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Topic: Tiling

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Squaring the square

Penrose tiling

Aperiodic tiling

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	Penrose tiling - Wikipedia, the free encyclopedia
		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).
		That it must be possible to tile the plane aperiodically was first proven as a general proposition in 1966 by Robert Berger, who shortly thereafter invented the first aperiodic set of tiles, consisting of 20426 distinct tile shapes.
		Aperiodic tiling was first considered only an interesting mathematical structure, but physical materials were later found where the atoms were arranged in the same pattern as a Penrose tiling.
	en.wikipedia.org /wiki/Penrose_tiling (690 words)

	Dissection Tiling
		Tilings have been used directly for constructing dissections (by overlaying two tilings with the same fundamental domain), but they also are useful for understanding n-to-one dissections in the limit as n grows large -- the number of pieces can be approximated by kn+O(sqrt n) where k is the average pieces/polygon in a dissection tiling.
		This dissection tiling of the nonagon [original, Dec 1995], involving three different strips with straight boundaries, must be aperiodic: the strips have incommensurate periods, and one must alternate the strips in an aperiodic way to achieve the correct density of each type of piece.
		Several different dissection tilings are possible for inputs consisting of a mixture of pentagons and decagons, depending on the ratio between the two shapes.
	www.ics.uci.edu /~eppstein/junkyard/distile (1634 words)

	Non Periodic Tiling of the Plane (Site not responding. Last check: 2007-09-05)
		Tiling a plane with single geometric shapes is an old exercise in geometry.
		The platonic solids were shapes that tile 3D space using regular polyhedra, in 2 dimensions examples of regular polygons that tile (without gaps or overlap) the plane are the equalateral triangle, square, and hexagon.
		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/nonperiodic (610 words)

	Reinhold Tiling - Wikipedia
		Nach Kriegsende verdiente sich Tiling seinen Lebensunterhalt zunächst als Kunstflieger.
		Tiling entwickelte wiederverwendbare Raketenflugzeuge, die als Rakete starten und mit ausklappbaren Flügeln landen sollten.
		Den notwendigen Schub und eine ausreichende Brenndauer erreichten seine Raketen durch eine bahnbrechende Verbesserung gebräuchlicher Pulverraketen.
	de.wikipedia.org /wiki/Reinhold_Tiling (550 words)

	NASAexplores 5-8 Lesson: (Teacher Sheets)
		Tiling is a repeating pattern of regular polygons.
		Since circles do not tile (there is a space between the circles) we will be using the polygons to simplify and ensure a consistent pattern.
		Tiling is like covering a floor with shapes; the pattern must cover the entire surface.
	www.nasaexplores.com /show_58_teacher_st.php?id=030103103647 (258 words)

	Tiles and tiling FAQ (Site not responding. Last check: 2007-09-05)
		Mosaic tiles on a sheet of paper are another option, but for permanently wet areas avoid the type of mosaic with the paper or cloth scrim stuck to the backs of the tesserae, as the adhesive used to do this is frequently not waterproof.
		Also available is a tile sawblade which fits a standard hacksaw frame and allows curves and holes to be cut in tiles.
		To cut a circular (or any other shape) hole in a tile, drill a hole in the middle of the bit to be removed (using a masonry drill with no hammer action), fit the saw blade through the hole, saw to the edge of the area to be removed, and follow round the periphery.
	www.axp.mdx.ac.uk /~john49/tilefaq.htm (3021 words)

	Tiling Terms (Site not responding. Last check: 2007-09-05)
		The tiling consisting of squares and octagons covering the plane is isogonal, as is the tiling consisting of alternating rows of triangles and squares.
		The tiling consisting of alternating rows of triangles and squares is equitransitive because all the squares are related by symmetry and so are all the triangles.
		This tiling is not normal (the tiles get arbitrarily thin) and is also not balanced, and it was precisely to eliminate paradoxes like this that Grunbaum and Shephard found it necessary to introduce the concepts of normality and balance.
	www.uwgb.edu /dutchs/symmetry/tilterm.htm (855 words)

	Polyomino, polyhex and polyiamond tiling
		Tiling the polyominoes of orders 1 through 6 is easy, and John Conway determined the tiling properties of the heptominoes with the aid of Conway’s criterion.
		Their tiling properties may be studied similarly to those of polyominoes; results through the hexahexes were published by Gardner from his correspondence.
		Their tiling properties may be studied similarly to those of polyominoes and polyhexes; results through the 9-iamonds were published by Gardner from his correspondence.
	www.srcf.ucam.org /~jsm28/tiling (1220 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)

