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Victoria Potter Amy Donato Renee Staudt

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Does there exist a non-periodic set of prototiles to create an aperiodic set in the plane? Tiling problems have been studied for years by computer scientists and exist in discrete and computational geometry. There are many rules when it comes to tiling problems and much of them involve symmetry restrictions (whether tiles can be flipped or rotated) according to certain rules. For many years it had been belief that no such set exists.

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Tiling-when you fit individual tiles together with no gaps or overlaps to fill a flat space (plane). Prototile-is a (finite or infinite) set of tiles that represent all the tiles of a tiling, or of a class of tiling's. Periodic-has translational symmetry, but must be in at least two non-parallel directions. Non-periodic means it lacks translational symmetry, a shifted copy will never match the original exactly. Aperiodic-a set of prototiles that tile the plane but never with translational symmetry.

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Prototiles

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This problem was first posed in 1961 by Hao Wang, when he used 4-way dominoes, now known as Wang tiles, to hypothesize that these dominoes can tile the plane if and only if it can do so periodically (in a pattern that repeats in two different non-parallel directions). Wang tried to find a method for deciding if any set of dominoes will tile non-periodically by putting the same colored sides against each other. Rotations and reflections not being allowed. Wang concluded that any set of tiles that can tile the plane will do so periodically, so no set of non-periodic tiles could possibly exist.

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In 1964, Robert Berger disproved Hao Wangs hypothesis and proved that sets of tiles can tile the plane non-periodically. He found an aperiodic set with 20,426 dominoes and later reduced that number to 104. Donald Knuth reduced the number to 92 dominoes. Raphael Robinson gave a set of just 6 tiles by putting projections and slots into the edges to get jigsaw pieces that work the same with the colors and created a non-periodic prototile.

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Roger Penrose was the first to give a set of variations in prototiles. First, he also found a set of 6 prototiles that cause non-periodicity in the plane. He then soon lowered that number to four. Then all of those led to Roger Penroses, along with help from Robinson, John Conway and Robert Ammanns, discovery of just two sets of tile-types that were not squares that cause non-periodicity.

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Roger Penrose was born in 1931 in Colchester, Essex, England, Father – Medical Geneticist Mother – Medical Doctor Penrose earned his Ph.D. in mathematics at Cambridge University in 1957 and married in 1959He argued that the human brain can carry out processes that no computer can do, running counter to the general tendency among other researchers in the field of artificial intelligence.

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Visited various universities in Brittan and the United States before settling down as a professor of applied mathematics at Birbeck College, London, in 1967. Penrose became the Ball Professor of Mathematics at Oxford University in 1973. Penrose retired from Oxford in 1998 Professor of Geometry at Gresham College in London Penrose and his father gained some popular recognition when they devised some geometrical figures later used by the Dutch surrealist artist M. C. Escher (1898– 1972).

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Penrose and Hawking shared the 1988 Wolf Prize in Physics for their work on black holes and relativity. Penrose became a fellow of the Royal Society in 1972, received the Royal Society Royal Medal in 1985, and was knighted in 1994. His interests turned to computers and artificial intelligence, and he published the best-selling The Emperors New Mind: Concerning Computers, Minds, and the Laws of Physics (1989) Shadows of the Mind: A Search for the Missing Science of Consciousness (1994).

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A Penrose tiling is a two tile-type, non-periodic tiling in a plane generated by an aperiodic set of prototiles. There are 3 types of Penrose tilings The original tiling The Kite and Dart tiling The Rhombus tiling

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Proposed in 1974 in a paper titled, Role of aesthetics in pure and applied research. Penrose got his inspiration from Johannes Kepler. Kepler in his book, Harmonices Mundi, discusses tilings built around pentagons. Penrose used this to discover that it could be expanded into a Penrose tile. When tiling the plane with regular pentagons it leaves gaps between the tiles, which breaks the rule of what tiling is.

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Penrose formed one specific tiling that could be filled with three different shapes. A star A boat A diamond (or thin rhombus) Earlier ideas of this had been traced back, but Penrose was the first to prove this idea.

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The Kite is a quadrilateral whose interior angles are 72°, 72°, 72°, and 144° The Dart is a non-convex quadrilateral whose interior angles are 36°, 72°, 36°, 216° Kite Dart

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The green and red arcs in the Kite and Dart restrict the placement of the tiles. When two tiles share an edge in a tiling, the patterns must match at these edges. First of the tilings that contained only two distinct tiles. The kite and dart can be bisected to form pairs of triangles that can be used for substitution tiling.

