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 , where windows do not overlap In computing, a tiled rendering , the process of subdividing an 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 that 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 Tessellation 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
Uniform hyperbolic tiles db Reg hyperbolic tiling stat table U73 0 In geometry , the heptagonal tiling is a regular hyperbolic tiling regular tiling of the Hyperbolic geometry hyperbolic plane . It is represented ... Poincare halfplane eptagonal hb.svg 480px BR Heptagonal tiling black lines in a Poincar half plane model Related polyhedra and tilings This tiling is topologically related as a part of sequence of regular ... tiling 6,3 align center File Uniform tiling 73 t0.png 100px BR 7,3 File Uniform tiling 83 t0.png 100px BR 8,3 File Uniform tiling 93 t0.png 100px BR 9,3 File H2 tiling 23i 1.png 100px BR ,3 The dual tiling is the order 7 triangular tiling . class wikitable File Uniform tiling 73 t0.png 100px br Heptagonal tiling File Uniform tiling 73 t2.png 100px br order 7 triangular tiling Wythoff constructions ... tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms. class wikitable Tiling Schl fli symbol Schl fli ... tiling t sub 0 sub 7,3 3 & 124 7 2 7 sup 3 sup Image Uniform tiling 73 t0.png 80px Truncated heptagonal tiling t sub 0,1 sub 7,3 2 3 & 124 7 3.14.14 Image Uniform tiling 73 t01.png 80px Rectified heptagonal tiling BR Triheptagonal tiling t sub 1 sub 7,3 2 & 124 7 3 3.7 sup 2 sup Image Uniform tiling 73 t1.png 80px Bitruncated heptagonal tiling BR Order 7 truncated triangular tiling t sub 1,2 sub 7,3 2 7 & 124 3 7.6.6 Image Uniform tiling 73 t12.png 80px Order 7 triangular tiling t sub 2 sub 7,3 7 & 124 3 2 3 sup 7 sup Image Uniform tiling 73 t2.png 80px Cantellated heptagonal tiling BR Rhombitriheptagonal tiling t sub 0,2 sub 7,3 7 3 & 124 2 3.4.7.4 Image Uniform tiling 73 t02.png 80px Omnitruncated heptagonal tiling BR Truncated triheptagonal tiling t sub 0,1,2 sub 7,3 7 3 2 & 124 4.7.14 Image Uniform tiling 73 t012.png 80px Snub heptagonal tiling s 7,3 & 124 7 3 2 3.3.3.3.7 Image Uniform tiling 73 snub.png 80px Hurwitz surfaces File Order 3 heptakis heptagonal tiling.png thumb The symmetry ... 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 ... tiling Hextille and triangular tiling deltille . There are 3 List of regular polytopes Euclidean ... There are two distinct uniform coloring s of a trihexagonal tiling. Naming the colors by indices ... 63 t1.png 100px File Uniform tiling 333 t01.png 100px align center Wythoff symbol 2 & 124 6 3 3 3 & 124 ... polyhedra and tilings This tiling is topologically part of sequence of Rectification geometry rectified ... 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 ... polyhedron uniform polyhedra there are eight uniform tiling s that can be based from the regular hexagonal tiling or the dual triangular tiling . Drawing the tiles colored as red on the original faces ... distinct. The truncated triangular tiling is topologically identical to the hexagonal tiling. class wikitable Parent Truncated Rectified Bitruncated Birectified BR dual Cantellated Omnitruncated ... sup 3.12.12 3.6.3.6 6.6.6 3 sup 6 sup 3.4.6.4 4.6.12 3.3.3.3.6 align center Hexagonal tiling Image Uniform tiling 63 t0.png 64px BR Hexagonal tiling 6,3 Image Uniform tiling 63 t01.png 64px BR Truncated hexagonal tiling 3.12.12 Image Uniform tiling 63 t1.png 64px BR Trihexagonal tiling 3.6.3.6 Image Uniform tiling 63 t12.png 64px BR Hexagonal tiling 6.6.6 Image Uniform tiling 63 t2.png 64px BR Triangular tiling 3,6 Image Uniform tiling 63 t02.png 64px BR Small rhombitrihexagonal tiling 3.4.6.4 ... 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 5,3 , having 3 pentagons around reach vertex. In the hyperbolic plane , there are tilings of regular 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 hyperbolic 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 ... 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 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 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 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 250px See also Rhombitrihexagonal tiling Order 3 heptagonal tiling Tilings of regular polygons List of uniform tilings Kagome lattice ... urlname HyperbolicTiling title Hyperbolic tiling MathWorld urlname PoincareHyperbolicDisk title Poincar hyperbolic disk http bork.hampshire.edu bernie hyper Hyperbolic and Spherical Tiling Gallery ... Tessellations, Don Hatch DEFAULTSORT Rhombitriheptagonal Tiling Category Tessellation geometry stub ... more details