	Labyrinth Tiling
		Many tilings, including the famous Penrose tiling, are generated by a subdivision process of the following sort: starting with a single tile from a given finite set, repeatedly replace each tile by a collection of smaller tiles, drawn from the same set.
		At first glance, this tiling looks quite uniform: it is formed by a regular pattern of 60-120 rhombuses, each subdivided by a diagonal into two triangles.
		It turns out that, given any two points in the tiling, either an infinite horizontal or vertical ray in the tiling crosses the line segment between them, or there is a unique sequence of triangles that forms a path connecting them and is not crossed by any horizontal or vertical edges.
	www.ics.uci.edu /~eppstein/junkyard/labtile (1114 words)

	Aperiodic Tilings (Site not responding. Last check: 2007-09-05)
		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)

	Reinhold Tiling
		Tiling entwickelte wiederverwendbare Raketenflugzeuge die als Rakete starten und mit ausklappbaren Flügeln landen Dieses Prinzip wird bis heute von der NASA für Flüge des Space Shuttle verwendet.
		Den notwendigen Schub und eine ausreichende erreichten seine Raketen durch eine bahnbrechende Verbesserung Pulverraketen.
		Trotz der durch Freunde und Gönner hatte Tiling jedoch Zeit seines Schaffens mit finanziellen zu kämpfen.
	www.uni-protokolle.de /Lexikon/Reinhold_Tiling.html (498 words)

	Combinatorial Tiling Theory (Site not responding. Last check: 2007-09-05)
		Dress and Huson 1991), the classification of all four-colorable tilings of the plane in "A Four-Color Theorem for Periodic Tilings" (Huson 1994) or the classification of all tile-transitive tilings of an infinite ribbon in "Ribbon Tilings From Spherical Ones" (Huson 1995).
		In "Tiling Space By Platonic Solids I" (Delgado and Huson 1997), we show that there exist precisely 46, 58, and 914 equivariant types of tile-transitive tilings of by topological tetrahedra, octahedra, and cubes, falling in to 9, 3, and 11 topological families, respectively.
		In particular, we have clarified the relationship between the geometry of periodic tilings, the topology of orbifolds, and the combinatorics of parameterized Coxeter matrices or Delaney symbols.
	www.mathematik.uni-bielefeld.de /~huson/approach.html (3565 words)

	Quasicrystals and Aperiodic Tiling
		A tiling has translational symmetry if the tiling can be shifted by a given distance in a given direction (think vector addition) such that it exactly matches itself.
		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°.
		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)

	Genome-wide transcription analyses in rice using tiling microarrays - Nature Genetics
		2), suggesting that tiling array detection was consistent with transcriptional activity reflected by the signal probes.
		Tiling microarray design and experimental data are available in the NCBI Gene Expression Omnibus (http://www.ncbi.nlm.nih.gov/projects/geo/index.cgi) under series GSE3452.
		Tiling microarray analysis of rice chromosome 10 to identify the transcriptome and relate its expression to chromosomal architecture.
	www.nature.com /ng/journal/v38/n1/full/ng1704.html (4037 words)

	Excel Tiling Supplies, North Yorkshire, United Kingdom.
		The use of tiles in the home is becoming increasingly popular and here at Excel Tiling Supplies we cater for this trend.
		The Excel tiling team combine years of hard-earned specialist experience and expertise with an approach that is totally professional.
		We are a family run firm who have 30 years of tiling experience in the Yorkshire area and we have established ourselves as a major source of supply for wall and floor coverings in the York area.
	www.exceltiling.co.uk (285 words)

	[No title]
		Introduction\endheading In 1907 Minkowski \cite5 conjectured that all the extremal lattices for the supremum norm were of a certain simple form and observed that this conjecture had a geometric interpretation: in any lattice tiling of $\Bbb R^n$ with unit $n$-cubes there must exist two cubes having a complete facet $((n-1)$-face) in common.
		Keller's conjecture is then false for all $n\ge 10$ because a counterexample tiling in $\Bbb R^n$ gives one in $\Bbb R^{n+1}$ by ``stacking'' layers of this tiling with suitable translations made between adjacent layers.
		For a Szab\'o-type tiling, two cubes having coordinates $(m_1,\dotsc, m_n)$ and $(m'_1,\dotsc, m'_n)$ in $G^*_n$ have a $k$-dimensional face in common if $m_i-m'_i=0$ or 2 for all $i$, and exactly $k$ values $m_i-m'_i=0$.
	www.ams.org /journals/bull/pre-1996-data/199227-2/Lagarias (2370 words)

