
	Where results make sense

Topic: Wang tile

	Wang tile - Wikipedia, the free encyclopedia
		Wang tiles (or Wang dominoes), first proposed by Hao Wang in 1961, are equal-sized squares with a color on each edge which give rise to a simple undecidable decision problem.
		This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal.
		Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense.
	en.wikipedia.org /wiki/Wang_tile (638 words)

	[No title]
		Wang Tiles are sets of 2D patches that may be fit together to non-periodically tile a plane.
		Each edge of a Wang Tile is assigned a color; one lays Wang Tiles on a plane such that the colors of adjacent edges match.
		The type of parametric surface determines the number of Wang Tiles needed to tile the surface--in general one needs (colors)^edges tiles to uniquely tile the surface, where edges is the maximum number of edges that need to be matched at any location and colors is the number of colors in use.
	www.cs.unc.edu /~beastwoo/courses/comp236/WangTiles/index.php (670 words)

	Wang Hao - Wikipedia, the free encyclopedia
		Hao Wang, also Wang Hao (Chinese: 王浩; Pinyin: Wáng Hào; 20 May 1921 – 13 May 1995) was a Chinese-American logician, philosopher and mathematician.
		One of the most important contribution of Wang was the invention of Wang tile.
		Noted the first example of aperiodic tiling is a set of Wang tiles, which Wang had once conjectured its nonexistence, was discovered by Robert Berger in 1966.
	en.wikipedia.org /wiki/Hao_Wang (268 words)

	Penrose Tilings (Site not responding. Last check: 2007-10-20)
		A periodic tiling is one on which you can outline a region of the tiling and tile the plane by translating copies of that region (without rotating or reflecting).
		An aperiodic tiling is one where if we repeat the exercise with the transparent paper we will not find another position where the outlines of the tiles will match with the tiles underneath except for the starting position.
		Wang conjectured that there exists a procedure for deciding whether any set of dominoes will tile by placing them such that adjacent edges are the same color.
	www.math.ubc.ca /~cass/courses/m308-02b/projects/schweber/tilings.html (360 words)

	Simonsays.com > SimonSays > The Good Earth (Trade Paperback) > Read an Excerpt (Site not responding. Last check: 2007-10-20)
		Wang Lung dipped some into a bowl, and then, after a moment, he opened a glazed jar that stood upon a ledge of the stove and took from it a dozen or so of the curled dried leaves and sprinkled them upon the surface of the water.
		Wang Lung was incensed at such impudence and he would have risen except that when he thought of going into the great House of Hwang and of asking there for a woman, sweat broke out over his whole body as though he were working in a field.
		Wang Lung nodded and left her and did not see her again until the guests came crowding in, his uncle jovial and sly and hungry, his uncle's son an impudent lad of fifteen, and the farmers clumsy and grinning with shyness.
	www.simonsays.com /content/book.cfm?sid=33&pid=499367&agid=2 (7118 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Wang tilings have the property that using arbitrary many copies of each tile, an infinite plane can be tiled so that the tiling is aperiodic.
		This set of Wang tiles has a remarkable property that there are infinitely many aperiodic tilings of an infinite Euclidean plane.
		Then, at the end, I'll use a small trick used by Culik to reduce the size of the set to 13 tiles with the same properties and conclude with a conjecture that a tile set of 12 tiles also exists that is aperiodic.
	www.cs.uiowa.edu /~sriram/arg/imranAbstract.html (273 words)

	Dr. Matrix at the Scientium: Programming Challenge! (Site not responding. Last check: 2007-10-20)
		A periodic tiling is one on which you can outline a region that tiles the plane by translation, that is, by shifting the position of the region without rotating or reflecting it.
		Wang's problem was to find a procedure for deciding whether any given set of dominoes will tile by placing them so that abutting edges are the same color.
		Wang conjectured that any set of tiles which can tile the plane can tile it periodically and showed that if this is the case, there is a decision procedure for such tiling.
	scientium.com /drmatrix/puzzles/progchal.htm (3899 words)

