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	Borromean Rings
		A form of the Borromean link known as Odin's Triangle or the Walknot ("knot of the slain") was used by the Norse folk of Scandinavia in two variants: a set of Borromean triangles and a unicursal curve that makes a trefoil knot.
		The Borromean Rings are commonly used as a symbol of the Christianity Trinity.
		Borromean Rings can be also be found on Japanese family emblems, at a Japanese Shinto shrine north of Sakurai in the province of Nara, and in the sculptures of the Australian artist John Robinson.
	www.daviddarling.info /encyclopedia/B/Borromean_Rings.html (555 words)

	Borromean rings - Wikipedia, the free encyclopedia
		The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript.
		The name "Borromean rings" comes from their use in the coat of arms of the aristocratic Borromeo family in Italy.
		The psychoanalyst Jacques Lacan famously found inspiration in the Borromean rings as a model for his topology of the human mind, with each ring representing a fundamental Lacanian component of reality (the "real", the "imaginary", and the "symbolic").
	en.wikipedia.org /wiki/Borromean_knot (414 words)

	Basic polyhedron 6* (Site not responding. Last check: 2007-09-10)
		Such a three-component link is named "Borromean rings" after the Borromeas, an Italian family from the Renaissance that used them as their family crest symbolizing the value of collaboration and unity.
		Borromean circles are impossible: B.Lindström and H.O.Zetterström [39] proved using geometrical arguments that three flat circles cannot be Borromean, but that Borromean triangles are possible.
		Borromean rings you could find in geometry as the regular octahedron {3,4}, in Venn diagrams, in DNA, and in other various areas.
	www.mi.sanu.ac.yu /vismath/sl/l30.htm (429 words)

	Cook's Borromean Ring Links
		To the Norse people of Scandinavia, a drawing of the Borromean rings using triangles is known as "Odin's triangle" or the "Walknot" (or "valknut" -- the knot of the slain).
		Call it P. The power of the point P (the product of the two distances to a given circle on any line through P and the circle) will always be larger with respect to the swallower than with respect to the swallowee, since P is inside both discs.
		Chasing this inequality around the three circles gives a contradiction, so it is impossible for three (or more) discs to share a common point if each one swallows the next.
	paradise.caltech.edu /~cook/Workshop/Math/Borromean/Borrring.html (870 words)

	MODULARITY IN ART
		In the same way, for the well-known Borromean rings the visual arguments are not sufficient, so we need the mathematical proof of their impossibility (this means, that they could not be realized by three flat rings).
		Such a three-component link is named "Borromean rings" after the Borromeas, an Italian family from the Renaissance that used them as their family crest.
		From the mathematical point of view, a knot is a homeomorphic image of a circle, this means, some placement of a circle in 3D space, and a link is such a placement of several circles.
	members.tripod.com /~modularity/d3.htm (4109 words)

	Constructing Borromean Links (Site not responding. Last check: 2007-09-10)
		Borromean is named after an Italian family of the Renaissance period who had a pattern of three, inter-linked circles as their crest.
		For example, in the Tait series, the Borromean Rings are the element of the series where m = 1.
		Borromean Links without the Brunnian property contain at least one nontrivial sublink, whereas those with the Brunnian property have no nontrivial sublinks.
	www.ma.utexas.edu /~katerman/a/essays/math321/borromean.html (1458 words)

	Threefold Pattern
		Each of its three circles would fit exactly in the triangle - i.e., their diameter was half that of the big, surrounding circle: that is the key to solving it.
		Three interlinked circles are called, 'Borromean rings.' For their rich history, see eg www.liv.ac.uk/~spmr02/rings/- none of which are, one is bound to say, quite as good as the Circlemakers' design.
		If three circles touch, with a fourth at their centre, and they are all the same size, then an enveloping hexagon around the centre circle can be drawn through the three centres.
	www.astro3.demon.co.uk /cropcirc/chap3/Three.htm (1239 words)

