File Penrose Tiling Rhombi .svg thumb 240px This form of the Penrose tiling has two prototiles, a fat rhombus shown blue in the figure and a thin rhombus green . In the mathematical theory of tessellation s, a prototile is one of the shapes of a tile in a tessellation. A tessellation of the plane or of any other space is a cover of the space by closed set closed shapes, called tiles, that have disjoint sets disjoint interior topology interiors . Some of the tiles may be congruence geometry congruent to one or more others. If mvar S is the set of tiles in a tessellation, a set mvar R of shapes is called a set of prototiles if no two shapes in mvar R are congruent to each other, and every tile in mvar S is congruent to one of the shapes in mvar R . It is possible to choose many different sets of prototiles for a tiling translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality , so the number of prototiles is well defined. A tessellation is said to be monohedral if it has exactly one prototile. A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling . The three dimensional Schmitt Conway Danzer tile is the prototile of a monohedral aperiodic tiling of three dimensional Euclidean space , but it remains open whether there is a monohedral aperiodic prototile for the plane. References citation page 7 title Introductory Tiling Theory for Computer Graphics series Synthesis Lectures on Computer Graphics and Animation first Craig S. last Kaplan publisher Morgan & Claypool Publishers year 2009 isbn 9781608450176 url http books.google.com books?id OPtQtnNXRMMC&pg PA7 . citation page 174 title A Course in Modern Geometries series Undergraduate Texts in Mathematics first Judith N. last Cederberg edition 2nd publisher Springer Verlag year 2001 isbn 9780387989723 url http books.google.com books?id Fo9tqL99jdMC&pg PA174 . Category Tiling Geometr ... more details
T 1, T 2, dots, T m math as prototiles. A placement of a prototile math T i math is a pair math T i ... math is called the placement s region. A tiling T is a set of prototile placements whose regions ... more details
File Self replication of sphynx hexidiamonds.svg thumb 200px The sphinx polyiamond reptile. Four copies of the sphinx can be put together as shown to make a larger sphinx. In the geometry of tessellation s, a shape that can be dissection geometry dissected into smaller copies of the same shape is called a reptile or rep tile . Solomon W. Golomb coined the term for self replicating tilings. The shape is labelled as rep n if the dissection uses n copies. Such a shape necessarily forms the prototile for a tiling of the plane, in many cases an aperiodic tiling . A shape that tiles itself using different sizes is called an irregular rep tile or irreptile. If the tiling uses n copies, the shape is said to be irrep n . If all these sub tiles are of different sizes then the tiling is additionally described as perfect. A shape that is rep n or irrep n is trivially also irrep kn &minus k n for any k 1, by replacing the smallest tile in the rep n dissection by n even smaller tiles. The order of a shape, whether using rep tiles or irrep tiles is the smallest possible number of tiles which will suffice. Examples image pinwheel 2.gif 250px thumb right Defining an aperiodic tiling the pinwheel tiling by repeatedly dissecting and inflating a rep tile. Every square , rectangle , parallelogram , rhombus , or triangle is rep 4. The sphinx polyiamond hexiamond illustrated is also rep 4 and is the only known self replicating pentagon. The Gosper curve Properties Gosper island is rep 7. The Koch snowflake is irrep 7 six small snowflakes of the same size, together with another snowflake with three times the area of the smaller ones, can combine to form a single larger snowflake. A right triangle with side lengths in the ratio 1 2 is rep 5, and its rep 5 dissection forms the basis of the aperiodic pinwheel tiling . The international standard ISO 216 defines sizes of paper sheets using the Lichtenberg ratio , in which the long side of a rectangular sheet ... more details
cubes volume 36 year 1984 . ref Burns and Rigby found several prototile s, including the Koch snowflake , that may be used to tile the plane only by using copies of the prototile in two or more ... more details
Ammann discovered several new sets in 1977. In 1988, Peter Schmitt discovered a single aperiodic prototile in 3 dimensional Euclidean space. While no tiling by this prototile admits a translation as a symmetry ... and Ludwig Danzer to a convex set convex aperiodic prototile, the Schmitt Conway Danzer tile . Because ... more details
Image AmmannBeenker.jpg 300px thumb right Ammann Beenker tiling In geometry , an Ammann Beenker tiling is a nonperiodic tessellation tiling generated by an aperiodic tiling aperiodic set of prototile s named after Robert Ammann , who first discovered the tilings in the 1970s, and after Beenker F. P. M. Beenker who discovered them independently and showed how to obtain them by the cut and project method. Because all tilings obtained with the tiles are non periodic, Ammann Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns ref name ReferenceA Branko Gr nbaum B. Gr nbaum and G.C. Shephard, Tilings and Patterns , Freemann, NY 1986 ref . The Ammann Beenker tilings have many properties similar to the more famous Penrose tiling s, most notably They are nonperiodic, which means that they lack any translational symmetry . Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, if one looks only at finite patches. They are quasicrystal line implemented as a physical structure an Ammann Beenker tiling will produce Bragg diffraction the diffractogram reveals both the underlying eightfold symmetry and the long range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called deflation or inflation. Various methods to construct the tilings have been proposed matching rules, substitutions, cut and project schemes ref Beenker FPM, Algebric theory of non periodic tilings of the plane by two simple building blocks a square and a rhombus, TH Report 82 WSK 04 1982 , Technische Hogeschool, Eindhoven ref and coverings ref F. Ga hler, in Proceedings of the 6th International Conference on Quasicrystals, edited by S. Takeuchi and T. Fujiwara, World Scientific, Singapore, 1998, p. 95. ref ref h ... more details
confusing date August 2010 File Fund un prim cell.svg 250px thumb right A truncated trihexagonal tiling periodic tiling with a fundamental unit triangle and a primitive cell hexagon highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge to edge to a neighboring triangle. Thus a triangular tiling of fundamental units will be generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be Translation geometry translated to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this. alt In geometry , a Tessellation tiling is a family of shapes called prototile tile s that cover the plane or any other geometric setting without gaps or overlaps. ref name Tilings by Regular Polygons Citation doi 10.2307 2689529 author Gr nbaum B., Shephard G. C. title Tilings by Regular Polygons journal Math. Mag. volume 50 issue 5 year 1977 pages 227 247 url http vohweb.chem.ucla.edu voh classes 5Cspring10 5CM117 20HNRS 20M180ID22 5CGrunbaumShephardTilingByRegPolygons.pdf postscript . archived at http www.webcitation.org 5t6mxDTfO WebCite ref Such a tiling might be Constructibility constructible from a single Fundamental domain fundamental unit or primitive cell and is then called periodic. ref Edwards S., http www.spsu.edu math tile defs fundamental.htm Fundamental Regions and Primitive cells archived at http www.webcitation.org 5smoLyN1Y WebCite ref An example of such a tiling is shown in the diagram to the right see the image description for more information . Every periodic tiling has a primitive cell that can generate it. A tiling that cannot be constructed from a single primitive cell is called nonperiodic. ... more details
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