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	Penrose triangle - Wikipedia, the free encyclopedia
		The Penrose triangle, also known as the tribar is an impossible object.
		The mathematician Roger Penrose independently devised and popularised it in the 1950s, describing it as "impossibility in its purest form".
		The resulting waterfall, forming the short sides of both triangles, drives a water wheel.
	en.wikipedia.org /wiki/Penrose_triangle (276 words)

	Roger Penrose - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-03)
		Roger Penrose is well-known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane aperiodically.
		Penrose and Stuart Hameroff have constructed a theory in which human consciousness is the result of quantum gravity effects in microtubules.
		Roger Penrose is the son of scientist Lionel S. Penrose, and the brother of mathematician Oliver Penrose and chess grandmaster Jonathan Penrose.
	encyclopedia.worldsearch.com /roger_penrose.htm (970 words)

	Roger Penrose Encyclopedia Article, Information, History and Biography @ Fburg.com (Site not responding. Last check: 2007-11-03)
		Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the University of Oxford.
		Roger Penrose is the son of scientist Lionel S. Penrose and Margaret Leathes, and the brother of mathematician Oliver Penrose and chess grandmaster Jonathan Penrose.
		In 1965 at Cambridge, Penrose and physicist Steven Hawking proved that singularities (such as fl holes) could be formed from the gravitational collapse of dying immense stars.
	www.fburg.com /encyclopedia/Roger_Penrose (1433 words)

	Penrose triangle
		In 1954, Roger Penrose, after attending a lecture by the artist M. Escher, rediscovered the impossible triangle and drew it in its most familiar form, which he published in a 1958 article in the British Journal of Psychology, coauthored with his father Lionel.
		Penrose was also unfamiliar with the work of Reutersvärd, Piranesi, and others who had created impossible figures previously.
		Penrose's impossible triangle, unlike Reutersvärd's earlier version, was drawn in perspective, which added an additional size paradox to the object.
	www.daviddarling.info /encyclopedia/P/Penrose_triangle.html (234 words)

	Encyclopedia: Penrose triangle (Site not responding. Last check: 2007-11-03)
		Penrose's "Colchester Survey" of 1938 was the earliest serious attempt to study the genetics of learning disability.
		In British psychiatry, the so-called 'Penrose's Law' states that the population size of prisons and psychiatric hospitals are inversely related, although this is generally viewed as an oversimplification.
		Penrose is the father of mathematician Oliver Penrose, scientist Roger Penrose (with whom he co-authored papers on the Penrose triangle) and chess grandmaster Jonathan Penrose.
	www.nationmaster.com /encyclopedia/Penrose-triangle (930 words)

	books - Page 1 at Think Bling
		Roger Penrose has mentioned the initiating of a readers' interest of a topic, not the complete understanding of all the applied scientific information data provided,seen could be a practical outcome from this work of him.
		Penrose describes the National debt-sized expenses, the half a lifetime timescales and the consequential scarcity of experiments.
		Penrose is a pure mathematician and his very welcome iconoclasms can be clearly put down to his non-physics training, which must make him rather irritating to physicists because he also happens to be a very great mathematician.
	www.thinkbling.com /detail.php?ASIN=0679454438 (1167 words)

	Quasi-crystals
		In their simplest form "Penrose's tiles" represent a nonrandom set of the diamond-shaped figures of two types, one of them is with the interior angle of 36°, the other one with the angle of 72°.
		In such triangle the acute angles at the vertex of E and B are equal to 36° and the obtuse angle at the vertex of K is equal to 108°.
		Penrose's idea about dense filling of the plane with the help of the "golden" rhombuses of the kinds of (a) and (b) was transformed on the three-dimensional space.
	www.goldenmuseum.com /1603QuasiCrystals_engl.html (1228 words)

	World of Escher - Roger Penrose
		Roger Penrose, a professor of mathematics at the University of Oxford in England, pursues an active interest in recreational math which he shared with his father.
		Penrose began to work on the problem of whether a set of shapes could be found which would tile a surface but without generating a repeating pattern (known as quasi-symmetry).
		Penrose was raised in a family with strong mathematical interests: his mother was a doctor, his father, a medical geneticist, used math in his work as well as his recreation, one brother is a mathematician, another was ten times British chess champion.
	www.worldofescher.com /misc/penrose.html (644 words)

	Roger Penrose Biography / Biography of Roger Penrose World of Mathematics Biography
		A modern bridge between the worlds of physics and mathematics, Sir Roger Penrose was an important contributor to theories on fl holes in the 1960s and today is a fixture in the arcane world of recreational mathematics.
		Penrose is famous for his invention, with his father, of the Penrose impossible staircase and the Penrose triangle (also called the "tribar").
		Penrose was born in Colchester, England on August 8, 1931, the son of a geneticist who was an expert on inherited mental defects.
	www.bookrags.com /biography-roger-penrose-wom (241 words)

