Wiktionary Tiling may refer to The physical act of laying tile s The mathematics of tessellation s The compiler optimization of loop tiling In computing a Tiling window manager tiling window manager is one where windows do not overlap. In computing a tiled rendering is the process of subdividing image by regular grid. People with the surname Tiling Reinhold Tiling , German rocket pioneer Heinrich Sylvester Theodor Tiling , Physician, botanist Disambiguation ... more details
A quasiperiodic tiling is a tessellation tiling of the plane which exhibits local periodicity under some transformations we can slide or rotate it such that a finite number of tiles overlap perfectly, yet the entire tiling will not. See Aperiodic tiling and Penrose tiling for a mathematical viewpoint. Quasicrystal for a physics viewpoint. Category Tiling disambig ... more details
Unreferenced date December 2009 Puzzles Tiling puzzles are puzzle s involving two dimensional packing problem s in which a number of flat shapes have to be assembled into a larger given shape without overlaps and often without gaps . Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask you to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzle s. Tiling puzzles may be made from wood , metal , Corrugated fiberboard cardboard , plastic or any other sheet material. Many tiling puzzles are now available as computer game s. Tiling puzzles have a long history. Some of the oldest and most famous are jigsaw puzzle s and the Tangram puzzle. Image Chinese Sunset.jpg thumb 160px right Example of a tiling puzzle that utilizes negative space. Other examples of tiling puzzles include Conway puzzle Domino tiling , of which the mutilated chessboard problem is one example Eternity puzzle Puzz 3D Squaring the square Tantrix T puzzle Pentomino es Many three dimensional mechanical puzzle s can be regarded as three dimensional tiling puzzles. See also Dissection puzzle Polyform s Sliding puzzle Tessellation Wang tile DEFAULTSORT Tiling Puzzle Category Tiling puzzles eo Kahelara enigmo ... more details
A tiling with rectangles is a tessellation tiling which uses rectangle s as its parts. Tilings with non congruent rectangles The smallest square that can be cut into m x n rectangles, such that all m and n are different integers, is the 11 x 11 square, and the tiling uses five rectangles sup 1 sup . br The smallest rectangle that can be cut into m x n rectangles, such that all m and n are different integers, is the 9 x 13 rectangle, and the tiling uses five rectangles sup 1 sup . See also tessellation , tiling puzzle References Journal of Recreational Mathematics , 28 1, p.64. Category Tiling geometry stub eo Kahelaro el ortanguloj ... more details
Uniform hyperbolic tiles db Uniform hyperbolic tiling stat table U73 02 In geometry , the rhombitriheptagonal tiling is a semiregular tiling of the Hyperbolic space hyperbolic plane . At each vertex geometry vertex of the tiling there is one triangle and one heptagon , alternating between two square geometry square s. The tiling has Schl fli symbol t sub 0,2 sub 7, 3 . Related polyhedra and tilings This tiling is topologically related as a part of sequence of Cantellation geometry cantellated polyhedra with vertex figure 3.4.n.4 , and continues as tilings of the Hyperbolic space hyperbolic plane . These vertex transitive figures have n32 reflectional Orbifold notation symmetry . class wikitable Image Uniform polyhedron 33 t02.png 100px BR Cuboctahedron 3.4.3.4 BR 332 BR CDD node 1 3 node 3 node 1 Image Uniform polyhedron 43 t02.png 100px BR Rhombicuboctahedron 3.4.4.4 BR 432 BR CDD node ... 532 BR CDD node 1 5 node 3 node 1 Image Uniform polyhedron 63 t02.png 100px BR rhombitrihexagonal tiling 3.4.6.4 BR 632 BR CDD node 1 6 node 3 node 1 Image Uniform tiling 73 t02.png 100px BR rhombitriheptagonal tiling 3.4.7.4 BR 732 BR CDD node 1 7 node 3 node 1 Image Uniform tiling 83 t02.png 100px BR 3.4.8.4 BR 832 BR CDD node 1 8 node 3 node 1 Dual tiling The dual tiling is called a deltoidal triheptagonal tiling , and consists of congruent kite geometry kites . It is formed by overlaying an order 3 heptagonal tiling and an order 7 triangular tiling . Image Deltoidal triheptagonal til.png ... 10 Regular honeycombs in hyperbolic space See also Rhombitrihexagonal tiling Order 3 heptagonal tiling Tilings of regular polygons List of uniform tilings Kagome lattice External links MathWorld urlname HyperbolicTiling title Hyperbolic tiling MathWorld urlname PoincareHyperbolicDisk title Poincar hyperbolic disk http bork.hampshire.edu bernie hyper Hyperbolic and Spherical Tiling Gallery http ..., Don Hatch geometry stub Category Tiling eo Malgranda rombo tri seplatera kahelaro ... more details