	The Geometry Junkyard: Tilings
		Tilings can be divided into two types, periodic and aperiodic, depending on whether they have any translational symmetries.
		The regular tiling by hexagons can be repeatedly subdivided and recombined into a tiling by hexagons 1/7 the size of the original, to form an interesting recursive structure.
		Some planar tilings generated by the lattice projection method (of which the Penrose tiling is a special case) by Andrew Lewis, Queens U. SpaceBric building blocks and Windows software based on a tiling of 3d space by congruent tetrahedra.
	www.ics.uci.edu /~eppstein/junkyard/tiling.html (1767 words)

	DNA Computing by Self-Assembly (Site not responding. Last check: 2007-10-20)
		For each tiling, the set of shapes must be finite; for example, the tile set could consist of an octagon and a square, both with unit-length sides.
		One motivation for studying tiling is that the tiles correspond to the periodic arrangement of atoms in crystals.
		In fact, the tiles Wang used were all essentially square, distinguished only by labels on their sides that had to match up when the tiles were juxtaposed.
	www.nae.edu /nae/bridgecom.nsf/weblinks/MKUF-5UZJFP?OpenDocument (3525 words)

	TeX2HTML - Converting TeX to HTMLA Difficult Job in the Tiling Industry
		A tiling of the plane with periodic squares is said to be a periodic tiling.
		Wang conjectured that any set of tile types which could be used to tile the plane could be used to tile a periodic square.
		Wang's Conjecture remained an open question for several years but, unfortunately for your tiling business, was shown to be false in the thesis work of one of Wang's students, Robert Berger [1].
	www.users.drew.edu /~kmadden/paper.htm (3325 words)

	CSEE Colloquium \| OGI School of Science & Engineering (Site not responding. Last check: 2007-10-20)
		For theoretical explorations of this connection we use the Tile Assembly Model, a model of pseudo-crystalline self-assembly in which self-assembling objects are abstracted as Wang tiles--square tiles with four edge labels that represent potential binding interactions.
		Such tile systems are Turing Universal; given any computer program a set of tiles may be designed such that self-assembly of the tiles corresponds to execution of the computer program.
		Under the Tile Assembly Model this complexity undergoes a dramatic decrease from N^2 tiles to O(log N) tiles, as the parameter T is increased from 1 (modelling noncooperative intermolecular bonds) to 2(modelling cooperative bonds).
	www.cse.ogi.edu /colloquia/event/183.html (563 words)

	tile-based texture mapping - Beyond3D Forum (Site not responding. Last check: 2007-10-20)
		Tiles are used mirrored and rotated as well, and vary in brightness.
		What aths did is offline generation of tile textures, which of course hits the texture size limit of the hardware quite fast.
		I basically took the tile generation method from this well-known paper by Cohen et al., dropped the overlap area in the middle, and got away with huge memory savings by removing redundant tile parts where possible.
	www.beyond3d.com /forum/showthread.php?t=23341 (1581 words)

	DNA-Based Computation (Site not responding. Last check: 2007-10-20)
		The tiles can form a mosaic, such as the one shown at the bottom, according to the rule that each edge in the mosaic is flanked by the same color.
		If one counts columns, one can see that there is a special tile in the top row in the 5th and 9th column, and a somewhat different special tile in the top row of the 14th column, corresponding to the sum of 5 + 9.
		To ensure that the impact of both sticky ends was felt, longer sticky ends were assigned to those tiles connecting the X and C tiles, than to those involving the Y tiles; hence, the X and C tiles assembled first in a cooling protocol.
	seemanlab4.chem.nyu.edu /XOR.html (861 words)

	Nat' Academies Press, Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2003 NAE Symposium on ...
		Tiling Theory A tiling is an arrangement of a few basic shapes (called tiles) that fit to- gether perfectly in the infinite plane.
		One motivation for studying tiling is that the tiles corre- spond to the periodic arrangement of atoms in crystals.
		The problem, then, is to ensure that the growth process results in tile ar- rangements in which all tiles match with their neighbors.
	www.nap.edu /books/030909139X/html/103.html (6084 words)

	News & Features: DNA-rule-driven fractals probe biochemical self assembly - computationally & in vitro
		Because this nucleating strand serves as the bottom of these tiles, only four strands are needed to assemble the input tiles, and an additional capping strand is used to form a double-helix between input tiles on the nucleating strand.
		In the preformed tile approach, each tile is prepared separately by mixing a stoichiometric amount of each component strand in the hybridization buffer and then annealing from 90 °C to room temperature over the course of several hours.
		Frequently, as in Figure 5E, the identity of obscured or missing tiles was deduced from the neighboring tiles by assuming correct information propagation (the imperfection often being caused by sample preparation or by interaction with the AFM tip rather than by errors during assembly).
	www.iscid.org /boards/ubb-get_topic-f-1-t-000215.html (4500 words)