	Borromean Rings \|\| The Liberal Arts \|\| Dean's Welcome \|\| College of Arts and Sciences \|\| George Mason University (Site not responding. Last check: 2007-09-10)
		The three interlocking rings at the center of the logo of the College of Arts and Sciences are known as the Borromean Rings.
		If the Borromean Rings are stretched or pulled, they cease to be circles, but their special linking property remains.
		The Borromean Rings at the center of the College logo remind us also that the division of areas of study according to disciplines is an artificial means of organizing learning.
	cas.gmu.edu /deans_welcome/what_are_the_liberal_arts/borromean_rings (513 words)

	Hypermedia Joyce Studies, 3.1 (2002), Louis Armand
		In the chapter of her biography of Lacan entitled 'Mathemes and Borromean Knots,' Elizabeth Roudinesco describes how in his later years Lacan was known to be quite passionately concerned with Borromean knots, and discusses his interest in topological puzzles in collaboration with Pierre Soury, Rene Thom and several other French mathematicians.
		What the topological metaphor of the Borromean knot suggests, then, is the synthetic nature of the psychoanalytic subject, which, as subject, is the unique "solution" to the problem of the incomensurability of what is named by these three terms.
		As a consequence, it is necessary to posit the Borromean knot in a doubly fourfold manner: as the symptomatic topos of the encounter of the imaginary, symbolic and real, and as their tropological linkage.
	www.geocities.com /hypermedia_joyce/armand.html (4340 words)

	SummeResearch
		A set of Borromean Rings is a set of three interlocking noncircular rings (it has been proved that circular Borromean Rings cannot exist) interlocked so that any two are not interlocked while the shape as a whole is locked (i.e.
		Because the circles are not tangent, the length AJ is not equal to AG, AD, DJ, and GJ (the sides of the rhombus, each of which are exactly equal to the diameter of each of the circles) as I had thought.
		I computed the slope of the tangent (to circle A) at point B and took the negative reciprocal and took its tangent inverse to find the proper angle for the extruded face.
	maven.smith.edu /~orourke/DTS/Sum05/WebPages/Gillian/SummerBlog.htm (5913 words)

	Sculpture Maths - Borromean Rings
		But no individual circle links with any one other, it is only the figure as a whole which cannot be disentangled.
		The theorem stating Borromean rings to be impossible with flat circles is proved rigorously in the article "Borromean circles are impossible," Amer.
		This material may be used freely for educational, artistic and scientific purposes, but may not be used for commercial purposes, for profit or in texts without the permission of the publishers.
	www.popmath.org.uk /sculpmath/pagesm/borings.html (546 words)

	Introduction (Site not responding. Last check: 2007-09-10)
		B.Lindström and H.O.Zetterström [1] proved that "Borromean circles are impossible": three flat circles cannot construct them, but by triangles they can.
		P.Cromwel recognized it in a picture-stone from Gotland [2], and it are exactly these and other symmetrical combinations of three and four hollow triangles that were considered by H.S.M.Coxeter [3].
		In geometry, Borromean rings appear as the regular octahedron {3,4} [4], in Venn diagrams [5], in DNA [6], and in other various areas [7].
	www.mi.sanu.ac.yu /vismath/bor/bor1.htm (149 words)

	Chemical Knot: Scientists assemble legendary symbol by interlocking molecules: Science News Online, May 29, 2004 (Site not responding. Last check: 2007-09-10)
		In a feat of chemistry imitating art, researchers have created a molecular version of a Borromean knot, an attractive pattern of three interlocking rings that commonly adorned Viking art and Renaissance architecture.
		Making molecular versions of Borromean rings poses formidable challenges for chemists because no pair of rings is linked unless the third ring is present.
		When X-ray crystallographic analysis confirmed that the 2.5-nanometer-wide molecular structures were indeed Borromean rings, Cantrill and the rest of the UCLA team were elated.
	www.sciencenews.org /articles/20040529/fob7.asp (609 words)

	Olympic Circles (Site not responding. Last check: 2007-09-10)
		The geometric shape thus, for european road olympic circles finder.
		inscribed circle george hirthler is recognized as the world’s.
		At antwerp, olympic circles 1920 role in bringing the 2010 winter.
	eurohockey.search11.info /Olympic-Circles (179 words)