Uniform hyperbolic tiles db Uniform hyperbolic tiling stat table U73 1 In geometry , the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a Rectification geometry rectified Order 3 heptagonal tiling . There are two triangle s and two heptagon s alternating on each vertex geometry vertex . It has Schl fli symbol of t sub 1 sub 7,3 . Compare to trihexagonal tiling with vertex configuration 3.6.3.6 . Images class wikitable width 480 valign top File Uniform tiling 73 t1 klein.png 240px BR Klein disk model of this tiling preserves straight lines, but distorts angles Image Order73 qreg rhombic til.png 240px BR The dual tiling is called an Order 7 3 rhombille tiling , made from rhombic faces, alternating 3 and 7 per vertex. Related polyhedra and tilings This tiling is topologically part of sequence of Rectification geometry rectified polyhedra with vertex figure 3.n.3.n and n32 reflectional Orbifold notation symmetry . class wikitable Image Uniform polyhedron 33 t1.png 100px BR Octahedron 3.3.3.3 BR 332 and 432 Image Uniform polyhedron 43 t1.png 100px BR Cuboctahedron 3.4.3.4 BR 432 Image Uniform polyhedron 53 t1.png 100px BR Icosidodecahedron 3.5.3.5 BR 532 Image Uniform polyhedron 63 t1.png 100px BR 3.6.3.6 BR 632 Image Uniform tiling 73 t1.png 100px BR 3.7.3.7 BR 732 Image Uniform tiling 83 t1.png 100px BR 3.8.3.8 BR 832 See also Trihexagonal tiling 3.6.3.6 tiling Rhombille tiling dual V3.6.3.6 tiling Tilings of regular polygons List of uniform tilings References John Horton Conway John H. Conway , Heidi Burgiel, Chaim Goodman Strass, The Symmetries of Things 2008, ISBN 978 1 56881 220 5 Chapter 19, The Hyperbolic Archimedean Tessellations The Beauty ... Hyperbolic tiling MathWorld urlname PoincareHyperbolicDisk title Poincar hyperbolic disk http bork.hampshire.edu bernie hyper Hyperbolic and Spherical Tiling Gallery http geometrygames.org KaleidoTile ... stub Category Tiling Category Quasiregular polyhedra cs Triheptagon ln dl d n eo Tri seplatera ... more details
Image Quaquaversal Tiling.svg 200px right The quaquaversal tiling is a aperiodic tiling nonperiodic tiling of the euclidean 3 space introduced by John H. Conway John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern which relies essentially on their previous construct, the pinwheel tiling . The rotations relating these tiles belong to the group G 6,4 generated by two rotations of order 6 and 4 whose axes are perpendicular to each other. clear References Conway J., Radin C., Quaquaversal tilings and rotations , Invent. math. 132 1998 , 179 188. Radin C., Sadun L., Subgroups of SO 3 associated with tilings , J. Algebra 202 1998 , 611 633. External links A http www.ma.utexas.edu users radin quaqua.html picture of a quaquaversal tiling http www.ma.utexas.edu users radin Charles Radin page at the University of Texas geometry stub Category discrete geometry Category tiling ... more details
No footnotes date March 2011 Uniform tiles db Reg tiling stat table Uh In geometry , the hexagonal tiling is a regular tiling of the Euclidean plane , in which three hexagon s meet at each vertex. It has Schl fli symbol of 6,3 or t 3,6 as a truncated triangular tiling . John Horton Conway Conway calls ... of the plane . The other two are the triangular tiling and the square tiling . Applications The hexagonal tiling is the densest way to sphere packing arrange circles in two dimensions. It also has the thinnest ... of wires for structural rigidity. Unlike the square tiling , which can be deformed into a parallelogram mic tiling, the hexagonal tiling is rigid. This structure exists naturally in the form of graphite ... 2d.svg The densest sphere packing circle packing is arranged like the hexagons in this tiling File Chicken ... povray.PNG carbon nanotube can be seen as a hexagon tiling on a Cylinder geometry cylindrical surface gallery The hexagonal tiling appears in many crystals. In three dimensions, the face centered ... distinct uniform coloring s of a hexagonal tiling, all generated from reflective symmetry of Wythoff ... Uniform tiling 63 t0.png 100px Image Uniform tiling 63 t12.png 100px Image Uniform tiling 333 t012.png 100px File Truncated rhombille tiling.png 100px File Hexagonal tiling 4 colors.png 100px align ... t6dateH The 3 color tiling is a tessellation generated by the order 3 permutohedron s. Topologically identical tilings Image Wallpaper group cmm 1.jpg thumb Bricks as nonregular hexagonal tiling The hexagonal tiling can be stretched and adjusted to other geometric proportions and different symmetries. The standard brick pattern can be considered a nonregular hexagonal tiling. Each rectangular brick ... a distorted hexagonal tiling. class wikitable 4 color hexagonal tilings Regular hexagons Hexagonal weave p6m 632 p6 632 File Hexagonal tiling 4 colors.png 240px File Weaved hexagonal tiling.png ... 240px File Herringbone pattern as hexagonal tiling.png 240px Related polyhedra and tilings This tiling ... more details
