
	Where results make sense

Topic: Penrose staircase

Related Topics

Penrose stairs

Penrose triangle

Optical illusion

Shepard tone

Hermann grid illusion

Impossible trident illusion

Cafe wall illusion

Stereogram

Dolphins (Dolphins in Phosphorescent Sea)

Missing square puzzle

Autostereogram

Fraser spiral illusion

Necker cube

Polyhedron

Magic Eye

	World of Escher - Roger Penrose
		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.
		Roger and his father are the creators of the famous Penrose staircase and the impossible triangle known as the tribar.
	www.worldofescher.com /misc/penrose.html (644 words)

	Roger Penrose Summary
		Penrose suspects that a greater understanding of the functioning of the human brain may depend on a fundamentally new understanding of physics, to be sought in a radical new theory of quantum gravity.
		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.
		Penrose and Stuart Hameroff have constructed a theory in which human consciousness is the result of quantum gravity effects in microtubules.
	www.bookrags.com /Roger_Penrose (5262 words)

	From quasicrystals to Kleenex (Site not responding. Last check: 2007-10-18)
		Penrose, who was knighted in 1994 for his services to science, is best known for his work on relativity theory and quantum mechanics, having shared the 1988 Wolf Prize for Physics with Stephen Hawking.
		Penrose first developed a set of six tiles in 1973 and then "began thinking about reducing the number; and by various operations of slicing and re-gluing, I was able to reduce it to two".
		Penrose tilings have also been put to rather different use: in 1997, Kleenex used a form of the rhomb design on their quilted toilet paper.
	pass.maths.org.uk /issue16/features/penrose/index.html (1540 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 (195 words)

	Review of Penrose, The Emperor's New Mind
		Penrose sees that he has no hope of overthrowing the case for strong AI unless he can dislodge one of the most imperturbable objects in the intellectual universe: something I will call the Cathedral of Science.
		Penrose neglects to provide any argument to show what those limits are, and this is surprising, since this is where most of the attention is focussed in artificial intelligence today.
		I may well have missed a crucial ingredient in Penrose's argument that somehow obviates this criticism, but it is disconcerting that he does not even address the issue, and often writes as if an algorithm could have only the powers it could be proven mathematically to have in the worst case.
	ase.tufts.edu /cogstud/papers/penrose.htm (2883 words)

	P.P. Cook's Tangent Space: A Penrose Universe
		Penrose points out that there is a contradiction in the entropy increase picture of the big bang, which is that the background radiation matches a model that is in thermal equilibrium.
		Penrose suggested his Weyl curvature hypothesis in 1979 as a physical origin of the increasing entropy of the universe with time.
		Penrose argues that the appropriate conformally invariant verion of general relativity the spin-2 field picks up an inverse conformal factor when the conformal tranformation is applied to the metrc, while the Weyl curvature does not.
	ppcook.blogspot.com /2006/09/penrose-universe.html (2757 words)

	Roger Penrose: A Knight on the tiles (Site not responding. Last check: 2007-10-18)
		Penrose is unusual in believing that quantum mechanics will have to change in order to fit into such a unified theory.
		Penrose's ideas on consciousness are, to say the least, controversial in the AI community.
		Penrose shows himself an unabashed realist, by proclaiming that acting conscious is not the same as being conscious.
	plus.maths.org /issue18/features/penrose (2258 words)

	The Emperor's New Mind Pg3 - business & technology consultants
		The argument Penrose unfolds has more facets than my summary can report, and it is unlikely that such an enterprise would succumb to a single, crashing oversight on the part of its creator--that the argument could be "refuted" by any simple objection.
		And Penrose then goes to some length to argue that there could be no algorithm, or at any rate no practical algorithm, for insight.
		I was struck as never before by the gleeful staircase of human artifices--diagrams, mappings, formalisms--piled one on top of the other over the years, permitting our species so much as to entertain such audacious hypotheses about the world we live in.
	www.stty.co.uk /new_mind3.htm (1616 words)

	Penrose Victorian Inn Bed and Breakfast, Haydenville, Massachusetts - (Northampton area) - Lodging - Country Inns
		PENROSE VICTORIAN INN is an elegant experience and only 5 miles from Northampton, Massachusetts.
		Penrose has been featured on a Christmas television special along with the Mark Twain House.
		Penrose has a fountain garden, antiques, 6 fireplaces, and unique chandeliers, lamps and welcoming candles in all the windows.
	bnblist.com /ma/penrose (241 words)

	Dr. Matrix at the Scientium: Programming Challenge! (Site not responding. Last check: 2007-10-18)
		The irrationality of the ratio underlies a proof by Penrose that the tiling is nonperiodic because if it were periodic, the ratio clearly would have to be rational.
		Penrose and Conway independently proved that whenever a curve closes, it has a pentagonal symmetry, and the entire region within the curve has a fivefold symmetry.
		As Penrose discovered (it was later found independently by Clive Bach), some of the seven vertex figures force the placing of tiles that are not joined to the immediately forced region.
	scientium.com /drmatrix/puzzles/progchal.htm (3897 words)

