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Topic: Aperiodic tiling

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	Penrose tiling
		The most elegant of Penrose tilings use two rhombi, a thick one called a "kite" and a thin one called a "dart," which are fitted together so that no two tiles are aligned to form a single parallelogram (otherwise, a single rhombus could be used to make a periodic tiling).
		The only rules for assembling the tiles to ensure an aperiodic tiling are that two adjacent vertices must be of the same color, and two adjacent edges must have arrows pointing in the same direction or no arrows at all.
		In a correct Penrose tiling, the ratio of kites to darts is always the same and is equal to the golden ratio.
	www.daviddarling.info /encyclopedia/P/Penrose_tiling.html (357 words)

	Tiling - Biocrawler (Site not responding. Last check: 2007-09-10)
		In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well.
		A fault line or breaking line of a tiling is a straight line from one point of the boundary of S to another point of the boundary of S such that the line has no point in common with the interior of any tile of the tiling.
		He often modified the polygons in his tilings slightly to turn them into shapes of animals etc. Some of his tilings have an interesting morphing property; e.g., a friese may start as a tiling using fish shapes and slowly turn into a tiling using bird shapes as you go from left top right.
	www.biocrawler.com /encyclopedia/Tiling (2570 words)

	Aperiodic tiling - Wikipedia, the free encyclopedia
		An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern.
		Aperiodic tiling is also relevant in the formation of quasicrystals.
		The question of aperiodic tiling first arose in 1961, when logician Hao Wang tried to determine whether the tiling problem was decidable: i.e.
	en.wikipedia.org /wiki/Aperiodic_tiling (358 words)

	Informat.io on Penrose Tiling
		A Penrose tiling is pattern of tiles, discovered by Roger Penrose and Robert Ammann, which could completely cover an infinite plane, but only in a pattern which is non-repeating (aperiodic).
		The tiles are put together with one rule: no two tiles can be touching so as to form a single parallelogram.
		The images show tilings which on top of that have five-fold rotational symmetry with respect to one center point, and also mirror-image symmetry with respect to a symmetry line (and hence 5 of these) through the center.
	www.informat.io /?title=penrose-tiling (673 words)

	Aperiodic Tilings
		The first known aperiodic set was developed by Berger, and consisted of 20426 tile shapes(!), although he soon reduced this to 104 tiles.
		In fact, it's not much of an exaggeration to say that every living creature is an aperiodic tiling of cells, produced by a two-phase process of uniform growth alternating with sub-divisions.
		Each blue tile that is bounded by yellow tiles is either encircled by the three interiors of the three "V"s or else by the sides of three V's, two of which are pointing on one direction and one in the other direction.
	www.mathpages.com /home/kmath540/kmath540.htm (1801 words)

	Quasicrystals and Aperiodic Tiling
		As one might assume from this definition, there are sets of tiles which will only tile in a nonperiodic way, and some which tile both periodically and nonperiodically, such as the set consisting of the single isosceles triangle whose interior angles measure 30°, 75°, and 75°.
		The most famous example of an aperiodic set of tiles is the pair of Penrose tiles, known as the kite and the dart.
		Two tiles with mirror image faces may be matched up along those faces, unless the matched faces contain a blue (Zometool color) edge, in which case there is an additional restriction that the tiles themselves must be mirror images of each other.
	www.ms.uky.edu /~lee/zerhusen/quasi.html (1307 words)

	Penrose-tiling : Balázs Faa
		By rule: a tiling is periodic if outlining a part of it and cutting it, the part can be refitted by shifting into the tiling.
		There is a variation of the tiling with the squares: each square intersected by a straight line, which is not rectangular to the edge of the shape.
		The tiles had no really exciting shape (at least from my point of view), they were squares with different gaps and horns at the edges.
	www.faa.hu /english/publications/penrose.html (543 words)

	The Geometry Junkyard: Tilings
		Tilings can be divided into two types, periodic and aperiodic, depending on whether they have any translational symmetries.
		An aperiodic set of Wang cubes, J. Culik and Kari describe how to increase the dimension of sets of aperiodic tilings, turning a 13-square set of tiles into a 21-cube set.
		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 (1702 words)

	Penrose Tilings and Wang Tilings
		Aperiodic Tilings in 2, 3, and 4 dimensions can be thought of as Irrational Slices of an 8-dimensional E8 Lattice and its sublattices, such as E6.
		The 2-dimensional Penrose Tiling in the above image was generated by Quasitiler as a section of a 5-dimensional cubic lattice based on the 5-dimensional HyperCube shown in the center above the Penrose Tiling plane.
		Since the Tiling Problem is undecidable, if the set of 64 hexagrams is identified with the aperiodic set of 64 Wang tiles, they can simulate the operation of any Turing machine, and therefore process information, so that the traditional role of the I Ching hexagrams is confirmed by the mathematics of Penrose-Wang tilings.
	www.valdostamuseum.org /hamsmith/pwtile.html (1543 words)