Uniform tiles db Reg tiling stat table Us In geometry , the square tiling or square grid is a regular tiling of the Euclidean plane. It has Schl fli symbol of 4,4 , meaning it has 4 Square geometry squares ... tiling and the hexagonal tiling . Uniform colorings There are 9 distinct uniform coloring s of a square tiling. Naming the colors by indices on the 4 squares around a vertex 1111, 1112 i , 1112 ii , 1122 ... symmetry. Image Square tiling uniform colorings.png 240px Related polyhedra and tilings This tiling ... Image Uniform polyhedron 43 t0.png 100px BR Cube 4,3 Image Uniform tiling 44 t0.png 100px BR 4,4 Image Uniform tiling 45 t0.png 100px BR Order 5 square tiling 4,5 Image Uniform tiling 46 t0.png 100px BR 4,6 Image Uniform tiling 47 t0.png 100px BR 4,7 Image Uniform tiling 48 t0.png 100px BR 4,8 ... Wythoff constructions from square tiling Like the Uniform polyhedron uniform polyhedra there are eight List of uniform tilings uniform tilings that can be based from the regular square tiling. Drawing ... unique topologically forms square tiling , truncated square tiling , snub square tiling . class wikitable ... Vertex figure Image Parent t sub 0 sub 4,4 4 & 124 2 4 4 sup 4 sup Image Uniform tiling 44 t0.png 80px Truncation geometry Truncation t sub 0,1 sub 4,4 2 4 & 124 4 Truncated square tiling 4.8.8 Image Uniform tiling 44 t01.png 80px Rectification geometry Rectification t sub 1 sub 4,4 2 & 124 4 4 4.4 sup 2 sup Image Uniform tiling 44 t1.png 80px Truncation geometry Bitruncation t sub 1,2 sub 4,4 2 4 & 124 4 Truncated square tiling 4.8.8 Image Uniform tiling 44 t12.png 80px Dual polyhedron Dual t sub 2 sub 4,4 4 & 124 2 4 4 sup 4 sup Image Uniform tiling 44 t2.png 80px Cantellation geometry Cantellation t sub 0,2 sub 4,4 4 4 & 124 2 4.4.4.4 Image Uniform tiling 44 t02.png 80px Omnitruncation geometry Omnitruncation t sub 0,1,2 sub 4,4 2 4 4 & 124 Truncated square tiling 4.8.8 Image Uniform tiling 44 t012.png 80px Snub geometry Snubbing s 4,4 & 124 2 4 4 Snub square tiling 3.3.4.3.4 ... more details
File PentagonTilings.svg thumb 300px The 14 known pentagon tilings In geometry , a pentagon tiling is a tessellation tiling of the plane by pentagon s. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108 is not a divisor of 360. There are fourteen known types of convex pentagon that tile the plane it is not known if this list is complete. Dual uniform tilings There are 3 isohedral pentagonal tilings generated as Dual polyhedron duals of the uniform tiling s class wikitable File Tiling Dual Semiregular V3 3 4 3 4 Cairo Pentagonal.svg 150px BR Cairo pentagonal tiling File Tiling Dual Semiregular V3 3 3 3 6 Floret Pentagonal.svg 150px BR Floret pentagonal tiling File Tiling Dual Semiregular V3 3 3 4 4 Prismatic Pentagonal.svg 150px BR Prismatic pentagonal tiling Regular pentagonal tilings in non Euclidean geometry A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere , with Schlafli symbol ... pentagons, for instance order 4 pentagonal tiling , with Schlafli symbol 5,4 , having 4 pentagons ..., ending in 5,&infin . class wikitable Sphere colspan 5 Hyperbolic plane File Uniform tiling 532 t0.png 120px BR Dodecahedron BR 5,3 File Uniform tiling 54 t0.png 120px BR order 4 pentagonal tiling BR 5,4 File Uniform tiling 55 t0.png 120px BR order 5 pentagonal tiling BR 5,5 File Uniform tiling 56 t0.png 120px BR order 6 pentagonal tiling BR 5,6 File Uniform tiling 57 t0.png 120px BR order 7 pentagonal tiling BR 5,7 valign bottom ... 5,&infin Irregular hypebolic plane pentagonal tilings There are an infinite ... tiling Order 3 snub heptagonal tiling Dual tiling 7 3 8 3 9 3 ... 5 4 6 4 7 4 ... 5 5 valign bottom ... location New York isbn 0 7167 1925 8 chapter Tiling with Convex Polygons External links MathWorld title Pentagon Tiling urlname PentagonTiling http demonstrations.wolfram.com PentagonTilings Pentagon ... Category Tiling fr Pavage pentagonal ... more details
Infobox face uniform tiling Image File Rhombic star tiling.png Type List of uniform tilings Dual semiregular tiling Face List Rhombus 30 60 rhombus Symmetry Group 632 Face Type V3.6.3.6 Dual Trihexagonal tiling Property List edge transitive face transitive In geometry , the rhombille tiling ref John ... as a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon ... tiling . ref MathWorld urlname DualTessellation title Dual tessellation See comparison ... or wall tiling, sometimes with one fatter rhombus or square and two more narrow rhombi. http www.alibaba.com ... uniform tiling all the rhombi are the same, but with larger symmetry fundamental domains, there are many ... colors File rhombic star tiling 1.png 120px File rhombic star tiling 2.png 120px File Rhombic star tiling 0.png 120px File Rhombic star tiling 3.png 120px File Rhombic star tiling 4.png 120px Related polyhedra and tilings This tiling is a part of a sequence of rhombic polyhedra and tilings with n,3 .... class wikitable width 500 colspan 3 Polyhedra Euclidean tiling colspan 2 Hyperbolic tiling 3,3 4,3 ... tiling 433 t01 yellow.png 100px Similarly it relates to the infinite series of 3 color tilings with the face ... V3636.svg 210px br V3.6.3.6 br Euclidean plane tiling br Rhombille tiling align center valign top Image Uniform dual tiling 433 t01.png 210px br V3.8.3.8 br Hyperbolic plane tiling br Drawn in a Poincar ... tilings, and the triangular tiling . The translational rhombic and zig zag rhombic tilings, which are topologically equivalent to the square tiling , can be constructed from rhombi or parallelogram s of any angle and lengths, while the rhombille tiling is limited to 60 degrees. class wikitable width ... kah 3 6 nd.png 120px br Triangular tiling Triangular Image kah 3 6 romb.png 120px br Translation ... lattice.png 120px BR Rhombille class wikitable width 400 valign top File Snub square rhombic tiling 2.png 200px BR A variation of snub square tiling with merged triangles File Penrose Tiling Rhombi ... more details