	Penrose dodecahedrons, in preference permutations.
		Penrose says one is here and the other is far away (perhaps galaxies away) implying that light speed influences are not at work.
		Re Penrose's 'non-locality' proof: The vertices are given Penrose's letters: A to E are the vertices of a pentagon facet.
		An electoral interpretation of the Penrose dodecahedrons depends on the geometry of regular solids serving as illustrations of permutatons of choice.
	www.voting.ukscientists.com /penrose.html (2236 words)

	Penrose stairs - Wikipedia, the free encyclopedia
		The Penrose stairs is an impossible object devised by Lionel Penrose and his son Roger Penrose and can be seen as a variation on his Penrose triangle.
		It is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher.
		The staircase had also been discovered previously by the Swedish artist Oscar Reutersvärd, but neither Penrose nor Escher were aware of his designs.
	en.wikipedia.org /wiki/Penrose_stairs (193 words)

	Penrose stairs (Site not responding. Last check: 2007-11-03)
		The Penrose stairs is an impossible object devised by Lionel Penrose and his Roger Penrose and can be seen as a on his Penrose triangle.
		It is a two-dimensional depiction of staircase in which the stairs make four turns as they ascend or descend yet a continuous loop so that a person climb them forever and never get any This is clearly impossible in three dimensions; two-dimensional figure achieves this paradox by distorting
		Escher where it is incorporated into a monastery where several monks do penance by ascending continuously but are allowed turn around and descend occasionally.
	www.freeglossary.com /Penrose_stairs (223 words)

	penrose_triangle (Site not responding. Last check: 2007-11-03)
		The Penrose Triangle was discovered by geneticist Lionel Penrose and his son, mathematician and physicist Roger Penrose in 1958.
		C. Escher used three interconnected penrose triangles create the illusion of a perpetually flowing water mill.
		Penrose triangles also occur in the mathematical branch of knot theory
	www.todoso.co.uk /shock/front/penrose_triangle.html (76 words)

	Penrose Tilings
		Penrose tilings can also be generated using a substitution method.
		The penrose tiling above was generated using this method with the postcript Penrose tiler described later.
		It uses a set of 4 tiriangular tiles that are generated by bisecting the thin rhomb along its short diagonal and the thick rhomb along its long diagonal.
	www.math.ubc.ca /~cass/courses/m308-02b/projects/schweber/penrose.html (621 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.
		And the isosceles triangle can be subdivided into two similar isosceles triangles and an equilateral triangle, all three having the same area, by trisecting its hypotenuse and connecting the resulting points to the opposite vertex.
		Eventually the large triangles in the codes get so large that there is only room for two possible configurations: either the two points are in a single large triangle, or they are in two adjacent triangles.
	www.ics.uci.edu /~eppstein/junkyard/labtile (1114 words)

	Two-dimensional Geometry and the Golden section
		Putting the compass point at the top point of the triangle and opening it out to point B (so it has a radius along the right-angle line) mark out a point on the diagonal which will also be half as long as the original line.
		Since the triangles AEB and DEC are similar (the angles are the same in each triangle) then their sides are in the same ratio (proportion) to each other.
		Penrose Tiles to Trapdoor Ciphers, 1997, chapters 1 and 2 are on Penrose Tilings and, as with all of Martin Gardner's mathematical writings they are a joy to read and accessible to everyone.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/phi2DGeomTrig.html (8108 words)

	roger penrose (Site not responding. Last check: 2007-11-03)
		Penrose and Hameroff have constructed a theory of human consciousness in which human consciousness is the result of quantum gravity effects in microtubules.
		Roger Penrose is the son of scientist Lionel S Penrose and the brother of mathemetician Prof Oliver Penrose and chess Grand Master Jonathan Penrose
		Two theories for the formation of quasicrystals resembling Penrose tilings are here: [1]
	www.yourencyclopedia.net /Roger_Penrose.html (696 words)

	The Road to Reality : A Complete Guide to the Laws of the Universe, Roger Penrose
		The bulk of the book is then devoted to quantum physics, cosmological theories (including Penrose's favored ideas about string theory and universal inflation), and what we know about how the universe is held together.
		And the particularly important advances made by means of the revolutionary theories of relativity and quantum mechanics have deeply altered our vision of the cosmos and provided us with models of unprecedented accuracy.
		Third, Penrose comes to the arresting conclusion–as he explores the compatibility of the two grand classic theories of modern physics–that Einstein’s general theory of relativity stands firm while quantum theory, as presently constituted, still needs refashioning.
	www.marked4sale.com /product_book/the_road_to_reality_0679454438 (1584 words)

	iqexpand.com (Site not responding. Last check: 2007-11-03)
		A Penrose tiling is pattern of tiles, discovered by Roger Penrose, which could completely cover an infinite surface, but only in a pattern which is non-repeating (aperiodic).
		In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).Two additional types of Penrose...
		Penrose Tilings Kepler was one of the first to investigate tilings with 5-fold symmetry.
	penrose_tiling.iqexpand.com (726 words)