A tiling with rectangles is a tessellation tiling which uses rectangle s as its parts. Congruent rectangles Some tiling of rectangles include class wikitable File Stacked bond.png 150px br Stacked bond File Wallpaper group cmm 1.jpg 150px br Running bond File Wallpaper group p4g 1.jpg 150px br Basket weave File Basketweave bond.svg 150px br Basket weave File Herringbone bond.svg 150px br Herringbone pattern 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. ref name x Journal of Recreational Mathematics , 28 1, p.64 ref 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. ref name x See also Squaring the square tessellation tiling puzzle Notes reflist Category Tessellation geometry stub eo Kahelaro el ortanguloj ... 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 p6m, 6,3 , 632 BR p3m1, 3 sup 3 sup , 333 Face Type V3.6.3.6 Dual Trihexagonal tiling Property List edge transitive face transitive In geometry , the rhombille tiling ref John H. Conway, Heidi Burgiel, Chaim Goodman Strass, The Symmetries ... and one with six rhombi. It can be seen as a hexagonal tiling with each hexagon divided into three .... Dual tiling This is the Dual polyhedron dual of the trihexagonal tiling . ref MathWorld urlname DualTessellation ... used as a floor or wall tiling, sometimes with one fatter rhombus or square and two more ... As a face transitive dual uniform tiling all the rhombi are the same, but with larger symmetry fundamental ... colors colspan 3 Three 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 class wikitable width 400 align center File Rhombic star tiling 2 vertices.png 200px BR 2 colors 632 File Rhombic star tiling 3 vertices.png 200px BR 3 colors 333 colspan 2 BR The single colored rhombus tiling has 632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a 333 symmetry. Related polyhedra and tilings This tiling is a part of a sequence .... class wikitable width 500 colspan 3 Spherical polyhedron Spherical polyhedra Euclidean tiling colspan 2 Hyperbolic tiling 3,3 4,3 5,3 6,3 7,3 8,3 V3.3.3.3 V3.4.3.4 V3.5.3.5 V3.6.3.6 V3.7.3.7 V3.8.3.8 ... til.png 100px File Uniform dual tiling 433 t01 yellow.png 100px Similarly it relates to the infinite ... center valign top Image Tile 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 disk model Other rhombic tilings It has the same vertex arrangement ... 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 Cite book title The Beauty of Geometry Twelve Essays year 1999 publisher Dover Publications lccn 99035678 ... urlname HyperbolicTiling title Hyperbolic tiling MathWorld urlname PoincareHyperbolicDisk title Poincar hyperbolic disk http bork.hampshire.edu bernie hyper Hyperbolic and Spherical Tiling Gallery ... 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 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 Conway, 2008, p288 table ref It can be considered a Cantellation cantellated or Expansion geometry expanded hexagonal tiling by Norman Johnson mathematician Johnson s operational language. There are 3 List of regular polytopes Euclidean tilings ... 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 ... 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 ... polyhedra there are eight uniform tiling s that can be based from the regular hexagonal tiling or the dual triangular tiling . Drawing the tiles colored as red on the original faces, yellow at the original ... distinct. The truncated triangular tiling is topologically identical to the hexagonal tiling. class ... 6.6.6 3 sup 6 sup 3.4.6.4 4.6.12 3.3.3.3.6 align center Hexagonal tiling Image Uniform tiling 63 t0.png 64px BR Hexagonal tiling 6,3 Image Uniform tiling 63 t01.png 64px BR Truncated hexagonal tiling 3.12.12 Image Uniform tiling 63 t1.png 64px BR Trihexagonal tiling 3.6.3.6 Image Uniform tiling 63 t12.png 64px BR Hexagonal tiling 6.6.6 Image Uniform tiling 63 t2.png 64px BR Triangular tiling 3,6 Image Uniform tiling 63 t02.png 64px BR Small rhombitrihexagonal tiling 3.4.6.4 Image Uniform tiling 63 t012.png 64px BR Great rhombitrihexagonal tiling 4.6.12 Image Uniform tiling 63 snub.png 65px BR Snub hexagonal tiling 3.3.3.3.6 Gallery class wikitable Image Wallpaper group p6m 4.jpg 166px ... 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 ... 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 Q sigma ... 