	The Geometry Junkyard: Circles and Spheres
		Circle fractal based on repeated placement of two equal tangent circles within each circle of the figure.
		Ford circles and circle inversions, Kleinian groups, more Kleinian groups, even more Kleinian groups, 3d sphere packings, spiral circle packings, and sphere inversion fractals.
		Miquel's pentagram theorem on circles associated with a pentagon.
	www.ics.uci.edu /~eppstein/junkyard/sphere.html (1021 words)

	[GR] > The Shield of the Blessed Trinity
		The eternal distinction between the three Divine Persons is taught by reading around the outer circle: The Father is not (non est) the Son, the Son is not (non est) the Holy Spirit, and the Holy Spirit is not (non est) the Father.
		One of the most common symbols of the Trinity is the Trefoil, generated from three interlocking Circles.
		In the diagram on the left, we have three overlapping circles with each circle representing one of the primes from the Unity Holograph generating set (2, 13, 43).
	www.biblewheel.com /GR/GR_Unity_Shield.asp (3624 words)

	» Blog Archive » BORROMEAN RINGS
		The new state behaves like a Borromean ring, a symbol of three interlocking circles significant in Medieval Italy, the researchers explained.
		The Borromean concept is also important in some other aspects of physics, chemistry and mathematics.
		The Borromean rings were added to the coat of arms of the Borromeo family, Milanese aristocrats, in the 15th century.
	blog.dogdoright.com /?p=55 (707 words)

	List of circle topics - Wikipedia, the free encyclopedia
		This list of circle topics is not intended for metaphorical circles, but rather for topics related to the geometric shape.
		Thus, for example, a link to inner circle does not belong here.
		This page was last modified 22:01, 26 June 2006.
	en.wikipedia.org /wiki/List_of_circle_topics (73 words)

	Runes and Norse Symbology
		At the time of the Middle Ages, when thousands were being murdered on the charge of witchcraft, there were many (including high dignitaries of the Christian Church) who engaged in the practice of magick quite openly.
		A form of the Borromean link was used by the Norse people of Scandinavia.
		The origins of the Germanic writing system known as the fuþark is a hotly debated issue in scholarly circles.
	www.symbols.net /runes (209 words)

	Crop Circle Geometry: Chapter 3
		Adding a Circle A year later, the fine symmetry of 'Meonstoke' appeared www.lucypringle.co.uk/photos/1999/uk1999dk.html, having the same basic geometry as the 'sphere within.' Do you agree that its inner circle just touches the central 'latent triangle'?
		for one who has difficulty grasping that the circle around an equilateral triangle has twice the radius of that inscribed within it, this diagram could be just what is needed.
		With a circle divided into 18, draw arcs within it, setting the compass to four of its divisions, without removing the pen from the paper, as it were.
	www.hypermaths.org /cropcircles/chapter3/index.html (1319 words)

	The Geometry Junkyard: All Topics
		Geoff Mess relates a proof that the Borromean ring configuration (in which three loops are tangled together but no pair is linked) can not be formed out of circles.
		The ancient Greeks performed similar constructions for certain lunules as an approach to squaring the circle.
		Zonohedron generated by 30 vectors in a circle, and another generated by 100 random vectors, Paul Heckbert, CMU.
	www.math.ntnu.edu.tw /~jcchuan/all.html (7425 words)

	Spring 2004 MAA NC State Dinner (Site not responding. Last check: 2007-09-10)
		The Borromean Rings, pictured at right, are a special type of link, in which the components are unknotted loops that cannot be separated from each other, but such that as soon as any component is deleted the remaining components can all be.
		Links with this property, such as the Borromean Rings, are called Brunnian links.
		Even though in the picture it appears that the Borromean rings are constructed out of round circles we will prove that this is impossible.
	math.elon.edu /maadinner2004/talk.html (273 words)