	The Penrose tilings (Site not responding. Last check: 2007-10-20)
		A periodic tiling of the plane is one on which you can outline a region that tiles the plane by translation.
		In other words, if the plane is divided into tiles, then the tiling is periodic if you can slide the plane onto itself by a translation so that the tiles of the original copy fit exactly the tiles of the new copy.
		His tiles (Wang dominoes) were unit squares with edges colored in different ways, and tilings must be constructed by placing then side by side so that colors match.
	www.csun.edu /~ac53971/courses/math623/spring02/penrose.html (391 words)

	Rendering of a sidewalk - Beyond3D Forum (Site not responding. Last check: 2007-10-20)
		Inspired by Xmas' general idea to compose a large texture out of smaller tiles, I tried to do that for a sidewalk, since it is also a sort of compression of photographed textures, which is the main topic for my diploma thesis.
		Wang Tiles for Image and Texture Generation is close to what you're doing, but you might also be interested in: Graph Cut Textures
		Nice, I think the cleaning of tile edges is a good idea, but what about combining a normal map, that may sort out both areas.
	www.beyond3d.com /forum/showthread.php?t=18751 (2210 words)

	Boehm DNA Study - WiseNano
		Every molecular (Wang) tile is a DNA-double crossover molecule having four “sticky ends” (single strand overhangs comprised of five bases) operating as binding domains that are programmable.
		The assembly of a row of input tiles is guided by a long single oligonucleotide which serves as a scaffold.
		The bottoms of these tiles is provided by the nucleating strand, and an extra capping strand is used to construct a duplex between tiles on the nucleating strand.
	wise-nano.org /w/Boehm_DNA_Study (14697 words)

	Diary, February 2003
		With the Wang Tile the sides of each square have to match, where in the square grids the corners must be matching.
		Yesterday evening, I continued working on a program for establishing minimal requirements for a set of tiles to be able to tile an infinite square grid.
		It is the set of four tiles where each tile has three corners with 0 and one with 1.
	www.iwriteiam.nl /D0302.html (4034 words)

	flipCode - Game Development News & Resources (Site not responding. Last check: 2007-10-20)
		Wang Tiles for Image and Texture Generation by Michael Cohen (Microsoft Research), Jonathan Shade (Wild Tangent), Stefan Hiller, and Oliver Deussen (Dresden University of Technology) is a texture synthesis method that can reduce tiling artifacts on terrains.
		The Wang Tile paper describes how to fit a set of related textures together to form an arbitrarily large plane that has no repetitive pattern and no seams.
		View-Dependent Displacement Mapping is a technique for true per-pixel displacement of surfaces by Lifeng Wang, Xi Wang, Xin Tong, Stephen Lin, Shimin Hu, Baining Guo, and Heung-Yeung Shum of Microsoft Research Asia and Tsinghua University.
	www.flipcode.com /misc/siggraph2003.shtml (3301 words)

	DNAcomp2.html (Site not responding. Last check: 2007-10-20)
		If you do, you are being reminded of a game known as Wang Tiles.
		Thus, this tile method could harness the raw computational power of DNA with efficiency.
		If all goes well a single test tube of DNA tiles is predicted to perform at a rate almost 1 million times faster than an electronic computer or about 10 trillion additions per second.
	www.udel.edu /physics/scen103/CGZ/DNAcomp2.html (226 words)

	WQFS
		Water sampling frequency effects on estimates of nitrate-N losses from subsurface tile drains.
		Pesticide transport to subsurface tile drains in humid regions of North America.
		Pesticide and nutrient movement into subsurface tile drains on a silt loam in Indiana.
	www.agry.purdue.edu /water/fieldstn/Publications.htm (439 words)