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. class wikitable align center 1111 1112 i 1212 1213 align center Image Square tiling uniform coloring 1.png 80px Image Square tiling uniform coloring 2.png 80px Image Square tiling uniform coloring 7.png 80px Image Square tiling uniform coloring 8.png 80px align center colspan 4 p4m ... align center Image Square tiling uniform coloring 3.png 80px Image Square tiling uniform coloring 6.png 80px Image Square tiling uniform coloring 4.png 80px Image Square tiling uniform coloring 5.png 80px Image Square tiling uniform coloring 9.png 80px align center colspan 2 c2 BR &infin ,2 sup sup ... polyhedra and tilings This tiling is topologically related as a part of sequence of regular polyhedra ... , p 3,4,5... class wikitable align center File Uniform tiling 432 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 ... from the regular square tiling. Drawing the tiles colored as red on the original faces, yellow at the original ... identically, there are only three unique topologically forms square tiling , truncated square tiling , snub square tiling . class wikitable 4 4 2 Parent Truncated Rectified Bitruncated Birectified ... Image BR Vertex figure Image Uniform tiling 44 t0.png 64px BR Square tiling 4,4 Image Uniform tiling 44 t01.png 64px BR Truncated square tiling 4.8.8 Image Uniform tiling 44 t1.png 64px BR Square tiling ... 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 ... of the greek letter delta &Delta . The triangular tiling is roughly the kishextile. It is one of List ... tiling and the hexagonal tiling . This vertex arrangement is called the A2 lattice . ref ... honeycomb . Uniform colorings There are 9 distinct uniform coloring s of a triangular tiling. Naming ... 111111 121212 121314 121213 align center Coloring Image Uniform tiling 63 t2.png 60px Image Uniform tiling 333 t1.png 60px Image Uniform tiling 333 snub.png 60px File Uniform tiling 63 h12.png 60px ... Uniform triangular tiling 111222.png 60px File Uniform triangular tiling 112122.png 60px File Uniform triangular tiling 111112.png 60px File Uniform triangular tiling 111212.png 60px File Uniform triangular tiling 111213.png 60px align center List of planar symmetry groups Symmetry 2 22 BR cmm BR ... 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 Wythoff constructions from hexagonal and triangular tilings Like the Uniform polyhedron uniform polyhedra there are eight uniform tiling s that can be based from the regular hexagonal tiling or the dual triangular tiling . Drawing the tiles ... edges, there are 8 forms, 7 which are topologically distinct. The truncated triangular tiling is topologically identical to the hexagonal tiling. class wikitable Parent Truncated Rectified Bitruncated ... align center Hexagonal tiling Image Uniform tiling 63 t0.png 64px BR Hexagonal tiling 6,3 Image ... 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. The Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal ... File Kissing 2d.svg The densest sphere packing circle packing is arranged like the hexagons in this tiling ... nanotube zigzag 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 ... There are 3 distinct uniform coloring s of a hexagonal tiling, all generated from reflective ... uniform colspan 2 2 uniform colspan 2 3 uniform Picture Image 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 File Hexagonal tiling 2 1.png 100px File Hexagonal tiling 7 colors.png 100px align center Colors 1 2 3 2 4 2 7 align center h,k 1,0 colspan 2 1,1 ... 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 ... pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted ... 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 colspan 2 ... pattern as hexagonal tiling.png 240px Related polyhedra and tilings This tiling is topologically ... more details
Pinwheel tilings are aperiodic tiling non periodic tilings defined by Charles Radin and based on a construction due to John H. Conway John Conway . They are the first known non periodic 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 ... of triangles which defines the Conway tiling of the plane. By suitably rescaling and translating rotating ... all made of isometric 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 ... tiling tiles can be grouped five by five thick lines to form a new Pinwheel tiling up to rescaling ... property holds. Given any Pinwheel tiling math P math , there is a Pinwheel tiling math ... form up to rescaling a new 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 ... proposed a three dimensional 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 ... Some Generalizations of the Pinwheel Tiling journal Discrete and Computational Geometry volume 20 issue ... fa ades Federation Square , a building complex in Melbourne, Australia features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub framing for the facades .... The pinwheel tiling system was based on the single triangular element, composed of zinc, perforated ... of its construction is based on pre fabrication and repetition. The same pinwheel tiling system is used ... pinwheel Pinwheel at the Tilings Encyclopedia DEFAULTSORT Pinwheel Tiling Category Discrete geometry ... more details