	Graphics Archive Author Index (Science U)
		Borromean Ring Complement Manifold 1 [Special Topics:Hyperbolic Geometry]
		Borromean Ring Complement Manifold 2 [Special Topics:Hyperbolic Geometry]
		Borromean Ring Complement Manifold 3 [Special Topics:Hyperbolic Geometry]
	www.scienceu.com /library/graphics/authors.html (308 words)

	Venn diagram illustrations (Site not responding. Last check: 2007-09-10)
		Below are two other illustrations of the Borromean rings.
		The one on the left is not a minimal projection (and isn't a Venn diagram).
		A Venn diagram uses circles or rings to show how two or more things are similar or...
	www.ebs-schreibservice.de /89240-venndiagramillustrations (268 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		Therefore a link of 3 round circles which have linking numbers zero is split.
		In R^3, of course.) a) three mutually unlinked hoops, b) two linked hoops and the third far away, c) a chain of three hoops, d) a circular chain of mutually linked hoops, and e) left to the reader as a puzzle.
		All circular chains of three hoops seem to lie in the same component of H(1;3) (and there is a theorem that the Borromean rings cannot be represented by round circles).
	www.ics.uci.edu /~eppstein/junkyard/borromeo.html (670 words)

	borromean rings (Site not responding. Last check: 2007-09-10)
		These three rings taken together are inseparable, but if you remove any one of them then the other two fall apart.
		It has been proven that the borromean rings cannot be constructed with flat circles.
		The borromean rings have been and are still used as symbols and emblems; they appear in many logos.
	www.ac-noumea.nc /maths/amc/polyhedr/borromea_.htm (64 words)

	alpha omega christian symbols best resources and information. (Site not responding. Last check: 2007-09-10)
		Pages in category "Christian Christian symbolism; A. Alpha and Omega B. Borromean rings; Burning bush I am the Alpha and Omega I am the beginning and the end " says the LORD "Who is and who was and who is to come the Almighty." Revelation NIV.
		Christian Circle City Codes Color Conlangs Crop Circles Danger Da Vinci Code Dictionaries Dreams Fraternity Symbols.
		The Epsilon chapter of AGO is located at the Symbols of Christian concepts and Christian religious figures.
	alphaomega.webinfoacquire.info /alpha-omega-christian-symbols (458 words)

	Diamond wedding rings
		Long before crop circles caught the headlines there were fairy rings.
		GORDON RUTTER explores the legends and lore of this mysterious phenomenon, as well the various explanations which have been offered to account for it.
		Borromean Rings from MathWorld Borromean Rings from MathWorld The Borromean rings, also called the Borromean links (Livingston 1993, p.
	jewelry.businest.us /rings/diamond-wedding-rings.html (1363 words)

	Crop Circles - medieval & folklore guide books from Wessex Books
		I cannot think of a more inspirational way of awakening interest in geometry and how it affects us, than to have the exquisite forms behind the crop circle phenomenon placed before us, as the author has done in this book.
		The Hidden Form shows a new way of experiencing the crop circle mystery, and it also gives us insights into structures of the natural world.
		A kind of hidden language is being revealed that the universe seems to be showing to us, but it is a language which cannot easily be expressed in words.
	www.wessexbooks.co.uk /cropcircles-hiddenform.htm (357 words)

	Symbol trinity (Site not responding. Last check: 2007-09-10)
		Today, the Borromean rings are commonly used as a symbol of the Trinity.
		Triangle and Circle signifies the eternal nature of the Holy Trinity.
		Trinity Symbol, The three circles represent the Father, Son and Holy Spirit.
	symbol-trinity.wd29.com (1626 words)

	Symbolic Sculpture and Borromean Rings (Site not responding. Last check: 2007-09-10)
		John Robinson found that if you use squares instead of circles then you can make a real figure.
		William W.Chernoff at the University of New Brunswich, Canada has experimented with "Five Borromean Square Frames", and more complex figures.
		The University of Michigan School of Information has adopted the Borromean Rings as a logo.
	www.atractor.pt /bangor/SculMath/image/borings.htm (736 words)

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