	Aperiodic Tilings
		The tiles are based on work done by Hao Wang in 1961, which originally involved tiling the plane with squares having different-colored edges.
		They can be prevented from tiling periodically by putting notches and tabs on the edges of the tiles, but a more aesthetic approach is to color the tiles as shown and require the edges to match.
		The stubby tile is colored in shades of blue and purple, the elongate tile in shades of yellow and green.
	www.uwgb.edu /dutchs/symmetry/aperiod.htm (967 words)

	sam.ufm Helpfile
		These colorings can be used to frame the tiles produced with the transoforms Aperiodic Tiling I, III and IV and Aperiodic Tiling II.
		Note about Gradient for Aperiodic Tiling I and II : "Trapezoid" and "Rhombus" are to be used with Aperiodic Tiling I and "L" with Aperiodic Tiling II.
		Tiling II doesn't need a frame mode because there is only one type of tiles.
	www.p-gallery.net /help/aperiodic.htm (501 words)

	Aperiodic Tilings Within Conventional Lattices (Site not responding. Last check: 2007-09-10)
		Recently, an aperiodic tiling involving two tiles, and matching rules for those tiles, was derived from the first of these two simple recursive tilings by Chaim Goodman-Strauss.
		The two tiles used are called the trilobite and the cross, and the matching rule involves not colored areas that touch along a line, but rather those that are on opposite sides where four of the sharp points of the pieces meet.
		This tiling was described in a paper published in 1971, one year before the publication of the Penrose tilings, and this was the first tiling known which forced aperiodicity with only six tiles.
	www.quadibloc.com /math/til03.htm (483 words)

	Aperiodic Tiling
		The tiles in these pictures and in the flip and POV files have all been shrunk by 30% toward their centers of mass to allow us to see the internals of the vertex clusters.
		Vertex 1 of the level 0 tile is placed at the origin, vertex 2 is on the positive x-axis, and vertex 3 is in the x-y plane.
		The lengths of the sides of the level 0 tile and the vertex numbers are those given by Danzer.
	www.cs.williams.edu /~98bcc/tiling (875 words)

	The Binary Tiling
		And aperiodicity is not guaranteed, in the absence of additional matching rules, even if all three kinds of pieces are used; thus, the periodic background to that page, made from a repeated rectangular section of a Keplerian tiling, can be made into an HBS tiling.
		The importance of the HBS tiling is not simply that it provides a clear formal definition of the type of pentagonal tiling with which this section began, as long as extra pieces such as the double-sized pentagon and the decagon are excluded.
		This suggests that those edges might form the boundaries of shapes which tile the plane in a tiling that is the dual of a given HBS tiling, in the same sense, or nearly the same sense, that the octahedron is the dual of the cube, or the dodecahedron is the dual of the icosahedron.
	www.quadibloc.com /math/pen02.htm (968 words)

	Penrose Tilings (Site not responding. Last check: 2007-09-10)
		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).
		As an example, the tilings in the previous figures are periodic.
		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.
	www.math.ubc.ca /~cass/courses/m308-02b/projects/schweber/tilings.html (360 words)

	Tilings (Site not responding. Last check: 2007-09-10)
		Aperiodic tilings have the attribute that they do not repeat patterns.
		Penrose tilings apparently mimic the behaviors of quasicrystals, make for a comfortable wipe on toilet paper, and are used in the design of a better no-stick coating for cookware.
		The really interesting thing to Penrose about the tilings (besides the fact they bring in income as a patented pattern) is that the solution to the tiling pattern is noncomputational.
	ffden-2.phys.uaf.edu /212_fall2003.web.dir/Pauline_Fusco/Tilings.htm (194 words)

	Theses Completed with Duane Bailey
		Penrose tiles are infinite tilings that cannot tile the plane in a periodic manner.
		These tilings are of interest to researchers in the natural sciences because they can be used to model and understand quasicrystals, a recently-discovered type of matter which bridges the gap between glass, which has no regular structure, and crystals, which demonstrate translational symmetry and certain rotational symmetries.
		Penrose tilings are tilings of the plane which use sets of two prototiles (either kites and darts, or thin rhombs and thick rhombs) whose edges are marked in order to indicate allowed arrangements of the tiles.
	www.cs.williams.edu /~bailey/theses.html (2576 words)

	Graphics Archive - Special Topics:Tilings (Site not responding. Last check: 2007-09-10)
		There are regular tilings of the plane using squares or hexagons (like the tiles on your bathroom floor), or the regular patterns of bricks on a wall.
		Tilings are closely related to symmetries, and are often studied together.
		The tilings of three-space have been studied by crystalographers, and are the subject of current research into quasicrystals.
	www.geom.uiuc.edu /graphics/pix/Special_Topics/Tilings (103 words)