File Bundesarchiv Bild 102 11541, Osnabr ck, Erste deutsche Postrakete.jpg thumb Tiling s post office rocket Reinhold Tiling June 13, 1893 &ndash October 11, 1933 was a Germany German engineer , aviator pilot and a rocket pioneer. Biography Tiling was born in Absberg , Kingdom of Bavaria , as the son of a pastor. Shortly after he began the study of mechanical engineering and electro technology he found himself in war service at the beginning of the First World War . In 1915 he volunteered as a fighter ... expansion User Master Of Ninja Master Of Ninja 00 40, 12 Feb 2005 UTC In 1926 Tiling became flight ... he started his first experiments in 1928. Tiling developed re usable rocket planes, which start as a rocket ... and burning duration for flight. In 1929 Gisbert Freiherr von Ledebur 1899 1980 allowed Tiling the use ... which missiles reached a height of 1,000 metres. On March 13, 1931 Tiling and his coworker Karl Poggensee ... through experiment occurred on April 15, 1931 when Tiling demonstrated a post office rocket, which ... the support of friends and sponsors, Tiling s work was beset by financial difficulties. On October 10, 1933, the overheating of the powder needed to power the rocket created an explosion in Tiling s workshop in Ahrenshorst Tiling, his assistant Angela Buddenboehmer and his mechanic Friedrich Kuhr suffered heavy burns. They succumbed to these injuries on the following day, with Tiling dying in Osnabr ck . A crater on the backside of the moon is named after Reinhold Tiling, located at coordinates 53 06 S 132 36 W. See also Rocket mail External links Commonscat Reinhold Tiling http technikatlas.de ta16 index.html Reinhold Tiling Pionier der Raketentechnik German Persondata Metadata see Wikipedia Persondata . NAME Tiling, Reinhold ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH June 13, 1893 PLACE OF BIRTH DATE OF DEATH October 11, 1933 PLACE OF DEATH DEFAULTSORT Tiling, Reinhold Category ... Category Aerobatic pilots de Reinhold Tiling ru ... more details
Uniform tiles db Uniform tiling stat table Uht In geometry , the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex geometry vertex . It has Schl fli symbol of t sub 1 sub 6,3 its edges form an infinite arrangement of lines . John Horton Conway Conway calls it a hexadeltille , combining alternate elements from a hexagonal tiling Hextille and triangular tiling deltille . There are 3 List of regular polytopes Euclidean tilings regular and 8 List of uniform tilings semiregular tilings in the plane. Uniform colorings There are two distinct uniform coloring s of a trihexagonal tiling. Naming the colors by indices on the 4 faces around a vertex 3.6.3.6 1212, 1232. class wikitable Coloring File Uniform polyhedron 63 t1.png 100px File Uniform tiling 333 t01.png 100px align center Wythoff symbol 2 & 124 6 3 3 3 & 124 3 align center Coxeter Dynkin diagram CDD node 6 node 1 3 node CDD node split1 branch 11 Related polyhedra and tilings This tiling is topologically part of sequence of Rectification geometry rectified polyhedra with vertex figure 3.n.3.n and n32 reflectional Orbifold notation symmetry . class wikitable Image Uniform polyhedron 33 t1.png 100px BR Octahedron 3.3.3.3 BR 332 and 432 Image Uniform polyhedron 43 t1.png 100px BR Cuboctahedron 3.4.3.4 BR 432 Image Uniform polyhedron 53 t1.png 100px ... Uniform tiling 73 t1.png 100px BR Triheptagonal tiling 3.7.3.7 BR 732 Image Uniform tiling 83 t1.png 100px BR 3.8.3.8 BR 832 This tiling is also topologically part of sequence of polyhedra and tilings ... Uniform polyhedron 33 t02.png 100px BR 3.4.3.4 BR 233 Image Uniform tiling 333 t01.png 100px BR 3.6.3.6 BR 333 Image Uniform tiling 433 t01.png 100px BR 3.8.3.8 BR 433 A tiling with alternate large and small triangles is topologically identical to the trihexagonal tiling. The hexagons are distorted ... Tiling Category Quasiregular polyhedra eo Tri seslatera kahelaro fr Pavage trihexagonal ... more details
