![]() Homework: Make the cutting process more algorithmic to create a non-periodic tilingġ7 Homework Find other examples of non-periodic tilings by copies of a single triangle Can non-periodic tilings be created using copies of a single square? What about rectangles?ġ8 Source of more interesting examples: substitution tilingsĪ partial tiling of the plane consisting of finitely many tiles is called a patch Let S be a finite set of distinct tiles and S’ is a set of bigger (inflated) tiles, similar to those from S under the same rescaling Suppose that each tile in S’ can be cut into a finite number of tiles that belong to S Let P be a patch consisting of tiles from S Rescale (inflate) P and then cut each tile in P to produce bigger patch that still uses tiles from SĢ0 Why is the armchair tiling non-periodic? ![]() edge to edge matching) Matching rules can be enforced in a number of ways, including: vertex labeling or coloring edge labeling or coloring edge modificationsġ1 Examples Edge modification Vertex coloring Homework:ġ2 Homework Any triangle can tile the planeĪny quadrilateral (even non-convex) can tile the plane Which pentagons can tile the plane? Find at least one Find a convex tile that can tile the plane in exactly one way Can a regular tetrahedron tile the space? What about other regular polyhedra? What about a non-regular tetrahedron?Ī tiling is called periodic if it can be shifted to perfectly align with itself in at least two non-parallel directions A tiling is called non-periodic if it cannot be shifted to perfectly align with itself Do non-periodic tilings exist?ġ5 Trivial example Is there a non-periodic tiling of the plane consisiting of identical tiles?ġ6 Less Trivial example Yes: cut squares “randomly” of Computer Science, University of Utrecht, Utrecht, 1982 (submitted for publication).Presentation on theme: "Aperiodic Tilings Alexandre Karassev."- Presentation transcript:Ģ Tilings A tiling (or tessellation) is a cover of the plane (or space) by nonoverlaping regionsĨ Tiles A tile is a polygonal region of the plane (not necessarily convex) Two tiles are called identical (congruent) if one can be transformed to the other by shift and rotation of the same type (or similar), if one is a rescaling of the other Tiles of the same type Identical tilesĩ Example: two different tilings by squaresġ0 Matching rules Matching rules specify a way of joining individual tiles (e.g. van Leeuwen, Periodic versus arbitrary tessellations of the plane using polyominos of a single type, Techn. Wang, H., Games, logic and computers, Scientific American 213 (1965) 98–106. Thurber, K.J., Large scale computer architecture: parallel and associative processors, Hayden Book Comp., Rochelle Park NJ, 1976. Shapiro, H.D., Theoretical limitations on the efficient use of parallel memories, IEEE Trans. Levi, P., Sur une generalisation du theorème de Rolle, Comp. Golomb, S.W., Tiling with sets of polyominos, J. Centre Tracts 106, Mathematisch Centrum, Amsterdam, 1979, pp. Schrijver (ed.), Packing and covering in combinatorics, Math. Göbel, F., Geometrical packing and covering problems, in: A. ![]() Gardner, M., Mathematical games, articles in the Scientific American: january 1977, pp. Gardner, M., Mathematical games, articles in the Scientific American: december 1975, pp. Gardner, M., Mathematical games, articles in the Scientific American: august 1975, pp. Gardner, M., Mathematical games, articles in the Scientific American: july 1975, pp. Kuck, The organisation and use of parallel memories, IEEE Trans. ![]() This process is experimental and the keywords may be updated as the learning algorithm improves.īudnik, P. These keywords were added by machine and not by the authors. It is also proved that there is a polynomial time algorithm to decide whether a polyomino tessellates the plane, assuming the polyominos in the tessellation should all have an equal orientation. (Periodicity implies a rapid technique to locate data elements.) The proof shows that when a polyomino P tessellates the plane without rotations or reflections, then it can tessellate the plane periodically, i.e., with the instances of P arranged in a lattice. We settle a conjecture of Shapiro and prove that for polyominos P a valid skewing scheme exists if and only if there exists a valid periodic skewing scheme. Shapiro proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. Array embeddings that allow for this are called skewing schemes and have been studied in connection with vector-processing and SIMD machines. Given N parallel memory modules, we like to distribute the elements of an (infinite) array in storage such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. ![]()
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