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The kite and dart are one of the most well known and popular tiling patterns. These are two of the most used kite and dart prototiles.

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The thin rhombus has angles 36°, 144°, 36°, 144°and can be bisected along its short diagonal to form a pair of triangles that can be used for substitution. The thick rhombus has angles 72°, 108°, 72°, 108° and can be bisected along its long diagonal to form a pair of triangles that can be used for substitution.

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Ordinary rhombus-shaped tiles can be used to tile the plane, but they tile periodically. There are specific rules that you must follow when placing rhombus tiles so they tile non- periodically. Two tiles cannot be put together to form a single parallelogram. Only particular sides can be put together with one another. They must be assembled so that the curves match in color and position. Must be constructed so the slots and projections on the edges fit together. Rhombus tiling is the other most popular Penrose tiling.

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Repeated generations of deflation produce a tiling of the original axiom shape with smaller and smaller shapes. Original Generation 1 Generation 2 Generation 3 Kite (half) Dart (half) Sun Star

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Drop City artist Clark Richert used Penrose rhombi in artwork in 1970. In more recent times Computer artist Jos Leys has produced numerous variations on the Penrose theme Art historian Martin Kemp has commented a contemporary decoration which used Penrose tiles and observed that Albrecht Durer has sketched similar motifs of a rhombus tiling

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The rhombus tiling is the most famous of all Penrose tiles. To construct a rhombus tiling you must follow rules on construction of putting rhombs together. The point of a Penrose tile is that it is non- periodic. There are many ways to construct periodic rhombus tilings that are obvious. There is more to tiling then just the shapes, it has to do with using the rhombs and rules.

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When constructing a Penrose tiling, two adjacent vertices must both be blank or must both be black. If two edges lie next to each other they must both be blank, or both have an arrow. If the two adjacent edges have arrows, both arrows must point in the same direction.

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The inner section in purple is surrounded blue sections making a decagon. The outer section is made up of two parts. The ten blue spokes and ten yellow sectors.

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Penrose tilings are known to be in 2D. It seems that Penrose tilings could be extended to a 3D case, but just the 2D case is proven to agree with the structure and algebraically.

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"Aperiodic Tiling's.". N.p.. Web. 19 Nov 2012. http://www.math.cornell.edu/~mec/2008- 2009/KathrynLindsey/PROJECT/Page5.htm. http://www.math.cornell.edu/~mec/2008- 2009/KathrynLindsey/PROJECT/Page5.htm Bartlomiej Kozakowski, Janusz Wolny. Faculty of Physics and Applied Computer Science, AGH University of Science and Technologyhttp://www.uwgb.edu/dutchs/symmetry/penro se.htm.http://www.uwgb.edu/dutchs/symmetry/penro se.htm Eric G. Swedin, editor. Science in the Contemporary World: An Encyclopedia. 2005. http://rave.ohiolink.edu/ebooks/ebc/CSCIENE http://rave.ohiolink.edu/ebooks/ebc/CSCIENE "Penrose Kite-Dart." Tiling's Encyclopedia. N.p.. Web. 19 Nov 2012.

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"Penrose Tiles and Aperiodic Tessellations'.". N.p.. Web. 19 Nov 2012. http://math.uchicago.edu/~mann/penrose.pdfhttp://math.uchicago.edu/~mann/penrose.pdf "Penrose Tiling.". N.p., 31 2008. Web. 19 Nov 2012. http://www.cs.brandeis.edu/~storer/JimPuzzles/PACK/Czech Farms/PenroseTilingWikipedia.pdf. http://www.cs.brandeis.edu/~storer/JimPuzzles/PACK/Czech Farms/PenroseTilingWikipedia.pdf "Penrose Tiling." Science U. Geometry Technologies, 19 2005. Web. 19 Nov 2012. http://www.scienceu.com/geometry/articles/tiling/penrose. html. http://www.scienceu.com/geometry/articles/tiling/penrose. html "Robinson Triangle." Tiling's Encyclopedia. N.p.. Web. 19 Nov 2012. http://tilings.math.uni- bielefeld.de/substitution_rules/robinson_trianglehttp://tilings.math.uni- bielefeld.de/substitution_rules/robinson_triangle Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay. Last Update 11 August 1999. http://arxiv.org/ftp/cond-mat/papers/0503/0503464.pdf http://arxiv.org/ftp/cond-mat/papers/0503/0503464.pdf

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