	Search ScienceWorld
		It must be true that the sum of the interior angles divided by the number of sides is a divisor of 360 degrees.
		A regular tiling of polygons (in two dimensions), polyhedra (three dimensions), or polytopes (n dimensions) is called a tessellation.
		The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders).
	scienceworld.wolfram.com /search/index.cgi?as_q=Radin (323 words)

	The Geometry Junkyard: Penrose Tiling
		Penrose was not the first to discover aperiodic tilings, but his is probably the most well-known.
		Some planar tilings generated by the lattice projection method (of which the Penrose tiling is a special case) by Andrew Lewis, Queens U. Tessellations, a company which makes Puzzellations puzzles, posters, prints, and kaleidoscopes inspired in part by Escher, Penrose, and Mendelbrot.
		Ivars Peterson reports on a new proof by Tom Sibley and Stan Wagon that the rhomb version of the tiling is 3-colorable; A proof of 3-colorability for kites and darts was recently published by Robert Babilon [Discrete Mathematics 235(1-3):137-143, May 2001].
	www.ics.uci.edu /~eppstein/junkyard/penrose.html (798 words)

	Trilobite - Cross aperiodic tiling (Site not responding. Last check: 2007-09-10)
		The tiling is only possible as an aperiodic tile, that is, there are no periodic tilings of infinite extent on the plane using these two tiles and the joining rule.
		The second tile, the cross, got its name because it and the trilobite is often drawn rotated by 45 degree to those shown here.
		The next stage of the tiling can be constructed as in the chair tile.
	astronomy.swin.edu.au /~pbourke/texture/trilobite/index.html (247 words)

	Pinwheel Aperiodic Tiling (Melbourne Federation Square) (Site not responding. Last check: 2007-09-10)
		This infinite tiling of the plane is known to be aperiodic, that is, there is no parallelogram shaped region that can be repeated to form the same tiling.
		The properties of this tiling were discovered and named by Charles Radin of the University of Texas in 1991 and first published in 1994 in the Annals of Mathematics number 139, page 661-702.
		A periodic tiling is one where it is possible to make a parallelogram (generally larger than the tiles) that can be repeated to produce the same tiling.
	astronomy.swin.edu.au /~pbourke/texture/pinwheel (397 words)

	Caspar & Fontano (1996) PNAS 93, 14271-78. (Site not responding. Last check: 2007-09-10)
		Kepler's tiling with pentagons and decagons from Fig.
		An obvious strategy is that used by Penrose (3) to generate aperiodic tilings with pentagons: starting with a pentagonal array of pentagons, each pentagon was subdivided into six smaller pentagons and the gaps in the array between the larger pentagons were filled with the smaller ones; and this process was iterated.
		6: Superposition of Kepler's and Penrose's pentagon tilings.
	www.sb.fsu.edu /~caspar/201 (6673 words)

	Abstract, Nexus 98, Michael Ostwald: Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms (Site not responding. Last check: 2007-09-10)
		Michael Ostwald examined Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms at the Nexus 98 conference.
		The purpose of this binary analysis is not to critique Storey Hall or its use of aperiodic tiling but to use ARM's design as a catalyst for taking the first few steps in a greater analysis of Penrose tiling in the context of architectural form generation.
		Michael J. Ostwald "Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms", pp.
	www.nexusjournal.com /conferences/N1998-Ostwald.html (449 words)

	Tiling patterns - Hwmly (Site not responding. Last check: 2007-09-10)
		Up Aperiodic Tilings of 2D and 3D A fairlymathematics, non-periodic/aperiodic tiling took off as a serious branchin their X-ray diffraction patterns.
		Recording the tilings by drawing round shapes has theone could use the worksheet "Patterns" as a starting activity whateverwith concrete representations of tilings have been a source of fascination
		From new tiles to morphing patterns The methods used withtiles are applicable to changing a tiling, but there are more possibilities using these tiling motifs.
	www.hwmly.com /patterns/tiling-patterns.html (335 words)

	aperiodic tiling
		A tiling made from the same basic elements or tiles that can cover an arbitrarily large surface without ever exactly repeating itself.
		For a long time it was thought that whenever tiles could be used to make an aperiodic tiling, those same tiles could also be fitted together in a different way to make a periodic tiling.
		The most famous of these are the Penrose tilings.
	www.daviddarling.info /encyclopedia/A/aperiodic_tiling.html (150 words)

	Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms from the book Nexus II: Architecture and ... (Site not responding. Last check: 2007-09-10)
		Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms from the book Nexus II: Architecture and Mathematics
		He concluded his response to the invitation with an enigmatic postscript which records that he is currently working on "the single tile problem" and recently "found a tile set consisting of one tile together with complicated matching rule that can be enforced with two small extra pieces".
		The newly completed Storey Hall is literally covered in a particular set of giant, aperiodic tiles that were discovered by Roger Penrose in the 1970's and have since become known as Penrose tiles.
	www.leonet.it /culture/nexus/98/Ostwald.html (333 words)

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