	Puzzle: Up and Down the Staircase
		While Escher did indeed make the impossible staircase widely known to the world in his 1960 print, the original idea for the print came from the field of recreational mathematics.
		This staircase was drawn in a closed loop and had no bottom step when allowing the eye when allow in the eye to move around it in a clockwise direction.
		Penrose received his Ph.D. at Cambridge in algebraic geometry and is a professor of mathematics at Oxford University.
	www.aimsedu.org /puzzle/stairs/stairs.html (708 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)

	Impossible Staircase - Planet Perplex
		Escher didn't invent the endless staircase but he managed to draw it in a very appealing way.
		The inventors of the endless stairs were L.S. and Roger Penrose in 1958.
		Father and son L.S. and Roger Penrose were the first to publish its existence in 1958.
	www.planetperplex.com /en/impossible_staircase.html (312 words)

	Totally Tessellated: Related Advanced Mathematical Topics, page 3/5
		Penrose, a math professor at the Oxford University in England, began exploring tilings as a student at Cambridge University.
		A penrose tiling is an example of quasiperiodic tilings of the plane.
		Penrose created the "Penrose staircase" and the "tribar," both of which are used by Escher in Waterfall and Ascending and Descending.
	library.thinkquest.org /16661/advanced.math/3.html (355 words)

	Introduction to Quasitiler (Science U)
		Penrose tilings are well known because of their interesting and sometimes intriguing properties; for example, they are locally but not globally symmetric under 5-fold rotations, quasi-periodic with respect to translations, self-similar, and more.
		Now we project the staircase perpendicularly onto E. Adjacent edges of the staircase collapse to adjacent line segments in E. We have as result a tiling of this one-dimensional space.
		In the background is a Penrose tiling, which is the projection of the surface onto the generating plane E (see the previous section).
	www.scienceu.com /geometry/articles/tiling/quasiinfo.html (3070 words)

	Roger Penrose (Site not responding. Last check: 2007-10-18)
		In addition to the to Penrose tiles, Roger Penrose has made several other important contributions to mathematics for which he is well known.
		The Penrose staircase is a staircase that seems to be constantly going up.
		A true Penrose staircase is used in "Ascending and Descending" (bottom left), and a distorted version is used in "House of Stairs." "Waterfall" (bottom right) is based on the tribar.
	www.ccds.charlotte.nc.us /2005hpc05/roger_penrose.htm (107 words)

	IMA Update, Fall 2004
		The visual aspect of Penrose's creativity is well-known: the Penrose staircase, which he invented in collaboration with his father, is a fascinating perceptual puzzle, while the Penrose tiling is a mesmeric solution to a long-standing geometric problem.
		Penrose's sketches gave form to mathematical and physical concepts, from the five elements and their corresponding Platonic solids, to light cones and spin states, and in doing so allowed the audience to glimpse a great theorist's vision of the mathematical world.
		Professor Sir Roger Penrose, FRS OM, is Emeritus Rouse Ball Professor of the University of Oxford.
	www.ima.umn.edu /newsletters/updates/fall04/pub-lect.shtml (378 words)

	Geological Society of America - Penrose Conference - 1999
		Our plan was to assemble a multidisciplinary and international group of geologists and geophysicists to first characterize the geologic and seismic signature of active transition areas around the world and then to compare and contrast these characteristics to ancient transitions described from mountain belts.
		Strain partitioning, or the manifestation of transpressional plate motion into parallel and coeval systems of strike-slip and thrust faults, has been the topic of earlier Penrose Conferences and was addressed again at this session because of the common occurrence of transpressional suites of structures in all three types of transition area.
		The Penrose participants shared their impressions of seismic hazard issues to the visitors based on their experiences in this and other transition areas.
	www.geosociety.org /penrose/99pcrpt1.htm (3477 words)

	WHEN - History (Site not responding. Last check: 2007-10-18)
		The monks in the picture seem to be either forever ascending or descending the impossible staircase that leads to nowhere.
		This idea of the impossible staircase was not discovered by Escher himself, but its use in “Ascending and Descending” was actually inspired by the actual founders, Roger Penrose and his father Lionel.
		Both the triangle and staircase are conceptually impossible, which was what obviously peaked Escher’s interest after these discoveries were published in the British Journal of Psychology in 1958.
	ahsmail.uwaterloo.ca /~mgramsda/when.html (186 words)

	Our City's Voice: The Gazette's First 125 Years (Site not responding. Last check: 2007-10-18)
		Although Penrose was accused of using a raised ``a'' in the hotel's logo to belittle the competing Antlers Hotel, he insisted it was only to add a distinguishing touch to the name Broadmoor, which couldn't be copyrighted.
		The superb decorative achievements in the lobby, the grand staircase and the vaulted vestibule above never failed to bring gasps of admiration.
		Penrose, the work will not be considered complete until the Broadmoor hotel has been accepted as the greatest tourist hotel in America.''
	www.co-ol.com /news/anniversary/bwww.html (406 words)