	Stephen Collins - Penrose Tiling Generator (Site not responding. Last check: 2007-09-05)
		The number of rhombi in any given tiling is larger than its predecessor by a factor converging on (approximately) 2.6 as the number of deflations increases.
		Another characteristic of deflation is that as the tiling grows, it often surrounds "holes", typically near the perimeter of the tiling, which can then persist from one deflation to the next.
		The simplest way to create a new geodesic walk within a given tiling is to first specify the starting point of the walk by double-clicking the right-hand mouse button on the targeted rhombus.
	www.stephencollins.net /web/penrose (1797 words)

	Hyperbolic and Spherical Tiling Gallery
		In this tiling, each vertex is surrounded by two 7-gons and two triangles.
		This is supposed to be a hyperbolic tiling where each vertex is surrounded by a 7-gon and two hexagons.
		This is a spherical tiling of an icosahedron subdivided into a geodesic dome.
	bork.hampshire.edu /~bernie/hyper (292 words)

	Open Directory - Science: Math: Recreations: Polyominoes (Site not responding. Last check: 2007-09-05)
		Tiling a square with eight congruent polyominoes - Michael Reid's abstract of a paper in the "Journal of Combinatorial Theory, Series A".
		Tiling of Pythagorean triplets - Joe Fields suggests that L-decomposition of squares of Pythagorean triplets could always be tiled.
		Unbalanced anisohedral tiling - Joseph Myers and John Berglund found a polyhex that must be placed in two different ways in a tiling of a plane, such that one placement occurs twice as often as the other.
	www.dmoz.org /Science/Math/Recreations/Polyominoes (2005 words)

	Tiling photos
		But making a real tiling photo means finding a naturally tiling subject, and then fiddling around with the boundaries so that apart from the patching up of the seam, the whole thing is as the original might have been.
		Initially, this project was the result of an idea I had a long time ago, to photograph some of the variety of tilings in use in real life.
		Then it occurred to me that as well as regular repeating 'proper' tiles, there are irregularly repeating natural patterns, such as flowers and leaves, and I may as well beat some of these into shape too.
	www.imaginatorium.org /tiles/tiles.htm (479 words)

	Computer Software for Tiling
		User can build up a tiling from a collection of tiles in the same manner as done "by hand," interactively gluing patches of tiles edge-to-edge.
		Tilings are created by reflections in mirrors that surround a fundamental triangular region that has been divided into three subsections by line segments concurrent at a point in the region.
		Takes as input a "Delaney Symbol" and produces a picture of the corresponding periodic tiling of a 2-D surface (Euclidean, hyperbolic, or sphere surface), that can then be colored, manipulated, modified, or immediately changed to a related tiling with a similar Delaney symbol (possibly on a different surface).
	www.geom.umn.edu /software/tilings/TilingSoftware.html (702 words)

	Pinwheel Aperiodic Tiling (Melbourne Federation Square) (Site not responding. Last check: 2007-09-05)
		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.
		In this case the tile has been scaled down (click on the image for a larger version) and another triangle is used to form a rectangular region.
	astronomy.swin.edu.au /~pbourke/texture/pinwheel (397 words)

	The Penrose Tiling at Miami University
		A brass framework was fabricated in the outline of the tiling, and the colored tiles were poured individually.
		When they returned, it was discovered that the brass frame had been rotated 90 degrees so that the tiling's axis of symmetry did not coincide with the axis of symmetry of the building.
		We believe that this was the first Penrose tiling to be incorporated in the design of a public building and it may still be the only one executed in terrazzo.
	www.lib.muohio.edu /epub/tilings/doc.html (728 words)

	Perfecta Tiling - specialist tile contractors - high quality workmanship for commercial, industrial, retail and ...
		Expert tiling for your building contract or business premises, beautiful, exquisite tiling to your home, we have the expertise, experienced and qualified workforce to turn your cutting edge design ideas into reality, making us the perfect choice for your tiling contract.
		At Perfecta Tiling we can supply and fit all types of ceramic wall and floor tiles, for high specification work and customised designs.
		Of course, to complete the job may involve input from other trades, but this can be accommodated as our high quality workmanship ensures only the most reputable sub contractors are allowed to work with the company to ensure customer satisfaction.
	www.perfecta-tiling.co.uk (181 words)

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