Pinwheel tilings are aperiodic tiling s defined by Charles Radin and based on a construction due to John H. Conway John Conway . They are the first known aperiodic tilings to each have the property that their tiles appear in infinitely many orientations. The Conway tesselation image pinwheel 1.svg 250px thumb right Conway triangle decomposition into homothetic smaller triangles. Let math T math be the right ... the Conway tiling of the plane. By suitably rescaling and translating rotating, this operation ... copies of math T math . The union of all these triangles yields a tiling of the whole plane by isometric copies of math T math . In this tiling, isometric copies of math T math appears in infinitely .... The Pinwheel tilings image pinwheel 3.jpg 250px thumb right A Pinwheel tiling tiles can be grouped five by five thick lines to form a new Pinwheel tiling up to rescaling Radin relied on the above ... holds. Given any Pinwheel tiling math P math , there is a Pinwheel tiling math P math which, once ... Pinwheel tiling. The tiling constructed by Conway is a Pinwheel tiling, but there are uncountably many other different Pinwheel tiling. They are all locally undistinguishable i.e. , they have the same finite patches . They all share with the Conway tiling the property that tiles appear in infinitely ... analogue which was dubbed the quaquaversal tiling . ref Radin, C., Conway, J., Quaquaversal tiling and rotations, preprint, Princeton University Press, 1995 ref There are other variants and generalizations ... of the Pinwheel Tiling journal Discrete and Computational Geometry volume 20 issue 1 pages pp.79&ndash ... complex in Melbourne, Australia features the pinwheel tiling. In the project, the tiling pattern is used ... off site, in a factory and later erected to form the facades. The pinwheel tiling system was based ... and repetition. The same pinwheel tiling system is used in the development of the structural frame ... DEFAULTSORT Pinwheel Tiling Category Discrete geometry Category Tiling ... more details
In geometry, a tile substitution is a useful method for constructing highly ordered Tessellation tiling s. Most importantly, some tile substitutions generate aperiodic tiling s, which are tilings whose prototiles do not admit any tiling with translational symmetry . The most famous of these are the Penrose tiling s. Introduction A tile substitution is described by a Set mathematics set of prototiles ... math . Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling . Some substitution tilings are Periodic function periodic , defined as having translational symmetry . Among the nonperiodic substitution tilings are some aperiodic tiling s, those whose prototiles cannot be rearranged to form a periodic tiling usually if one requires in addition some matching rules . A simple example that produces a periodic tiling has only one prototile ... how this procedure yields a substitution tiling of the entire Plane mathematics plane . A mathematically ... aperiodic tiling s, which are objects of interest in many fields of mathematics , including automata ... and chemistry . In particular, the celebrated Penrose tiling is an example of an aperiodic substitution tiling. History In 1973 and 1974, Roger Penrose discovered a family of aperiodic tilings, now called Penrose tiling s. The first description was given in terms of matching rules treating the prototiles ... a tessellation tiling of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 Robert Ammann discovered a number of sets ... math is called the placement s region. A tiling T is a set of prototile placements whose regions have pairwise disjoint interiors. We say that the tiling T is a tiling of W where W is the union of the regions ... sigma math that maps each math T i math to a tiling of math Q T i math . The tile substitution math sigma math induces a map from any tiling T of a region W to a tiling math sigma bold T math of math ... more details
Uniform tiles db Reg tiling stat table Ut In geometry , the triangular tiling is one of the three regular tessellation tiling s of the Euclidean plane . Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schl fli symbol of 3,6 . John Horton Conway Conway calls it a deltille , named from the triangular shape of the greek letter delta &Delta . The triangular tiling is roughly the kishextile. It is one of List of regular polytopes Euclidean tilings three regular tilings of the plane . The other two are the square tiling and the hexagonal tiling . This vertex arrangement is called the A2 lattice . ref http www2.research.att.com njas lattices A2.html ref Uniform colorings There are 9 distinct uniform coloring s of a triangular tiling. Naming the colors by indices on the 6 triangles around a vertex 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314 Three of the colorings are generated by Wythoff construction s. Six of the nine distinct colorings can be made as reductions of the four coloring 121314. The remaining two, 111222 and 122122, have no Wythoff constructions. class wikitable Coloring BR indices 111111 111112 121314 111222 112122 Coloring Image Uniform tiling 63 t2.png 60px Image Uniform tiling 333 t1.png 60px Image Uniform tiling 333 snub.png 60px File Uniform triangular tiling 111222.png 60px File Uniform triangular tiling 112122.png 60px align center Orbifold ... define an icosahedron , octahedron , and tetrahedron respectively. This tiling is topologically ... tiling 73 t2.png 60px BR Order 7 triangular tiling 3,7 Image Uniform tiling 83 t2.png 60px BR 3,8 Image Uniform tiling 39 t0.png 60px BR 3,9 It is also topologically related as a part of sequence ... til.png 60px BR Order 7 truncated triangular tiling V7.6.6 See also Tilings of regular polygons ... O2 Category Tiling eo Triangula kahelaro fr Pavage triangulaire ... more details