	Technology Review: Emerging Technologies and their Impact (Site not responding. Last check: 2007-10-20)
		Among the most promising of these new approaches are smart "DNA tiles" invented by Erik Winfree, a 30-year-old computer scientist at California Institute of Technology (see "100 Young Innovators," TR November/December 1999).
		Winfree's brainstorm is to create nanoscopic building blocks out of DNA that not only can store data but are designed-Winfree likes to say "programmed"-to carry out mathematical operations by fitting together in specific ways.
		But Winfree's DNA tiles are made by knotting together three or more of these strands, forming "tiles" about 15 nanometers (billionths of a meter) on their longest side.
	www.technologyreview.com /BioTech/wtr_12108,312,p2.html (494 words)

	Who's Who In Nashville - Feature Stories, Spring/Summer 2004
		This would also mean Dr. Wang would never earn more than fifty cents a day, for the rest of his life, ensuring him a life of poverty.
		Like Dr. Wang, Tian Ma was all too familiar with hardship, losing his father to execution by the government.
		Today, Dr. Wang is one of the few LASIK surgeons in the world who holds a doctorate degree in laser physics.
	www.whoswhoinnashville.com /issue4_ss_2004/stories_ss_2004.htm (10672 words)

	The Geometry Junkyard: Recent Additions
		Marcin Malinowski still has some Escher-like tiling patterns on his home page, although his old Escheresque wallpaper group page seems to be irretrievably gone.
		Lecture notes from the Clay Math Institute, by Richard Stanley and Federico Ardila, discussing polyomino tilings, coloring arguments for proving the nonexistence of tilings, counting how many tilings a region has, the arctic circle theorem for domino tilings of diamonds, tiling the unit square with unit-fraction rectangles, symmetry groups, penrose tilings, and more.
		Waterman polyhedra, formed from the convex hulls of centers of points near the origin in an alternating lattice.
	www.ics.uci.edu /~eppstein/junkyard/recent.html (1090 words)

	The Geometry Junkyard: All Topics
		This shape, constructed by inscribing circular arcs in a spiral tiling of squares, resembles but is not quite the same as a logarithmic spiral.
		Heureka, the Finnish science center uses Penrose tiles to pave the area in front of its main entrance.
		William Thurston answers a question of Greg Kuperberg, on whether there is a constant C such that every convex body in the hyperbolic plane can be packed with density C. The answer is no -- long skinny bodies can not be packed efficiently.
	www.ics.uci.edu /~eppstein/junkyard/all.html (9756 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Used a set of 8 Wang tiles (from given paper) 4.
		Have a minimum of 3 and a maximum of 8 different styles (geometry and/or textures) of grass clumps, one for each tile (or shared by a subset of tiles if not using 8 distinct styles).
		Switching between different render modes, showing i) Just the terrain (no grass) with properly lit surface (can be flat shaded - does not have to be Gauraud shaded by averaging the normals).
	www.cs.up.ac.za /download.php/GRF780/Assignments/feature_checklist.txt (305 words)

	Active Skim View of: Biomolecular Computing
		quences given to the tiles' sticky ends could be used to program different periodic arrangements of tiles (Winfree et al., 1998a)
		A ~~ ~~_~ ~:~ ~~ ~~ ~~ 25 nanometers of I' ~_~A^~ ~~:~ ~~ FIGURE 3 DNA double-crossover molecules can implement abstract Wang tiles, producing a two-dimensional lattice of DNA with binding interactions dictated by the DNA sticky ends.
		Small tile sets have been designed for demultiplexers, such as the ones necessary to access a RAM memory (shown in Figure 5)
	www.nap.edu /nap-cgi/skimit.cgi?isbn=030909139X&chap=103-130 (959 words)

	Erik Winfree CS Talk : Algorithmic Self-Assembly of DNA (Site not responding. Last check: 2007-10-20)
		This talk will focus on a single biochemical mechanism, the self-assembly of DNA structures, that is theoretically sufficient for Turing-universal computation.
		The theory combines Hao Wang's purely mathematical Tiling Problem with the branched DNA constructions of Ned Seeman.
		I will introduce branched DNA structures, illustrate the connection to the Tiling Problem, discuss biophysical issues that will determine error rates in any real system, and finally show experimental results confirming the self-assembly of a two-dimensional lattice of DNA Wang tiles.
	www.wisdom.weizmann.ac.il /~naor/SEMINARS/winfree_march18.html (174 words)

Try your search on: Qwika (all wikis)