let T be a regular tiling of R which has the origin 0 as a vertex, and suppose that φ : R → R is a homeomorphism such that i) φ(0) = 0, ii) the image under φ of each tile of T is a union of tiles of T , and iii) the images under φ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertice...

This paper obtains some important properties of planar normal tiling and gives a purely combinatorial proof for Grünbaum’s Theorem. Key–Words: Normal tiling, Adjacent, Neighbor, Adjacent-graph, Patch