Image Quaquaversal Tiling.svg 200px thumb The quaquaversal tiling is a aperiodic tiling nonperiodic tiling of the euclidean 3 space introduced by John Horton 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. These rotations are dense in SO 3 . References citation last1 Conway first1 John H. author1 link John Horton Conway last2 Radin first2 Charles doi 10.1007 s002220050221 issue 1 journal Inventiones Mathematicae mr 1618635 pages 179 188 title Quaquaversal tilings and rotations volume 132 year 1998 . citation last1 Radin first1 Charles last2 Sadun first2 Lorenzo doi 10.1006 jabr.1997.7320 issue 2 journal Journal of Algebra mr 1617675 pages 611 633 title Subgroups of SO 3 associated with tilings volume 202 year 1998 . 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 Tessellation ... 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 fa ... more details
File Distorted truncated square tiling.png thumb A Pythagorean tiling File Jacob Ochtervelt Street Musicians ... tiling In geometry , the Pythagorean tiling or two squares tessellation is a tessellation ... four squares of the other size on its four sides. A tiling of this type may be formed by squares of any ... Tile Inc. date August 2008 . ref Topology and symmetry The Pythagorean tiling is the unique tiling ... each two squares of the same size can be mapped into each other by a symmetry of the tiling ... b39h2mar.ps.gz volume 39 year 1998 . ref Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagon s. ref citation last1 Gr nbaum first1 ... tiling are adjacent to four larger tiles, as are the squares in the truncated square tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate between large and small, just as the octagons in the truncated square tiling, however, the two tilings have different sets of symmetries the truncated square tiling has dihedral group dihedral symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic group cyclic ... tiling is a tiling in which each tile is a regular polygon and in which there is a symmetry, mapping ... triangles and regular hexagons, and one more, the Pythagorean tiling. ref harvtxt Gr nbaum ... and by Henry Perigal right This tiling is called the Pythagorean tiling because it has been used ... 1987 , p. 94. ref If the sides of the two squares forming the tiling are the numbers a and b , then the closest ..., the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple 5,12,13 . By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate ... Aperiodic tilings File Pythagorean tiling section.svg thumb 240px An aperiodic sequence generated from ... 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 cite journal last Reinhardt first Karl year 1928 title Zur Zerlegung der euklidischen R ume in kongruente Polytope journal Sitzungsberichte der Preussischen Akamemie der Wissenschaften Berlin, Physikalisch ... 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 ... ref Isohedral numbers The problem of anisohedral tiling has been generalised by saying that the isohedral ... in any tiling of that tile under the action of the symmetry group of that tiling, and that a tile ... strauss papers survey.pdf title Open Questions in Tiling accessdate 2007 09 09 last Goodman ... translation . http www.srcf.ucam.org jsm28 tiling References references External links John Berglund ... Tiling urlname AnisohedralTiling Joseph Myers, http www.srcf.ucam.org jsm28 tiling Polyomino, polyhex and polyiamond tiling Category Tessellation ... more details