lunar crater data latitude 53.1 N or S S longitude 132.6 E or W W diameter 38 km depth Unknown colong 132 eponym Reinhold Tiling Tiling is a small, undistinguished Impact crater crater on the Far side Moon far side of the Moon . It is located just over one crater diameter to the north northeast of the prominent crater Fizeau crater Fizeau . The outer rim of Tiling is worn and rounded, with a small craterlet along the western edge. The interior floor is relatively level and featureless, with only a small craterlet along the eastern edge and inner wall. To the north northeast of Tiling and running towards the northeast is a broken groove in the surface. Several such features lie in the irregular terrain to the east and northeast of Tiling, and these are all radial to the Mare Orientale impact basin farther to the northeast. Satellite craters By convention these features are identified on lunar maps by placing the letter on the side of the crater mid point that is closest to Tiling. class wikitable width 25 style background eeeeee Tiling width 25 style background eeeeee Latitude width 25 style background eeeeee Longitude width 25 style background eeeeee Diameter align center C align center 50.4 S align center 129.7 W align center 21 km align center D align center 52.0 S align center 131.2 W align center 34 km align center F align center 52.3 S align center 129.0 W align center 17 km align center G align center 53.0 S align center 128.6 W align center 14 km References Lunar crater references Category Impact craters on the Moon Moon crater stub ... more details
In geometry , a shape is said to be anisohedral if it admits a tessellation tiling , but no such tiling is isohedral tile transitive that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling . ref cite book last Gr nbaum first Branko authorlink Branko Gr nbaum coauthors Shephard, G. C. title Tilings and Patterns location New York publisher W. H. Freeman and Company year 1987 isbn 0 7167 1193 1 ref The second part of Hilbert s eighteenth problem asked whether there exists an anisohedral polyhedron in Euclidean 3 space Gr nbaum and Shephard suggest ref name Gr nbaum and Shephard, section 9.6 Gr nbaum and Shephard, section 9.6 ref that Hilbert was assuming that no such tile existed in the plane. Reinhardt answered Hilbert s problem in 1928 by finding examples of such polyhedra, and asserted that his proof that no such tiles exist in the plane would appear soon. ref ... unable to show there were no such convex pentagon s, while finding the five pentagon tiling types of convex pentagon tiling the plane isohedrally. ref name Gr nbaum and Shephard, section 9.6 Kershner ... The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the least number of group action orbits equivalence classes of tiles in any tiling of that tile under the action of the symmetry group of that tiling, and that a tile with isohedral number k ... Open Questions in Tiling accessdate 2007 09 09 last Goodman Strauss first Chaim date 2000 01 10 format ... tiling References references External links John Berglund, http www.angelfire.com mn3 anisohedral index.html Anisohedral Tilings Page MathWorld title Anisohedral Tiling urlname AnisohedralTiling Joseph Myers, http www.srcf.ucam.org jsm28 tiling Polyomino, polyhex and polyiamond tiling Category Tiling ... more details
Uniform tiles db Uniform tiling stat table Urth In geometry , the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane . There are one triangle , two Square geometry square s, and one hexagon on each vertex geometry vertex . It has Schl fli symbol of t sub 0,2 sub 3, 6 . John Horton Conway Conway calls it a rhombihexadeltille . ref John H. Conway, Heidi Burgiel, Chaim Goodman Strass, The Symmetries of Things 2008, ISBN 978 1 56881 220 5 http www.akpeters.com product.asp?ProdCode 2205 Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table ref There are 3 List of regular polytopes Euclidean tilings regular and 8 List of uniform tilings semiregular tilings in the plane. Uniform colorings There is only one uniform coloring in a rhombitrihexagonal tiling. Naming the colors by indices around a vertex 3.4.6.4 1232. File Uniform polyhedron 63 t02.png 100px Related polyhedra and tilings This tiling is topologically related as a part of sequence of Cantellation geometry cantellated polyhedra with vertex figure 3.4.n.4 , and continues as tilings of the Hyperbolic space hyperbolic plane . These vertex transitive figures have n32 reflectional Orbifold notation symmetry . class wikitable Image Uniform polyhedron 33 t02.png 100px BR Cuboctahedron 3.4.3.4 BR 332 BR CDD node 1 3 node 3 node 1 Image Uniform polyhedron 43 t02.png 100px BR Rhombicuboctahedron 3.4.4.4 BR 432 BR CDD node 1 4 node 3 node 1 Image Uniform polyhedron 53 t02.png 100px BR rhombicosidodecahedron 3.4.5.4 BR 532 BR CDD node 1 5 node 3 node 1 Image Uniform polyhedron 63 t02.png 100px BR 3.4.6.4 BR 632 BR CDD node 1 6 node 3 node 1 Image Uniform tiling 73 t02.png 100px BR rhombitriheptagonal tiling 3.4.7.4 BR 732 BR CDD node 1 7 node 3 node 1 Image Uniform tiling 83 t02.png 100px BR 3.4.8.4 BR 832 BR CDD node 1 8 node 3 node 1 Gallery class wikitable Image Wallpaper group ... Expand further date December 2009 Category Tiling eo Malgranda rombo tri seslatera kahelaro fr Pavage ... more details