	Science and Society Picture Library - Search
		This is the first model of the 'impossible staircase' one of a series of 'impossible figures' designed by English geneticist Lionel Penrose FRS (1898-1972) in the 1950s.
		Penrose's discovery of the 'impossible figure' laid the foundation for the scientific study of illusions and investigations into the psychology of human perception.
		The Dutch artist M C Escher (1898-1972) was heavily influenced by Penrose's discovery, and incorporated an 'impossible triangle' into his well known 'Waterfall' lithograph.
	www.scienceandsociety.co.uk /results.asp?image=10210224 (138 words)

	"Sir Roger Penrose" (Site not responding. Last check: 2007-10-18)
		In 1974, mathematical physicist Sir Roger Penrose of the University of Oxford discovered a remarkable aperiodic tileset that would cover a surface without forming a regularly repeating pattern.
		" in a 1977 issue of "Scientific American" Dr. Penrose and his father, L.S. Penrose, are also the creators of the puzzling "tribar" and "continuous staircase".
		Dr. Penrose corresponded with Escher for several years, and they met in 1962.
	www.miracerros.com /artwork/g_penrose_0.htm (95 words)

	The Principles of Artistic Illusions
		One particular example of the Reutersvard illusion is sometimes called the `Penrose' or `tribar' illusion.
		This illusion is often rendered as an isometric drawing, even in the Penrose paper.
		He placed the staircase on the roof of a building and structured the building below to convey an impression of conformity to strong (but inconsistent!) vanishing points.
	www.lhup.edu /~dsimanek/3d/illus1.htm (3214 words)

	all about spiral metal staircase (Site not responding. Last check: 2007-10-18)
		Salter Industries offers the finest in spiral stairs and staircase kits in metal, wood, and aluminum for interior, exterior and deck installations.
		Staircases - curved and spiral, crafted using wood and metal.
		spiral staircase metal staircase metal spiral staircase diy staircase diy spiral stairs metal spiral staircase bespoke spiral staircase...
	www.spiral-staircase-directory.com /spiral-metal-staircase.html (145 words)

	Mathematics Science Papers - The Impossible World of M. C. Escher
		Roger Penrose (the British mathematician) described the second type—impossible perspectives—as being “rather than locally unambiguous, but globally impossible, they are everywhere locally ambiguous, yet globally impossible” (Quoted from Coxeter, 154).
		This basis for this idea is not Escher’s—he found this (along with the staircase) in an article by Roger Penrose and his son, L.S. Penrose, in 1958 (Ernst, quoted in Coxeter 125).
		The tribar, or Roger Penrose’s triangle, is formed of three rectangular beams.
	www.123helpme.com /view.asp?id=28639 (2212 words)

	Penrose stairway (Site not responding. Last check: 2007-10-18)
		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 (175 words)

	Western Massachusetts Bed Breakfast Lodging Accomodations: Penroseinn Tour
		Penrose Victorian Inn brings you back in time to an era of elegance.
		A year round Christmas tree decorated with roses, cherubs and bows takes it's place in one corner while an exquisite staircase showcasing antique family photographs leads the way to the guest rooms in the other corner.
		Penrose Inn offers relaxation as well as an inviting array of things to do.
	www.penroseinn.com /common_rooms.htm (231 words)

	Ascending and Descending - Wikipedia, the free encyclopedia
		The lithograph depicts a large building roofed by a never-ending staircase.
		Two figures sit apart from the people on the endless staircase: one in a secluded courtyard, the other on a lower set of stairs.
		Ascending and Descending was influenced by, and is an artistic implementation of, the Penrose stairs, an impossible object; Lionel Penrose had first published his concept in the February, 1958 issue of the British Journal of Psychology.
	en.wikipedia.org /wiki/Ascending_and_Descending (230 words)

	Ascending and Descending (Site not responding. Last check: 2007-10-18)
		The graphic below shows the original Penrose drawing on the left and a sliced version of it on the right, which shows how the deception was accomplished.
		It can be seen that the staircase is in a horizontal plane, while the sections lie in a spiral.
		This is why slice #1 starts at the upper left and ends up in the lower left, instead of staying in the same horizontal plane.
	britton.disted.camosun.bc.ca /jbimpstair.htm (158 words)

	Tim Bahls (Site not responding. Last check: 2007-10-18)
		Roger Penrose (the British mathematician) described the second type—impossible perspectives—as being “rather than locally unambiguous, but globally impossible, they are everywhere locally ambiguous, yet globally impossible” (Quoted from Coxeter, 154).
		This basis for this idea is not Escher’s—he found this (along with the staircase) in an article by Roger Penrose and his son, L.S. Penrose, in 1958 (Ernst, quoted in Coxeter 125).
		h is pushing the staircase up on the page, while I want the stair case to meet itself.
	community.middlebury.edu /~schar/Courses/fs023.F02/paper2/tim.htm (1821 words)

Try your search on: Qwika (all wikis)