	The the Penrose Principles of Artistic Illusions
		This version is identical to that of the Penrose paper, except for its lack of shading.
		Penrose Parallelograms is a tiling system invented in 1974 by English physicist, Sir Roger Penrose.
		Penrose went on to ask, "What is the smallest number of shapes that could create an non-repeating infinite tiling?" Asking that question is remarkable.
	www.saxe-patterson.com /penrose.htm (569 words)

	Puzzles
		As shown on the Penrose Tiling page, phi is the basis for shapes called kites, darts and diamonds that can create five-fold symmetry, which was thought impossible until the 1970's.
		Two isosceles triangles have the interesting properties of forming ever larger models of themselves, and of modeling any parts of pentagon tilings.
		There are 4 triangle sizes each in 4 colors, which can build 4 separate little pentagons or one large one.
	www.goldennumber.net /products/puzzles.htm (715 words)

	Penrose Tiling
		The pattern is an area selected from a Penrose tiling.
		The Golden Ratio appears in the relative lengths of the triangle sides.
		There are two Penrose tilings with five-fold rotational symmetry -- they can be created by radially extending the two implied Penrose tilings in your quilt around a single point (the center of your quilt).
	dogfeathers.com /quilt/penrose.html (378 words)

	Search Results for 'Roger-Penrose' (Site not responding. Last check: 2007-11-03)
		The Right Honourable Lord Penrose, born George William Penrose, is a Scottish judge (from 1990) and member of the Privy Council (from 2000) who sits in the specialist court for commercial actions.
		Charles William Penrose (1832–1925) (commonly known as Charles W. Penrose) was a member of the Quorum of the Twelve Apostles of The Church of Jesus Christ of Latter Day Saints from July 7, 1904.
		Charles William Penrose was born on February 4, 1832 in London.
	www.worldhistory.com /wiki/R/Roger-Penrose.htm (951 words)

	Escher for Real
		The impossible shape conveyed by the Penrose triangle is the most well-known one.
		The rightmost image shows the Penrose triangle only, from above; the leftmost image shows the original Waterfall scene; and the middle image is a blend of the two.
		This trick is somewhat similar to the trick we used in the Penrose triangle but is somewhat simpler.
	www.cs.technion.ac.il /~gershon/EscherForReal/CodeArt.html (691 words)

	Penrose stairway
		An impossible figure named after by the British geneticist Lionel Penrose (1898-1972), father of Roger Penrose.
		It served as an inspiration for the staircase in M. Escher's famous print "Ascending and Descending." Although the Penrose stairway cannot be realized in three dimensions, this impossibility is not immediately perceived and, in fact, the paradox is not even apparent to many people at a quick glance.
		Although Escher and the Penroses made the Stairway famous, it was, unbeknownst to them, independently discovered and refined years before by the Swedish artist Oscar Reutersvard.
	www.daviddarling.info /encyclopedia/P/Penrose_stairway.html (185 words)

	Club History - prtcc.com
		The Penrose Hall, so named in memory of Charles Penrose stalwart of the Association, was erected during the Second World War on top of the Y.M.C.A. tennis courts at Reforne.
		The Red Triangle C.C. has the rare distinction of being the first truly civilian cricket club to be formed on Portland.
		Apparently even "Chippy" played, Red Triangle were never short of a cricket ball, because John Pearce insisted on using one of his personal collection (that is one of the many cricket balls he had been presented with for achievement on the field-of-play).
	www.prtcc.com /history.htm (1230 words)

	Penrose Tiling
		In the early 1970's, however, Roger Penrose discovered that a surface can be completely tiled in an asymmetrical, non-repeating manner in five-fold symmetry with just two shapes based on phi, now known as "Penrose tiles."
		This is accomplished by creating a set of two symmetrical tiles, each of which is the combination of the two triangles found in the geometry of the pentagon.
		The triangle shapes found within a pentagon are combined in pairs.
	goldennumber.net /penrose.htm (429 words)

	A family of impossible figures studied by knot theory
		Lionel and Roger Penrose (father and son) introduced the impossible tribar in 1958 [2] (Fig.
		A figure is called impossible when "a contradiction in our interpretation is noticed but does not result in our rejecting it in favour of a consistent one" [3].
		The Penrose triangle and a family of related figures.
	members.tripod.com /vismath8/cerf/index.html (1011 words)

	on Penrose and A.I. - IIDB
		Penrose is one of the greatest modern mathematician and he concluded that Artificial Intelligence through computers, as presently constructed, cannot in principle duplicate the workings of the human brain (the quote in the exemple is not exact).
		Mathematicians like Penrose (who was recently knighted for his outstanding contributions to mathematics) ARE well qualified to speak on the subject and are not the kind of people to speak (and write books) without solid logical, scientifical and calculable justifications.
		I'm not saying he`s right but I don`t agree that ''it is questionable whether he is well-qualified to speak on the subject''.
	www.iidb.org /vbb/showthread.php?t=1130 (370 words)

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