Tiling Arrays are a subtype of DNA microarray microarray chips. They function on a similar principle ... to unlabeled probes fixed on to a solid surface. Tiling arrays differ in the nature of the probes ... millions of copies of one probe. ref Mockler,T, Ecker,J Applications of DNA tiling arrays for whole ... use a few probes for each known or predicted gene. In contrast, tiling arrays can produce .... On top of individual gene expression analysis, other uses of tiling arrays are in transcriptome mapping ..., J, Gregory, B, Ecker, J Mapping the genome landscape using tiling array technology. Current Opinion in Plant Biology, 2007 10 534 542 ref Tiling arrays are quickly becoming one of the most powerful ... There are two main ways of synthesizing tiling arrays the first is photolithographic manufacturing ... tiling array chips is via mechanically printing probes onto the chip. This is done by using ... Liu Getting Started in Tiling Microarray Analysis, PLoS Computational Biology www.ploscompbiol.org, 1842 October 2007 , Volume 3, Issue 10 ref Three manufacturers of tiling arrays are Affymetrix, NimbleGen ... chip ChIP chip is one of the most popular usages of tiling arrays. Chromatin immunoprecipitation is the technique ..., designed to cover regions representative of the genome. However, with tiling arrays, overlapping ... resolution. Besides these advantages, tiling arrays show high reproducibility, and with overlapping probes spanning large segments of the genome, tiling arrays can still interrogate protein ... mapping Another popular use of tiling arrays is in finding expressed genes. Traditional methods ... genes that are only active in response to signals or specific to a time frame. Tiling arrays can ..., Gerstein, M, Snyder, M Applications of DNA tiling arrays to experimental genome annotation and regulatory pathway discover. Chromosome Research 2005 13 259 274 ref ref Liu, S Getting Started in Tiling ... done on chromosome 21 and 22 showed the power of tiling arrays for identifying transcription units ... more details
File Penrose Tiling Rhombi .svg 250px thumb right A Penrose tiling. alt A Penrose tiling is a non periodic tessellation tiling generated by an aperiodic tiling aperiodic set of prototile s named after ... tiles are non periodic, Penrose tiles are considered aperiodic tiles. A Penrose tiling may be constructed ... at the right. A Penrose tiling has many remarkable properties, most notably It is non periodic ... scales. Thus, the tiling can be obtained through inflation or deflation and any finite patch from the tiling occurs infinitely many times. It is a quasicrystal implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold ... Physics and Astronomy, Texas A&M University , standing on a floor with a Penrose tiling ... right thumb 150px Figure 1. Part of a periodic tiling. alt Penrose tilings are simple examples of aperiodic tiling s of the plane geometry plane . ref General references for this article include Harvnb ... Senechal 1996 pp 170&ndash 206 . ref A tessellation tiling is a covering of the plane by tiles ... s , and set mathematics a set of prototiles is said to admit a tiling or tile the plane if there is a tiling ... are periodic tiling periodic a perfect copy of the tiling can be obtained by translation mathematics ... a period of the tiling more informally, this means that a finite region of the tiling repeats itself in periodic intervals. If a tiling has no periods it is said to be non periodic . A set of prototiles is said to be aperiodic if it tiles the plane, but every such tiling is non periodic tilings .... ref Harvnb Gr nbaum Shephard 1987 p 584 ref The color matching required in a tiling by Wang dominoes ... Gr nbaum Shephard 1987 p 525 ref Development of the Penrose tilings anchor Figure 2 File Penrose Tiling P1 over P3 .svg 250px thumb left Figure 2. The pentagonal Penrose tiling P1 drawn in black on a colored rhombus tiling P3 with yellow edges. ref name S 173 174 Harvnb Senechal 1996 pp 173&ndash 174 ... more details
tiling is a tesselation tiling obtained from an aperiodic set of tiles. The term used loosely refers ... sets of tiles any given finite tiling is either periodic or non periodic. Stated more formally, a tiling of the plane is aperiodic if and only if it consists of copies of a finite set of tiles ... other geometric setting, admits a tiling if non overlapping copies of the tiles in the set can be fitted ... problem asked for a single polyhedron tiling Euclidean 3 space, such that no tiling by it is isohedral an anisohedral tiling anisohedral tile . The problem as stated was solved by Karl Reinhardt in 1928 .... ref The specific question of aperiodic tiling first arose in 1961, when logician Hao Wang academic ... for deciding if a given finite set of prototiles admits a tiling of the plane. Wang found ... that admits a tiling of the plane also admits a periodic tiling. but was it Wang or Berger that proved ... tilings of the plane and so are aperiodic. Hence, when in 1966 Robert Berger demonstrated that the tiling ... discovered a single aperiodic prototile in 3 dimensional Euclidean space. While no tiling by this prototile ... strongly aperiodic if it admits no tiling with an infinite cyclic group of symmetries, and that other ... strauss papers survey.pdf title Open Questions in Tiling accessdate 2007 03 24 last Goodman ... the normal notion of tiling . An aperiodic protoset consisting of just one tile in the Euclidean plane ... in the line either cannot be used to form a complete tiling, or can be used to form a periodic tiling. Aperiodicity requires two or more dimensions. Aperiodic hierarchical tilings To date, there is not a formal definition describing when a tiling has a hierarchical structure nonetheless, it is clear ... Knuth , Hans L uchli L uchli and Raphael Robinson Robinson . As with the term aperiodic tiling itself, the term aperiodic hierarchical tiling is a convenient shorthand, meaning something along the lines ... of tiles, in any tiling they admit, forces a particular hierarchical structure. In many later ... more details
Loop tiling , also known as loop blocking , or strip mine and interchange , is a loop optimization used by compiler s to make the execution of certain types of Control flow Loops loop s more efficient. Overview Loop tiling partitions a loop s iteration space into smaller chunks or blocks, so as to help ensure data used in a loop stays in the CPU Cache cache until it is reused. The partitioning of loop iteration space leads to partitioning of large array into smaller blocks, thus fitting accessed array elements into cache size, enhancing cache reuse and eliminating cache size requirements. An ordinary loop source lang c for i 0 i N i ... source can be blocked with a block size B by replacing it with source lang c for j 0 j N j B for i j i min N, j B i .... source where min is a function returning the minimum of its arguments. Example The following is an example of matrix vector multiplication. There are three arrays, each with 100 elements. The code does not partition the arrays into smaller sizes. source lang c int i, j, a 100 100 , b 100 , c 100 int n 100 for i 0 i n i c i 0 for j 0 j n j c i c i a i j b j source After we apply loop tiling using 2 2 blocks, our code looks like source lang c int i, j, x, y, a 100 100 , b 100 , c 100 int n 100 for i 0 i n i 2 c i 0 for j 0 j n j 2 for x i x min i 2, n x for y j y min j 2, n y c x c x a x y b y source The original loop iteration space is n by n . The accessed chunk of array a i, j is also n by n . When n is too large and the cache ... Z J, I Z J, I X K, I Y J, K After loop tiling B B DO K2 1, M, B DO J2 1, M, B DO I 1, M DO K1 K2, MIN ... what value of tiling size is optimal for one loop because it demands an accurate estimate of accessed ..., M. More Iteration Space Tiling . Supercomputing 89, pages 655 664, 1989. Wolf, M. E. and Lam, M ... Partitioning . POPL 88, pages 319 329, 1988. Xue, J. Loop Tiling for Parallelism . Kluwer Academic ... ru Loop tiling ... more details
Image Pavage domino.svg thumb right Domino tiling of a square A domino tiling of a region in the Euclidean plane is a tessellation of the region by domino mathematics domino s, shapes formed by the union of two unit square s meeting edge to edge. Equivalently, it is a Matching graph theory matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares. Height functions For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the nodes of the grid. For instance, draw a chessboard, fix a node math A 0 math with height 0, then for any node there is a path from math A 0 math to it. On this path define the height of each node math A n 1 math i.e. corners of the squares to be the height of the previous node math A n math plus one if the square on the right of the path from math A n math to math A n 1 math is black, and minus one else. More details can be found in harvtxt Kenyon Okounkov 2005 . Thurston s height condition William Thurston 1990 describes a test for determining whether a simply connected region, formed as the union of unit squares in the plane, has a domino tiling. He forms an undirected graph that has as its vertices the points x , y , z in the three dimensional integer lattice , where each such point is connected to four neighbors if x y is even, then x , y , z is connected to x 1, y , z 1 ... Dominoes tiling 8x8.svg thumb right Domino tiling of an 8 8 square using the minimum number of long ... . For the reduced Aztec diamond of order n with only one long middle row, there is only one tiling ... problem , a puzzle concerning domino tiling of a 62 square subset of the chessboard Tatami , floor ... W. P. title Conway s tiling groups journal American Mathematical Monthly volume 97 issue 8 year 1990 ... Recreational mathematics Category Tiling puzzles Category Matching ... more details
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