. An aperiodic tiling is a tesselation tiling obtained from an aperiodic set of tiles. A set ... such tilings are non periodic tiling periodic . Properly speaking, aperiodicity is a property of the set ... plane or some other geometric setting, admits a tiling if non overlapping copies of the tiles in the set ... polyhedron tiling Euclidean 3 space, such that no tiling by it is isohedral an anisohedral tiling anisohedral ... tiling first arose in 1961, when logician Hao Wang academic Hao Wang tried to determine whether ... finite set of prototiles admits a tiling of the plane. Wang found algorithms to enumerate the tilesets ... algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling. but was it Wang or Berger that proved the other direction? File Wang ... are aperiodic. Hence, when in 1966 Robert Berger mathematician Robert Berger demonstrated that the tiling ... in 3 dimensional Euclidean space. While no tiling by this prototile admits a translation as a symmetry ... Strauss suggested that a protoset be considered strongly aperiodic if it admits no tiling with an infinite ... in Tiling accessdate 2007 03 24 last Goodman Strauss first Chaim date 2000 01 10 format PDF ref ... 10.1007 BF00239998 ref this overlapping goes beyond the normal notion of tiling . An aperiodic protoset ... 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 ... aperiodic tiling itself, the term aperiodic hierarchical tiling is a convenient shorthand, meaning ... structure . Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure. In many later examples, this structure can be described as a substitution tiling system this is described below . No tiling admitted by such a set of tiles can be periodic, simply because ... tiles Image Robinson tiles.svg center thumbnail 200px The Robinson Tiles Any tiling by these tiles ... more details
coverage within tiling array applications. Tiling Arrays are a subtype of DNA microarray microarray ... labeled DNA or RNA target molecules to probes fixed onto a solid surface. Tiling arrays differ from ... of known or predicted genes which may be dispersed throughout the genome , tiling arrays probe ... unknown. Tiling arrays aid in transcriptome mapping as well as in discovering sites of DNA protein ... Gregory, BD, Ecker, JR title Mapping the genome landscape using tiling array technology. journal Current ..., S, Sundaresan, A, Chen, H, Jacobsen, SE, Ecker, JR title Applications of DNA tiling arrays for whole ... 28 May 2011 doi 10.1126 science.1088305 ref Tiling arrays are a useful tool in genome wide association ... of synthesizing tiling arrays the first is photolithographic manufacturing and the second is mechanically ..., each of which contains millions of copies of one probe. The other way of synthesizing tiling array ... last Liu first XS title Getting started in tiling microarray analysis. journal PLoS computational biology ... journal.pcbi.0030183 pmc 2041964 ref Three manufacturers of tiling arrays are Affymetrix, NimbleGen ... of the most popular usages of tiling arrays. Chromatin immunoprecipitation is the technique whereby ... regions representative of the genome. However, with tiling arrays, overlapping probes or probes in very ..., tiling arrays show high reproducibility, and with overlapping probes spanning large segments of the genome, tiling arrays can still interrogate protein binding sites, which harbor repeats .... Transcriptome mapping Another popular use of tiling arrays is in finding expressed genes. Traditional ... to a time frame. Tiling arrays can solve these issues in mapping the transcriptome. Due to the high ... of DNA tiling arrays to experimental genome annotation and regulatory pathway discovery. journal ... of tiling arrays for identifying transcription units. ref cite journal last Cawley first S coauthors ... of transcribed sequences in Arabidopsis thaliana by using high resolution genome tiling ... 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. Penrose tilings .... The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry , as in the diagram at the right. A Penrose tiling has many remarkable ... , so the same patterns occur at larger and larger 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 ... , standing on a floor with a Penrose tiling alt Background and history Periodic and aperiodic tilings anchor Figure 1 File RegularRhombs.svg 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 ... tiling is a covering of the plane by tiles with no overlaps or gaps the tiles normally have ... to admit a tiling or tile the plane if there is a tiling of the plane using only tiles congruence ... familiar tilings e.g., by squares or triangles are periodic tiling periodic a perfect copy of the tiling ... direction. Such a translation is called 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 ... such tiling is non periodic tilings by aperiodic sets of prototiles are called aperiodic tilings ... matching required in a tiling by Wang dominoes can easily be achieved by modifying the edges of the tiles ... 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 ... more details
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